resolve-lemma-equivalence-properties Let $S, s_i, S_i$ be as in (\ref{equation-equivalence}). If $f : X \to S$ corresponds to $g_i : Y_i \to S_i$ under $F$, then $f$ is separated, proper, finite, if and only if $g_i$ is so for $i = 1, \ldots, n$.


chow-lemma-lci-gysin-product-regular Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $f : X \to Y$ be a morphism of schemes locally of finite type over $S$ such that both $X$ and $Y$ are quasi-compact, regular, have affine diagonal, and finite dimension. Then $f$ is a local complete intersection morphism. Assume moreover the gysin map is defined for $f$. Then $$ f^!(\alpha \cdot \beta) = f^!\alpha \cdot f^!\beta $$ in $\CH^*(X) \otimes \mathbf{Q}$ where the intersection product is as in Section \ref{section-intersection-regular}.


dga-lemma-restriction-homotopy The functor (\ref{equation-restriction}) defines an exact functor $K(\text{Mod}_{(B, \text{d})}) \to K(\text{Mod}_{(A, \text{d})})$ of triangulated categories.


spaces-morphisms-lemma-birational Let $S$ be a scheme. Let $X$ and $Y$ be algebraic space over $S$ with $|X|$ and $|Y|$ irreducible. Then $X$ and $Y$ are birational if and only if there are nonempty open subspaces $U \subset X$ and $V \subset Y$ which are isomorphic as algebraic spaces over $S$.


derived-lemma-unbounded-left-derived Let $F : \mathcal{A} \to \mathcal{B}$ be a right exact functor of abelian categories. If \begin{enumerate} \item every object of $\mathcal{A}$ is a quotient of an object which is left acyclic for $F$, \item there exists an integer $n \geq 0$ such that $L^nF = 0$, \end{enumerate} Then \begin{enumerate} \item $LF : D(\mathcal{A}) \to D(\mathcal{B})$ exists, \item any complex consisting of left acyclic objects for $F$ computes $LF$, \item any complex is the target of a quasi-isomorphism from a complex consisting of left acyclic objects for $F$, \item for $E \in D(\mathcal{A})$ \begin{enumerate} \item $H^i(LF(\tau_{\leq a + n - 1}E) \to H^i(LF(E))$ is an isomorphism for $i \leq a$, \item $H^i(LF(E)) \to H^i(LF(\tau_{\geq b}E))$ is an isomorphism for $i \geq b$, \item if $H^i(E) = 0$ for $i \not \in [a, b]$ for some $-\infty \leq a \leq b \leq \infty$, then $H^i(LF(E)) = 0$ for $i \not \in [a - n + 1, b]$. \end{enumerate} \end{enumerate}


dualizing-lemma-well-defined Let $\varphi : R \to A$ be a finite type homomorphism of Noetherian rings. The functor $\varphi^!$ is well defined up to isomorphism.


formal-defos-theorem-minimal-smooth-prorepresentable-presentations Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$. Consider the following conditions \begin{enumerate} \item $\mathcal{F}$ admits a presentation by a normalized smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda$, \item $\mathcal{F}$ admits a presentation by a smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda$, \item $\mathcal{F}$ admits a presentation by a minimal smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda$, and \item $\mathcal{F}$ satisfies the following conditions \begin{enumerate} \item $\mathcal{F}$ is a deformation category. \item $\dim_k T\mathcal{F}$ is finite. \item $\dim_k \text{Inf}(\mathcal{F})$ is finite. \end{enumerate} \end{enumerate} Then (2), (3), (4) are equivalent and are implied by (1). If $k' \subset k$ is separable, then (1), (2), (3), (4) are all equivalent. Furthermore, the minimal smooth prorepresentable groupoids in functors which provide a presentation of $\mathcal{F}$ are unique up to isomorphism.


stacks-geometry-lemma-dimension-local-ring-pre Let $\mathcal{X}$ be a locally Noetherian algebraic stack. Let $U \to \mathcal{X}$ be a smooth morphism and let $u \in U$. Then $$ \dim(\mathcal{O}_{U, \overline{u}}) - \dim(\mathcal{O}_{R_u, e(\overline{u})}) = 2\dim(\mathcal{O}_{U, \overline{u}}) - \dim(\mathcal{O}_{R, e(\overline{u})}) $$ Here $R = U \times_\mathcal{X} U$ with projections $s, t : R \to U$ and diagonal $e : U \to R$ and $R_u$ is the fibre of $s : R \to U$ over $u$.


divisors-lemma-check-injective-on-weakass Let $X$ be a scheme. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of quasi-coherent $\mathcal{O}_X$-modules. Assume that for every $x \in X$ at least one of the following happens \begin{enumerate} \item $\mathcal{F}_x \to \mathcal{G}_x$ is injective, or \item $x \not \in \text{WeakAss}(\mathcal{F})$. \end{enumerate} Then $\varphi$ is injective.


proetale-lemma-compare-cohomology-big-small Let $f : T \to S$ be a morphism of schemes. For $K$ in $D(S_\proetale)$ we have $$ H^n_\proetale(S, \pi_S^{-1}K) = H^n(S_\proetale, K) $$ and $$ H^n_\proetale(T, \pi_S^{-1}K) = H^n(T_\proetale, f_{small}^{-1}K). $$ For $M$ in $D((\Sch/S)_\proetale)$ we have $$ H^n_\proetale(T, M) = H^n(T_\proetale, i_f^{-1}M). $$


algebra-lemma-flat-ML-over-ML-ring Let $R \to S$ be a ring map. Let $M$ be an $S$-module. If $S$ is Mittag-Leffler as an $R$-module, and $M$ is flat and Mittag-Leffler as an $S$-module, then $M$ is Mittag-Leffler as an $R$-module.


divisors-lemma-UFD-one-equation-CM Let $R$ be a Noetherian UFD. Let $I \subset R$ be an ideal such that $R/I$ has no embedded primes and such that every minimal prime over $I$ has height $1$. Then $I = (f)$ for some $f \in R$.


stacks-morphisms-lemma-change-of-base-separated Let $S$ be a scheme. The property of being quasi-DM over $S$, quasi-separated over $S$, or separated over $S$ (see Definition \ref{definition-absolute-separated}) is stable under change of base scheme, see Algebraic Stacks, Definition \ref{algebraic-definition-change-of-base}.


spaces-duality-lemma-more-base-change In diagram (\ref{equation-base-change}) assume in addition $g : Y' \to Y$ is a morphism of affine schemes and $f : X \to Y$ is proper. Then the base change map (\ref{equation-base-change-map}) induces an isomorphism $$ L(g')^*a(K) \longrightarrow a'(Lg^*K) $$ in the following cases \begin{enumerate} \item for all $K \in D_\QCoh(\mathcal{O}_X)$ if $f$ is flat of finite presentation, \item for all $K \in D_\QCoh(\mathcal{O}_X)$ if $f$ is perfect and $Y$ Noetherian, \item for $K \in D_\QCoh^+(\mathcal{O}_X)$ if $g$ has finite Tor dimension and $Y$ Noetherian. \end{enumerate}


more-morphisms-lemma-basic-example-pushout Let $A' \to A$ be a surjection of rings and let $B \to A$ be a ring map. Let $B' = B \times_A A'$ be the fibre product of rings. Set $S = \Spec(A)$, $S' = \Spec(A')$, $T = \Spec(B)$, and $T' = \Spec(B')$. Then $$ \vcenter{ \xymatrix{ S \ar[r]_i \ar[d]_f & S' \ar[d]^{f'} \\ T \ar[r]^{i'} & T' } } \quad\text{corresponding to}\quad \vcenter{ \xymatrix{ A & A' \ar[l] \\ B \ar[u] & B' \ar[l] \ar[u] } } $$ is a pushout of schemes.


obsolete-lemma-cosk-hom-deltak Let $\mathcal{C}$ be a category with finite coproducts and finite limits. Let $X$ be an object of $\mathcal{C}$. Let $k \geq 0$. The canonical map $$ \Hom(\Delta[k], X) \longrightarrow \text{cosk}_1 \text{sk}_1 \Hom(\Delta[k], X) $$ is an isomorphism.


algebra-lemma-nonzerodivisor-on-CM Let $R$ be a Noetherian local ring with maximal ideal $\mathfrak m$. Let $M$ be a finite $R$-module. Let $x \in \mathfrak m$ be a nonzerodivisor on $M$. Then $M$ is Cohen-Macaulay if and only if $M/xM$ is Cohen-Macaulay.


spaces-limits-lemma-limited-base-change Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ separated and of finite type. The following are equivalent \begin{enumerate} \item The morphism $f$ is proper. \item For any morphism $Y \to Z$ which is locally of finite presentation the map $|X \times_Y Z| \to |Z|$ is closed, and \item there exists a scheme $V$ and a surjective \'etale morphism $V \to Y$ such that $|\mathbf{A}^n \times (X \times_Y V)| \to |\mathbf{A}^n \times V|$ is closed for all $n \geq 0$. \end{enumerate}


algebra-lemma-silly-normal Let $R$ be a ring. Let $x, y \in R$ be nonzerodivisors. Let $R[x/y] \subset R_{xy}$ be the $R$-subalgebra generated by $x/y$, and similarly for the subalgebras $R[y/x]$ and $R[x/y, y/x]$. If $R$ is integrally closed in $R_x$ or $R_y$, then the sequence $$ 0 \to R \xrightarrow{(-1, 1)} R[x/y] \oplus R[y/x] \xrightarrow{(1, 1)} R[x/y, y/x] \to 0 $$ is a short exact sequence of $R$-modules.


duality-lemma-compactifyable-relative-dualizing In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$. If $f$ is flat, then $f^!\mathcal{O}_Y$ is (the first component of) a relative dualizing complex for $X$ over $Y$ in the sense of Definition \ref{definition-relative-dualizing-complex}.


spaces-descent-lemma-descending-property-quasi-compact-immersion The property $\mathcal{P}(f) =$``$f$ is a quasi-compact immersion'' is fpqc local on the base.


more-morphisms-lemma-normal-fppf-local-source-and-target The property $\mathcal{P}(f)=$``the fibres of $f$ are locally Noetherian and $f$ is normal'' is local in the fppf topology on the target and local in the smooth topology on the source.


coherent-lemma-thickening-quasi-affine Let $i : Z \to X$ be a closed immersion of Noetherian schemes inducing a homeomorphism of underlying topological spaces. Then $X$ is quasi-affine if and only if $Z$ is quasi-affine.


descent-lemma-morphism-source-faithfully-flat Let $f : X \to X'$ be a morphism of schemes over a base scheme $S$. Assume $X \to S$ is surjective and flat. Then the pullback functor of Lemma \ref{lemma-pullback} is a faithful functor from the category of descent data relative to $X'/S$ to the category of descent data relative to $X/S$.


spaces-resolve-lemma-dominate-by-normalized-blowing-up Let $S$ be a scheme. Let $X$ be a Noetherian Nagata algebraic space over $S$ with $\dim(X) = 2$. Let $f : Y \to X$ be a proper birational morphism. Then there exists a commutative diagram $$ \xymatrix{ X_n \ar[r] \ar[d] & X_{n - 1} \ar[r] & \ldots \ar[r] & X_1 \ar[r] & X_0 \ar[d] \\ Y \ar[rrrr]  & & & & X } $$ where $X_0 \to X$ is the normalization and where $X_{i + 1} \to X_i$ is the normalized blowing up of $X_i$ at a closed point.


constructions-lemma-spec In Situation \ref{situation-relative-spec}. The functor $F$ is representable by a scheme.


stacks-morphisms-lemma-etale-unramified An \'etale morphism is unramified.


sdga-lemma-cohomology-ext Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a differential graded $\mathcal{O}$-algebra. Let $\mathcal{M}$ be a differential graded $\mathcal{A}$-module. Let $n \in \mathbf{Z}$. We have $$ H^n(\mathcal{C}, \mathcal{M}) = \Hom_{D(\mathcal{A}, \text{d})}(\mathcal{A}, \mathcal{M}[n]) $$ where on the left hand side we have the cohomology of $\mathcal{M}$ viewed as a complex of $\mathcal{O}$-modules.


pione-proposition-lefschetz-equivalence Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{L})$. Let $Y = Z(s)$ be the zero scheme of $s$. Assume that for all $x \in X \setminus Y$ we have $$ \text{depth}(\mathcal{O}_{X, x}) + \dim(\overline{\{x\}}) > 2 $$ and that for $x \in X \setminus Y$ closed purity holds for $\mathcal{O}_{X, x}$. Then the restriction functor $\textit{F\'Et}_X \to \textit{F\'Et}_Y$ is an equivalence. If $X$ or equivalently $Y$ is connected, then $$ \pi_1(Y, \overline{y}) \to \pi_1(X, \overline{y}) $$ is an isomorphism for any geometric point $\overline{y}$ of $Y$.


spaces-perfect-lemma-cat-module-support-proper-over-base Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $\mathcal{F}$, $\mathcal{G}$ be finite type, quasi-coherent $\mathcal{O}_X$-module. \begin{enumerate} \item If the supports of $\mathcal{F}$, $\mathcal{G}$ are proper over $Y$, then the same is true for $\mathcal{F} \oplus \mathcal{G}$, for any extension of $\mathcal{G}$ by $\mathcal{F}$, for $\Im(u)$ and $\Coker(u)$ given any $\mathcal{O}_X$-module map $u : \mathcal{F} \to \mathcal{G}$, and for any quasi-coherent quotient of $\mathcal{F}$ or $\mathcal{G}$. \item If $Y$ is locally Noetherian, then the category of coherent $\mathcal{O}_X$-modules with support proper over $Y$ is a Serre subcategory (Homology, Definition \ref{homology-definition-serre-subcategory}) of the abelian category of coherent $\mathcal{O}_X$-modules. \end{enumerate}


spaces-over-fields-lemma-modification-iso-over-open Let $f : X' \to X$ be a modification as in Definition \ref{definition-modification}. There exists a nonempty open $U \subset X$ such that $f^{-1}(U) \to U$ is an isomorphism.


criteria-lemma-algebraic-morphism-to-algebraic Let $S$ be a scheme. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of stacks in groupoids over $(\Sch/S)_{fppf}$. If \begin{enumerate} \item $\mathcal{Y}$ is an algebraic stack, and \item $F$ is algebraic (see above), \end{enumerate} then $\mathcal{X}$ is an algebraic stack.


coherent-lemma-base-change-module-support-proper-over-base Consider a cartesian diagram of schemes $$ \xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ S' \ar[r]^g & S } $$ with $f$ locally of finite type. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_X$-module. If the support of $\mathcal{F}$ is proper over $S$, then the support of $(g')^*\mathcal{F}$ is proper over $S'$.


hypercovering-lemma-cech-spectral-sequence-verdier Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. Let $\mathcal{F}$ be an abelian sheaf. There is a spectral sequence $(E_r, d_r)_{r \geq 0}$ with $$ E_2^{p, q} = \check{H}^p(K, \underline{H}^q(\mathcal{F})) $$ converging to the global cohomology groups $H^{p + q}(\mathcal{F})$.


algebra-lemma-colimit-formally-unramified Let $R$ be a ring. Let $I$ be a directed set. Let $(S_i, \varphi_{ii'})$ be a system of $R$-algebras over $I$. If each $R \to S_i$ is formally unramified, then $S = \colim_{i \in I} S_i$ is formally unramified over $R$


topologies-lemma-characterize-sheaf-big Let $S$ be a scheme contained in a big Zariski site $\Sch_{Zar}$. A sheaf $\mathcal{F}$ on the big Zariski site $(\Sch/S)_{Zar}$ is given by the following data: \begin{enumerate} \item for every $T/S \in \Ob((\Sch/S)_{Zar})$ a sheaf $\mathcal{F}_T$ on $T$, \item for every $f : T' \to T$ in $(\Sch/S)_{Zar}$ a map $c_f : f^{-1}\mathcal{F}_T \to \mathcal{F}_{T'}$. \end{enumerate} These data are subject to the following conditions: \begin{enumerate} \item[(a)] given any $f : T' \to T$ and $g : T'' \to T'$ in $(\Sch/S)_{Zar}$ the composition $c_g \circ g^{-1}c_f$ is equal to $c_{f \circ g}$, and \item[(b)] if $f : T' \to T$ in $(\Sch/S)_{Zar}$ is an open immersion then $c_f$ is an isomorphism. \end{enumerate}


etale-cohomology-lemma-blow-up-square-equivalence Let $X$ be a scheme and let $Z \subset X$ be a closed subscheme cut out by a quasi-coherent ideal of finite type. Consider the corresponding blow up square $$ \xymatrix{ E \ar[d]_\pi \ar[r]_j & X' \ar[d]^b \\ Z \ar[r]^i & X } $$ Suppose given \begin{enumerate} \item an object $K'$ of $D^+(X'_\etale)$ with torsion cohomology sheaves, \item an object $L$ of $D^+(Z_\etale)$ with torsion cohomology sheaves, and \item an isomorphism $\gamma : K'|_E \to L|_E$. \end{enumerate} Then there exists an object $K$ of $D^+(X_\etale)$ and isomorphisms $f : K|_{X'} \to K'$, $g : K|_Z \to L$ such that $\gamma = g|_E \circ f^{-1}|_E$. Moreover, given \begin{enumerate} \item an object $M$ of $D^+(X_\etale)$ with torsion cohomology sheaves, \item a morphism $\alpha : K' \to M|_{X'}$ of $D(X'_\etale)$, \item a morphism $\beta : L \to M|_Z$ of $D(Z_\etale)$, \end{enumerate} such that $$ \alpha|_E  = \beta|_E \circ \gamma. $$ Then there exists a morphism $M \to K$ in $D(X_\etale)$ whose restriction to $X'$ is $a \circ f$ and whose restriction to $Z$ is $b \circ g$.


properties-lemma-reduced-is-locally-reduced Let $X$ be a scheme. Then $X$ is reduced if and only if $X$ is ``locally reduced'' in the sense of Definition \ref{definition-locally-P}.


resolve-lemma-embedded-resolution Let $X$ be a regular scheme of dimension $2$. Let $Z \subset X$ be a proper closed subscheme such that every irreducible component $Y \subset Z$ of dimension $1$ satisfies the equivalent conditions of Lemma \ref{lemma-resolve-curve}. Then there exists a sequence $$ X_n \to \ldots \to X_1 \to X $$ of blowups in closed points such that the inverse image $Z_n$ of $Z$ in $X_n$ is an effective Cartier divisor supported on a strict normal crossings divisor.


varieties-lemma-open-in-affine-curve-affine Let $X$ be an affine scheme all of whose local rings are Noetherian of dimension $\leq 1$. Then any quasi-compact open $U \subset X$ is affine.


algebra-lemma-jacobson-prime Let $R$ be a ring. If every prime ideal of $R$ is the intersection of the maximal ideals containing it, then $R$ is Jacobson.


algebra-lemma-lift-finite-projective-module Let $R$ be a ring. Let $I \subset R$ be a locally nilpotent ideal. Let $\overline{P}$ be a finite projective $R/I$-module. Then there exists a finite projective $R$-module $P$ such that $P/IP \cong \overline{P}$.


spaces-simplicial-lemma-compare-cohomology-hypercovering-X-simple-modules Let $\mathcal{C}$ be a site with fibre products and $X \in \Ob(\mathcal{C})$. Let $\mathcal{O}_\mathcal{C}$ be a sheaf of rings. Let $U$ be a hypercovering of $X$ in $\mathcal{C}$. Then we have a canonical isomorphism $$ R\Gamma(X, E) = R\Gamma((\mathcal{C}/U)_{total}, La^*E) $$ for $E \in D(\mathcal{O}_\mathcal{C})$.


spaces-more-morphisms-lemma-geometrically-connected-fibres-towards-normal Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item $f$ is proper, \item $Y$ is integral (Spaces over Fields, Definition \ref{spaces-over-fields-definition-integral-algebraic-space}) with generic point $\xi$, \item $Y$ is normal, \item $X$ is reduced, \item every generic point of an irreducible component of $|X|$ maps to $\xi$, \item we have $H^0(X_\xi, \mathcal{O}) = \kappa(\xi)$. \end{enumerate} Then $f_*\mathcal{O}_X = \mathcal{O}_Y$ and $f$ has geometrically connected fibres.


divisors-lemma-check-torsion Let $X$ be an integral scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $\mathcal{F}$ is torsion free if and only if $\mathcal{F}_x$ is a torsion free $\mathcal{O}_{X, x}$-module for all $x \in X$.


algebraization-lemma-interesting-case-final In Situation \ref{situation-algebraize} let $(\mathcal{F}_n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_U)$. Assume \begin{enumerate} \item $A$ is a local ring which has a dualizing complex, \item all irreducible components of $X$ have the same dimension, \item the scheme $X \setminus Y$ is Cohen-Macaulay, \item $I$ is generated by $d$ elements, \item $\dim(X) - \dim(Z) > d + 2$, and \item for $y \in U \cap Y$ the module $\mathcal{F}_y^\wedge$ is finite locally free outside $V(I\mathcal{O}_{X, y}^\wedge)$, for example if $\mathcal{F}_n$ is a finite locally free $\mathcal{O}_U/I^n\mathcal{O}_U$-module. \end{enumerate} Then $(\mathcal{F}_n)$ extends to $X$. In particular if $A$ is $I$-adically complete, then $(\mathcal{F}_n)$ is the completion of a coherent $\mathcal{O}_U$-module.


spaces-morphisms-theorem-chevalley Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-compact and locally of finite presentation. Then the image of every \'etale locally constructible subset of $|X|$ is an \'etale locally constructible subset of $|Y|$.


stacks-geometry-lemma-dims-of-images Let $f: \mathcal{T} \to \mathcal{X}$ be a locally of finite type morphism of Jacobson, pseudo-catenary, and locally Noetherian algebraic stacks, whose source is irreducible and whose target is quasi-separated, and let $\mathcal{Z} \hookrightarrow \mathcal{X}$ denote the scheme-theoretic image of $\mathcal{T}$. Then for every finite type point $t \in |T|$, we have that $\dim_t( \mathcal{T}_{f(t)}) \geq \dim \mathcal{T}  - \dim \mathcal{Z}$, and there is a non-empty (equivalently, dense) open subset of $|\mathcal{T}|$ over which equality holds.


stacks-morphisms-lemma-base-change-uniqueness The base change of a morphism of algebraic stacks which satisfies the uniqueness part of the valuative criterion by any morphism of algebraic stacks is a morphism of algebraic stacks which satisfies the uniqueness part of the valuative criterion.


spaces-chow-lemma-Gm-torsor-divisor-meromorphic-section In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Assume $X$ is integral. Let $s$ be a nonzero meromorphic section of $\mathcal{L}$. Let $q : T \to X$ be the morphism of Lemma \ref{lemma-Gm-torsor}. Then $$ q^*\text{div}_\mathcal{L}(s) = \text{div}_T(q^*(s)) $$ where we view the pullback $q^*(s)$ as a nonzero meromorphic function on $T$ using the isomorphism $q^*\mathcal{L} \to \mathcal{O}_T$


spaces-descent-lemma-finite-locally-free-descends Let $X$ be an algebraic space over a scheme $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\{f_i : X_i \to X\}_{i \in I}$ be an fpqc covering such that each $f_i^*\mathcal{F}$ is a finite locally free $\mathcal{O}_{X_i}$-module. Then $\mathcal{F}$ is a finite locally free $\mathcal{O}_X$-module.


obsolete-lemma-henselian Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring. The category of Algebraization of Formal Spaces, Equation (\ref{restricted-equation-modification}) for $A$ is equivalent to the category Algebraization of Formal Spaces, Equation (\ref{restricted-equation-modification}) for the henselization $A^h$ of $A$.


more-morphisms-lemma-Noetherian-approximation-geometrically-connected Let $f : X \to S$ be a morphism of affine schemes, which is of finite presentation with geometrically connected fibres. Then there exists a diagram as in Lemma \ref{lemma-Noetherian-approximation} such that in addition $f_0$ has geometrically connected fibres.


spaces-pushouts-lemma-glueable Let $(R \to R', f)$ be a glueing pair, see above. Let $Y$ be an algebraic space over $X$. The following are equivalent \begin{enumerate} \item there exists an \'etale covering $\{Y_i \to Y\}_{i \in I}$ with $Y_i$ affine and $\Gamma(Y_i, \mathcal{O}_{Y_i})$ glueable as an $R$-module, \item for every \'etale morphism $W \to Y$ with $W$ affine $\Gamma(W, \mathcal{O}_W)$ is a glueable $R$-module. \end{enumerate}


algebra-lemma-geometrically-connected-any-base-change Let $k$ be a field. Let $S$ be a geometrically connected $k$-algebra. Let $R$ be any $k$-algebra. The map $$ R \longrightarrow R \otimes_k S $$ induces a bijection on idempotents, and the map $$ \Spec(R \otimes_k S) \longrightarrow \Spec(R) $$ induces a bijection on connected components.


algebra-lemma-one-equation Suppose that $R$ is a Noetherian local ring and $x\in \mathfrak m$ an element of its maximal ideal. Then $\dim R \leq \dim R/xR + 1$. If $x$ is not contained in any of the minimal primes of $R$ then equality holds. (For example if $x$ is a nonzerodivisor.)


stacks-lemma-stack-in-setoids-equivalent Let $\mathcal{C}$ be a site. Let $\mathcal{S}_1$, $\mathcal{S}_2$ be categories over $\mathcal{C}$. Suppose that $\mathcal{S}_1$ and $\mathcal{S}_2$ are equivalent as categories over $\mathcal{C}$. Then $\mathcal{S}_1$ is a stack in setoids over $\mathcal{C}$ if and only if $\mathcal{S}_2$ is a stack in setoids over $\mathcal{C}$.


algebra-lemma-d-independent Let $R$ be a Noetherian local ring. Let $M$ be a finite $R$-module. \begin{enumerate} \item The degree of the numerical polynomial $\varphi_{I, M}$ is independent of the ideal of definition $I$. \item The degree of the numerical polynomial $\chi_{I, M}$ is independent of the ideal of definition $I$. \end{enumerate}


spaces-morphisms-lemma-scheme-theoretic-image-of-partial-section Let $S$ be a scheme. Let $f : X \to Y$ be a separated morphism of algebraic spaces over $S$. Let $V \subset Y$ be an open subspace such that $V \to Y$ is quasi-compact. Let $s : V \to X$ be a morphism such that $f \circ s = \text{id}_V$. Let $Y'$ be the scheme theoretic image of $s$. Then $Y' \to Y$ is an isomorphism over $V$.


local-cohomology-lemma-cd-blowup Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $b : X' \to X = \Spec(A)$ be the blowing up of $I$. If the fibres of $b$ have dimension $\leq d - 1$, then $\text{cd}(A, I) \leq d$.


etale-cohomology-proposition-h-cohomology-structure-sheaf Let $p$ be a prime number. Let $S$ be a quasi-compact and quasi-separated scheme over $\mathbf{F}_p$. Then $$ H^i((\Sch/S)_h, \mathcal{O}^h) = \colim_F H^i(S, \mathcal{O}) $$ Here on the left hand side by $\mathcal{O}^h$ we mean the h sheafification of the structure sheaf.


derham-lemma-chern-character There is a unique rule which assigns to every quasi-compact and quasi-separated scheme $X$ over $\mathbf{Q}$ a ``chern character'' $$ ch^{dR} : K_0(\textit{Vect}(X)) \longrightarrow \prod\nolimits_{i \geq 0} H_{dR}^{2i}(X/\mathbf{Q}) $$ with the following properties \begin{enumerate} \item $ch^{dR}$ is a ring map for all $X$, \item if $f : X' \to X$ is a morphism of quasi-compact and quasi-separated schemes over $\mathbf{Q}$, then $f^* \circ ch^{dR} =  ch^{dR} \circ f^*$, and \item given $\mathcal{L} \in \Pic(X)$ we have $ch^{dR}([\mathcal{L}]) = \exp(c_1^{dR}(\mathcal{L}))$. \end{enumerate}


spaces-cohomology-lemma-sections-with-support-acyclic Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$. Let $\mathcal{I}$ be an injective abelian sheaf on $X_\etale$. Then $\mathcal{H}_Z(\mathcal{I})$ is an injective abelian sheaf on $Z_\etale$.


spaces-limits-lemma-refined-valuative-criterion-universally-closed Let $S$ be a scheme. Let $f : X \to Y$ and $h : U \to X$ be morphisms of algebraic spaces over $S$. Assume that $Y$ is locally Noetherian, that $f$ and $h$ are of finite type, and that $h(U)$ is dense in $X$. If given any commutative solid diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & U \ar[r]^h & X \ar[d]^f \\ \Spec(A) \ar[rr] \ar@{-->}[rru] & & Y } $$ where $A$ is a discrete valuation ring with field of fractions $K$, there exists a unique dotted arrow making the diagram commute, then $f$ is proper.


stacks-morphisms-lemma-scheme-theoretic-image-of-partial-section Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is representable by algebraic spaces and separated. Let $\mathcal{V} \subset \mathcal{Y}$ be an open substack such that $\mathcal{V} \to \mathcal{Y}$ is quasi-compact. Let $s : \mathcal{V} \to \mathcal{X}$ be a morphism such that $f \circ s = \text{id}_\mathcal{V}$. Let $\mathcal{Y}'$ be the scheme theoretic image of $s$. Then $\mathcal{Y}' \to \mathcal{Y}$ is an isomorphism over $\mathcal{V}$.


etale-cohomology-lemma-zero-in-generic-point Let $X$ be a quasi-compact and quasi-separated scheme. Let $\eta \in X$ be a generic point of an irreducible component of $X$. \begin{enumerate} \item Let $\mathcal{F}$ be a torsion abelian sheaf on $X_\etale$ whose stalk $\mathcal{F}_{\overline{\eta}}$ is zero. Then $\mathcal{F} = \colim \mathcal{F}_i$ is a filtered colimit of constructible abelian sheaves $\mathcal{F}_i$ such that for each $i$ the support of $\mathcal{F}_i$ is contained in a closed subscheme not containing $\eta$. \item Let $\Lambda$ be a Noetherian ring and $\mathcal{F}$ a sheaf of $\Lambda$-modules on $X_\etale$ whose stalk $\mathcal{F}_{\overline{\eta}}$ is zero. Then $\mathcal{F} = \colim \mathcal{F}_i$ is a filtered colimit of constructible sheaves of $\Lambda$-modules $\mathcal{F}_i$ such that for each $i$ the support of $\mathcal{F}_i$ is contained in a closed subscheme not containing $\eta$. \end{enumerate}


algebraization-lemma-punctured-still-connected \begin{reference} \cite[Theorem 1.6]{Varbaro} \end{reference} Let $(A, \mathfrak m)$ be a Noetherian complete local ring. Let $I$ be a proper ideal of $A$. Set $X = \Spec(A)$ and $Y = V(I)$. Denote \begin{enumerate} \item $d$ the minimal dimension of an irreducible component of $X$, and \item $c$ the minimal dimension of a closed subset $Z \subset X$ such that $X \setminus Z$ is disconnected. \end{enumerate} Then for $Z \subset Y$ closed we have $Y \setminus Z$ is connected if $\dim(Z) < \min(c, d - 1) - \text{cd}(A, I)$. In particular, the punctured spectrum of $A/I$ is connected if $\text{cd}(A, I) < \min(c, d - 1)$.


decent-spaces-lemma-Jacobson-ft-points-lift-to-closed Let $S$ be a scheme. Let $X$ be a Jacobson algebraic space over $S$. For $x \in X_{\text{ft-pts}}$ and $g : W \to X$ locally of finite type with $W$ a scheme, if $x \in \Im(|g|)$, then there exists a closed point of $W$ mapping to $x$.


homology-lemma-differential-objects-ses Let $\mathcal{A}$ be an abelian category. Let $0 \to (A, d) \to (B, d) \to (C, d) \to 0$ be a short exact sequence of differential objects. Then we get an exact homology sequence $$ \ldots \to H(C, d) \to H(A, d) \to H(B, d) \to H(C, d) \to \ldots $$


obsolete-lemma-get-morphism-general Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $t$ be the minimal number of generators for $I$. Let $C$ be a Noetherian $I$-adically complete $A$-algebra. There exists an integer $d \geq 0$ depending only on $I \subset A \to C$ with the following property: given \begin{enumerate} \item $c \geq 0$ and $B$ in Algebraization of Formal Spaces, Equation (\ref{restricted-equation-C-prime}) such that for $a \in I^c$ multiplication by $a$ on $\NL_{B/A}^\wedge$ is zero in $D(B)$, \item an integer $n > 2t\max(c, d)$, \item an $A/I^n$-algebra map $\psi_n : B/I^nB \to C/I^nC$, \end{enumerate} there exists a map $\varphi : B \to C$ of $A$-algebras such that $\psi_n \bmod I^{m - c} = \varphi \bmod I^{m - c}$ with $m = \lfloor \frac{n}{t} \rfloor$.


sites-cohomology-lemma-compare-cohomology-LC With $X \in \Ob(\textit{LC}_{qc})$ and $a_X : \Sh(\textit{LC}_{qc}/X) \to \Sh(X)$ as above: \begin{enumerate} \item for an abelian sheaf $\mathcal{F}$ on $X$ we have $H^n(X, \mathcal{F}) = H^n_{qc}(X, a_X^{-1}\mathcal{F})$, \item for $K \in D^+(X)$ we have $H^n(X, K) = H^n_{qc}(X, a_X^{-1}K)$. \end{enumerate} For example, if $A$ is an abelian group, then we have $H^n(X, \underline{A}) = H^n_{qc}(X, \underline{A})$.


modules-lemma-section-direct-sum-quasi-compact Let $(X, \mathcal{O}_X)$ be a ringed space. Let $I$ be a set. For $i \in I$, let $\mathcal{F}_i$ be a sheaf of $\mathcal{O}_X$-modules. For $U \subset X$ quasi-compact open the map $$ \bigoplus\nolimits_{i \in I} \mathcal{F}_i(U) \longrightarrow \left(\bigoplus\nolimits_{i \in I} \mathcal{F}_i\right)(U) $$ is bijective.


more-algebra-lemma-normal-domain-absolutely-integrally-closed Let $A$ be a normal domain. Then $A$ is absolutely integrally closed if and only if its fraction field is algebraically closed.


homology-lemma-filtered-complex Let $\mathcal{A}$ be an abelian category. Let $A \to B \to C$ be a complex of filtered objects of $\mathcal{A}$. Assume $\alpha : A \to B$ and $\beta : B \to C$ are strict morphisms of filtered objects. Then $\text{gr}(\Ker(\beta)/\Im(\alpha)) = \Ker(\text{gr}(\beta))/\Im(\text{gr}(\alpha)))$.


algebra-lemma-epimorphism-injective-spec Let $R \to S$ be an epimorphism of rings. Then \begin{enumerate} \item $\Spec(S) \to \Spec(R)$ is injective, and \item for $\mathfrak q \subset S$ lying over $\mathfrak p \subset R$ we have $\kappa(\mathfrak p) = \kappa(\mathfrak q)$. \end{enumerate}


etale-cohomology-theorem-descent-modules If $A \to B$ is faithfully flat then descent data with respect to $A\to B$ are effective.


stacks-properties-lemma-map-into-reduction Let $\mathcal{X}$, $\mathcal{Y}$ be algebraic stacks. Let $\mathcal{Z} \subset \mathcal{X}$ be a closed substack Assume $\mathcal{Y}$ is reduced. A morphism $f : \mathcal{Y} \to \mathcal{X}$ factors through $\mathcal{Z}$ if and only if $f(|\mathcal{Y}|) \subset |\mathcal{Z}|$.


stacks-more-morphisms-lemma-make-section Let $f : \mathcal{X} \to Y$ be a morphism from an algebraic stack to an algebraic space. Let $V \subset Y$ be an open subspace. Assume \begin{enumerate} \item $f$ is separated and of finite type, \item $Y$ is quasi-compact and quasi-separated, \item $V$ is quasi-compact, and \item $\mathcal{X}_V$ is a gerbe over $V$. \end{enumerate} Then there exists a commutative diagram $$ \xymatrix{ \overline{Z} \ar[rd]_{\overline{g}} & Z \ar[l]^j \ar[d]_g \ar[r]_h & \mathcal{X} \ar[ld]^f \\ & Y } $$ with $j$ an open immersion, $\overline{g}$ and $h$ proper, and such that $|V|$ is contained in the image of $|g|$.


more-algebra-lemma-formal-fibres-CM Properties (A), (B), (C), (D), and (E) hold for $P(k \to R) =$``$R$ is Cohen-Macaulay''.


spaces-lemma-viewed-as-properties Let $\Sch_{fppf}$ be a big fppf site. Let $S \to S'$ be a morphism of this site. Let $F$ be an algebraic space over $S$. Let $T$ be a scheme over $S$ and let $f : T \to F$ be a morphism over $S$. Let $f' : T' \to F'$ be the morphism over $S'$ we get from $f$ by applying the equivalence of categories described in Lemma \ref{lemma-category-of-spaces-over-smaller-base-scheme}. For any property $\mathcal{P}$ as in Definition \ref{definition-relative-representable-property} we have $\mathcal{P}(f') \Leftrightarrow \mathcal{P}(f)$.


obsolete-lemma-double-dual-rational In Resolution of Surfaces, Situation \ref{resolve-situation-rational}. Let $M$ be a finite reflexive $A$-module. Let $M \otimes_A \mathcal{O}_X$ denote the pullback of the associated $\mathcal{O}_S$-module. Then $M \otimes_A \mathcal{O}_X$ maps onto its double dual.


obsolete-lemma-property-higher-rank Let $X$ be a Noetherian scheme. Let $\mathcal{P}$ be a property of coherent sheaves on $X$ such that \begin{enumerate} \item For any short exact sequence of coherent sheaves if two out of three of them have property $\mathcal{P}$ then so does the third. \item If $\mathcal{P}$ holds for a direct sum of coherent sheaves then it holds for both. \item For every integral closed subscheme $Z \subset X$ with generic point $\xi$ there exists some coherent sheaf $\mathcal{G}$ such that \begin{enumerate} \item $\text{Supp}(\mathcal{G}) = Z$, \item $\mathcal{G}_\xi$ is annihilated by $\mathfrak m_\xi$, and \item property $\mathcal{P}$ holds for $\mathcal{G}$. \end{enumerate} \end{enumerate} Then property $\mathcal{P}$ holds for every coherent sheaf on $X$.


algebra-lemma-ideal-in-localization \begin{slogan} Ideals in the localization of a ring are localizations of ideals. \end{slogan} Each ideal $I'$ of $S^{-1}A$ takes the form $S^{-1}I$, where one can take $I$ to be the inverse image of $I'$ in $A$.


quot-lemma-flat-geometrically-connected-fibres-with-section In Situation \ref{situation-pic} let $\sigma : B \to X$ be a section. Assume that $\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}$ is an isomorphism for all $T$ over $B$. Then $$ 0 \to \Pic(T) \to \Pic(X_T) \to \Picardfunctor_{X/B}(T) \to 0 $$ is a split exact sequence with splitting given by $\sigma_T^* : \Pic(X_T) \to \Pic(T)$.


sites-lemma-continuous-with-continuous-left-adjoint A continuous functor of sites which has a continuous left adjoint defines a morphism of sites.


schemes-lemma-locally-quasi-compact The underlying topological space of any scheme is locally quasi-compact, see Topology, Definition \ref{topology-definition-locally-quasi-compact}.


more-algebra-lemma-weakly-etale-finite-type Let $A \to B$ be a ring map such that $B \otimes_A B \to B$ is flat. \begin{enumerate} \item If $A \to B$ is of finite type, then $A \to B$ is unramified. \item If $A \to B$ is of finite presentation and flat, then $A \to B$ is \'etale. \end{enumerate} In particular a weakly \'etale ring map of finite presentation is \'etale.


spaces-perfect-lemma-map-from-pseudo-coherent-to-complex-with-support Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $T \subset |X|$ be a closed subset such that the complement $U \subset X$ is quasi-compact. Let $\alpha : P \to E$ be a morphism of $D_\QCoh(\mathcal{O}_X)$ with either \begin{enumerate} \item $P$ is perfect and $E$ supported on $T$, or \item $P$ pseudo-coherent, $E$ supported on $T$, and $E$ bounded below. \end{enumerate} Then there exists a perfect complex of $\mathcal{O}_X$-modules $I$ and a map $I \to \mathcal{O}_X[0]$ such that $I \otimes^\mathbf{L} P \to E$ is zero and such that $I|_U \to \mathcal{O}_U[0]$ is an isomorphism.


examples-lemma-non-effective-descent-projective There is an etale covering $X\to S$ of schemes and a descent datum $(V/X,\varphi)$ relative to $X\to S$ such that  $V\to X$ is projective, but the descent datum is not effective in the category of schemes.


moduli-lemma-pic-functor-curves-smooth Assume $f : X \to B$ has relative dimension $\leq 1$ in addition to the other assumptions in this section. Then $\Picardfunctor_{X/B} \to B$ is smooth.


more-morphisms-lemma-descending-property-perfect The property $\mathcal{P}(f) =$``$f$ is perfect'' is fpqc local on the base.


pione-lemma-ses-field Let $k$ be a field with algebraic closure $\overline{k}$. Let $X$ be a quasi-compact and quasi-separated scheme over $k$. If the base change $X_{\overline{k}}$ is connected, then there is a short exact sequence $$ 1 \to \pi_1(X_{\overline{k}}) \to \pi_1(X) \to \pi_1(\Spec(k)) \to 1 $$ of profinite topological groups.


constructions-lemma-tie-up-psi In Situation \ref{situation-relative-proj}. Let $(f : T \to S, d, \mathcal{L}, \psi)$ be a quadruple. Let $r_{d, \mathcal{L}, \psi} : T \to \underline{\text{Proj}}_S(\mathcal{A})$ be the associated $S$-morphism. There exists an isomorphism of $\mathbf{Z}$-graded $\mathcal{O}_T$-algebras $$ \theta : r_{d, \mathcal{L}, \psi}^*\left( \bigoplus\nolimits_{n \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(nd) \right) \longrightarrow \bigoplus\nolimits_{n \in \mathbf{Z}} \mathcal{L}^{\otimes n} $$ such that the following diagram commutes $$ \xymatrix{ \mathcal{A}^{(d)} \ar[rr]_-{\psi}  \ar[rd]_-{\psi_{univ}} & & f_*\left( \bigoplus\nolimits_{n \in \mathbf{Z}} \mathcal{L}^{\otimes n} \right) \\  & \pi_*\left( \bigoplus\nolimits_{n \geq 0} \mathcal{O}_{\underline{\text{Proj}}_S(\mathcal{A})}(nd) \right) \ar[ru]_\theta } $$ The commutativity of this diagram uniquely determines $\theta$.


algebra-lemma-etale-makes-unramified-closed-at-prime Let $R \to S$ be a ring map. Let $\mathfrak q$ be a prime of $S$ lying over $\mathfrak p \subset R$. Assume that $R \to S$ is of finite type and unramified at $\mathfrak q$. Then there exist \begin{enumerate} \item an \'etale ring map $R \to R'$, \item a prime $\mathfrak p' \subset R'$ lying over $\mathfrak p$. \item a product decomposition $$ R' \otimes_R S = A \times B $$ \end{enumerate} with the following properties \begin{enumerate} \item $R' \to A$ is surjective, and \item $\mathfrak p'A$ is a prime of $A$ lying over $\mathfrak p'$ and over $\mathfrak q$. \end{enumerate}


obsolete-lemma-equiv Assumptions and notation as in Simplicial, Lemma \ref{simplicial-lemma-section}. There exists a section $g : U \to V$ to the morphism $f$ and the composition $g \circ f$ is homotopy equivalent to the identity on $V$. In particular, the morphism $f$ is a homotopy equivalence.


properties-lemma-integral-normal Let $X$ be an integral scheme. Then $X$ is normal if and only if for every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is a normal domain.


derham-lemma-gysin-transverse-global Let $c \geq 0$ and $$ \xymatrix{ Z' \ar[d]_h \ar[r] & X' \ar[d]_g \ar[r] & S' \ar[d] \\ Z \ar[r] & X \ar[r] & S } $$ satisfy the assumptions of Lemma \ref{lemma-gysin-transverse} and assume in addition that $X \to S$ and $X' \to S'$ are smooth and that $Z \to X$ and $Z' \to X'$ are Koszul regular immersions. Then the diagram $$ \xymatrix{ H^q(Z, \Omega^p_{Z/S}) \ar[rr]_-{\gamma^{p, q}} \ar[d] & & H^{q + c}(X, \Omega^{p + c}_{X/S}) \ar[d] \\ H^q(Z', \Omega^p_{Z'/S'}) \ar[rr]^{\gamma^{p, q}} & & H^{q + c}(X', \Omega^{p + c}_{X'/S'}) } $$ is commutative where $\gamma^{p, q}$ is as in Remark \ref{remark-how-to-use}.


spaces-resolve-lemma-modification-of-dim-2-is-projective-over-complete If $(A, \mathfrak m, \kappa)$ is a complete Noetherian local domain of dimension $2$, then every modification of $\Spec(A)$ is projective over $A$.


modules-lemma-pullback-de-rham-complex Let $f : Y \to X$ be a continuous map of topological spaces. Let $\mathcal{A} \to \mathcal{B}$ be a homomorphism of sheaves of rings on $X$. Then there is a canonical identification $f^{-1}\Omega^\bullet_{\mathcal{B}/\mathcal{A}} = \Omega^\bullet_{f^{-1}\mathcal{B}/f^{-1}\mathcal{A}}$ of de Rham complexes.


spaces-more-morphisms-lemma-open-subspace-thickening Let $S$ be a scheme. Let $X \subset X'$ be a thickening of algebraic spaces over $S$. For any open subspace $U \subset X$ there exists a unique open subspace $U' \subset X'$ such that $U = X \times_{X'} U'$.


spaces-morphisms-lemma-trivial-implications Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is separated, then $f$ is locally separated and $f$ is quasi-separated.


spaces-groupoids-lemma-constructing-invariant-opens Let $B \to S$ as in Section \ref{section-notation}. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. \begin{enumerate} \item If $s$ and $t$ are open, then for every open $W \subset U$ the open $s(t^{-1}(W))$ is $R$-invariant. \item If $s$ and $t$ are open and quasi-compact, then $U$ has an open covering consisting of $R$-invariant quasi-compact open subspaces. \end{enumerate}


spaces-flat-lemma-relate-zero-affine-push In Situation \ref{situation-iso} suppose given an affine morphism $g : X \to X'$. Set $u' = f_*u : f_*\mathcal{F} \to f_*\mathcal{G}$. Then $F_{u, iso} = F_{u', iso}$, $F_{u, inj} = F_{u', inj}$, $F_{u, surj} = F_{u', surj}$, and $F_{u, zero} = F_{u', zero}$.


properties-lemma-catenary-local-rings-catenary Let $X$ be a scheme. The following are equivalent \begin{enumerate} \item $X$ is catenary, and \item for any $x \in X$ the local ring $\mathcal{O}_{X, x}$ is catenary. \end{enumerate}


curves-lemma-criterion-very-ample-bis Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_X) = k$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Assume \begin{enumerate} \item $\mathcal{L}$ is globally generated, \item $H^1(X, \mathcal{L}) = 0$, and \item $\mathcal{L}$ is ample. \end{enumerate} Then $\mathcal{L}^{\otimes 2}$ is very ample on $X$ over $k$.


pione-proposition-injective Let $p$ be a prime number. Let $i : Z \to X$ be a closed immersion of connected affine schemes over $\mathbf{F}_p$. For any geometric point $\overline{z}$ of $Z$ the map $$ \pi_1(Z, \overline{z}) \to \pi_1(X, \overline{z}) $$ is injective.


more-algebra-lemma-internal-hom-composition Let $R$ be a ring. Given $K, L, M$ in $D(R)$ there is a canonical morphism $$ R\Hom_R(L, M) \otimes_R^\mathbf{L} R\Hom_R(K, L) \longrightarrow R\Hom_R(K, M) $$ in $D(R)$ functorial in $K, L, M$.


categories-lemma-preserve-injective-maps Let $\mathcal{I}$ be an index category, i.e., a category. Assume that for every solid diagram $$ \xymatrix{ x \ar[d] \ar[r] & y \ar@{..>}[d] \\ z \ar@{..>}[r] & w } $$ in $\mathcal{I}$ there exists an object $w$ and dotted arrows making the diagram commute. Then \begin{enumerate} \item an injective morphism $M \to N$ of diagrams of sets over $\mathcal{I}$ gives rise to an injective map $\colim M_i \to \colim N_i$ of sets, \item in general the same is not the case for diagrams of abelian groups and their colimits. \end{enumerate}


flat-lemma-finite-type-flat-at-point-X Let $S$ be a scheme. Let $X$ be locally of finite type over $S$. Let $x \in X$ with image $s \in S$. If $X$ is flat at $x$ over $S$, then there exists an elementary \'etale neighbourhood $(S', s') \to (S, s)$ and an open subscheme $$ V \subset X \times_S \Spec(\mathcal{O}_{S', s'}) $$ which contains the unique point of $X \times_S \Spec(\mathcal{O}_{S', s'})$ mapping to $x$ such that $V \to \Spec(\mathcal{O}_{S', s'})$ is flat and of finite presentation.


spaces-divisors-lemma-dominate-admissible-blowups Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $U \subset X$ be a quasi-compact open subspace. Let $b_i : X_i \to X$, $i = 1, \ldots, n$ be $U$-admissible blowups. There exists a $U$-admissible blowup $b : X' \to X$ such that (a) $b$ factors as $X' \to X_i \to X$ for $i = 1, \ldots, n$ and (b) each of the morphisms $X' \to X_i$ is a $U$-admissible blowup.


varieties-lemma-galois-action-connected-components Let $k$ be a field, with separable algebraic closure $\overline{k}$. Let $X$ be a scheme over $k$. There is an action $$ \text{Gal}(\overline{k}/k)^{opp} \times \pi_0(X_{\overline{k}}) \longrightarrow \pi_0(X_{\overline{k}}) $$ with the following properties: \begin{enumerate} \item An element $\overline{T} \in \pi_0(X_{\overline{k}})$ is fixed by the action if and only if there exists a connected component $T \subset X$, which is geometrically connected over $k$, such that $T_{\overline{k}} = \overline{T}$. \item For any field extension $k \subset k'$ with separable algebraic closure $\overline{k}'$ the diagram $$ \xymatrix{ \text{Gal}(\overline{k}'/k') \times \pi_0(X_{\overline{k}'}) \ar[r] \ar[d] & \pi_0(X_{\overline{k}'}) \ar[d] \\ \text{Gal}(\overline{k}/k) \times \pi_0(X_{\overline{k}}) \ar[r] & \pi_0(X_{\overline{k}}) } $$ is commutative (where the right vertical arrow is a bijection according to Lemma \ref{lemma-separably-closed-field-connected-components}). \end{enumerate}


spaces-morphisms-lemma-universally-injective-representable Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$. Then $f$ is universally injective (in the sense of Section \ref{section-representable}) if and only if for all fields $K$ the map $X(K) \to Y(K)$ is injective.


spaces-chow-lemma-spell-out-degree-zero-cycle Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. Let $\alpha = \sum n_i[Z_i]$ be in $Z_0(X)$. Then $$ \deg(\alpha) = \sum n_i\deg(Z_i) $$ where $\deg(Z_i)$ is the degree of $Z_i \to \Spec(k)$, i.e., $\deg(Z_i) = \dim_k \Gamma(Z_i, \mathcal{O}_{Z_i})$.


more-morphisms-lemma-change-hypotheses Suppose that $f : X \to S$ is locally of finite type, $S$ locally Noetherian, $x \in X$ a closed point of its fibre $X_s$, and $U \subset X$ an open subscheme such that $U \cap X_s = \emptyset$ and $x \in \overline{U}$, then the conclusions of Lemma \ref{lemma-quasi-finite-quasi-section-meeting-nearby-open-X} hold.


moduli-lemma-pic-functor-uniqueness-part Assume the geometric fibres of $X \to B$ are integral in addition to the other assumptions in this section. Then $\Picardfunctor_{X/B} \to B$ is separated.


morphisms-lemma-affine-quasi-affine Let $S$ be a scheme. Let $X$ be an affine scheme. A morphism $f : X \to S$ is quasi-affine if and only if it is quasi-compact. In particular any morphism from an affine scheme to a quasi-separated scheme is quasi-affine.


artin-lemma-monomorphism Let $S$ be a locally Noetherian scheme. Let $a : F \to G$ be a transformation of functors $(\Sch/S)_{fppf}^{opp} \to \textit{Sets}$. Assume that \begin{enumerate} \item $a$ is injective, \item $F$ satisfies axioms [0], [1], [2], [4], and [5], \item $\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$, \item $G$ is an algebraic space locally of finite type over $S$, \end{enumerate} Then $F$ is an algebraic space.


stacks-morphisms-lemma-definition-separated Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Then \begin{enumerate} \item $f$ is separated if and only if $\Delta_{f, 1}$ and $\Delta_{f, 2}$ are universally closed, and \item $f$ is quasi-separated if and only if $\Delta_{f, 1}$ and $\Delta_{f, 2}$ are quasi-compact. \item $f$ is quasi-DM if and only if $\Delta_{f, 1}$ and $\Delta_{f, 2}$ are locally quasi-finite. \item $f$ is DM if and only if $\Delta_{f, 1}$ and $\Delta_{f, 2}$ are unramified. \end{enumerate}


more-morphisms-lemma-relative-finite-presentation Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. \begin{enumerate} \item If $f$ is locally of finite presentation, then $\mathcal{F}$ is of finite presentation relative to $S$ if and only if $\mathcal{F}$ is of finite presentation. \item The morphism $f$ is locally of finite presentation if and only if $\mathcal{O}_X$ is of finite presentation relative to $S$. \end{enumerate}


examples-defos-lemma-power-series-rings-TI In Lemma \ref{lemma-rings-TI} if $P = k[[x_1, \ldots, x_n]]/(f)$ for some nonzero $f \in (x_1, \ldots, x_n)^2$, then \begin{enumerate} \item $\text{Inf}(\Deformationcategory_P)$ is finite dimensional if and only if $n = 1$, and \item $T\Deformationcategory_P$ is finite dimensional if $$ \sqrt{(f, \partial f/\partial x_1, \ldots,  \partial f/\partial x_n)} = (x_1, \ldots, x_n) $$ \end{enumerate}


sdga-lemma-compose-pushforward-hom In the situation above, denote $RT : D(\mathcal{A}', \text{d}) \to D(\mathcal{B}, \text{d})$ the right derived extension of (\ref{equation-pushforward}). Then we have $$ RT(\mathcal{M}) = Rf_* R\SheafHom(\mathcal{N}, \mathcal{M}) $$ functorially in $\mathcal{M}$.


chow-lemma-section-smooth Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $p : P \to X$ be a smooth morphism of schemes locally of finite type over $S$ and let $s : X \to P$ be a section. Then $s$ is a regular immersion and $1 = s^! \circ p^*$ in $A^*(X)^\wedge$ where $p^* \in A^*(P \to X)^\wedge$ is the bivariant class of Lemma \ref{lemma-flat-pullback-bivariant}.


cohomology-lemma-mayer-vietoris-cup Let $(X, \mathcal{O}_X)$ be a ringed space. Set $A = \Gamma(X, \mathcal{O}_X)$. Suppose that $X = U \cup V$ is a union of two open subsets. For objects $K$ and $M$ of $D(\mathcal{O}_X)$ we have a map of distinguished triangles $$ \xymatrix{ R\Gamma(X, K) \otimes_A^\mathbf{L} R\Gamma(X, M) \ar[r] \ar[d] & R\Gamma(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} M) \ar[d] \\ R\Gamma(X, K) \otimes_A^\mathbf{L} (R\Gamma(U, M) \oplus R\Gamma(V, M)) \ar[r] \ar[d] & R\Gamma(U, K \otimes_{\mathcal{O}_X}^\mathbf{L} M) \oplus R\Gamma(V, K \otimes_{\mathcal{O}_X}^\mathbf{L} M)) \ar[d] \\ R\Gamma(X, K) \otimes_A^\mathbf{L} R\Gamma(U \cap V, M) \ar[r] \ar[d] & R\Gamma(U \cap V, K \otimes_{\mathcal{O}_X}^\mathbf{L} M) \ar[d] \\ R\Gamma(X, K) \otimes_A^\mathbf{L} R\Gamma(X, M)[1] \ar[r] & R\Gamma(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} M)[1] } $$ where \begin{enumerate} \item the horizontal arrows are given by cup product, \item on the right hand side we have the distinguished triangle of Lemma \ref{lemma-unbounded-mayer-vietoris} for $K \otimes_{\mathcal{O}_X}^\mathbf{L} M$, and \item on the left hand side we have the exact functor $R\Gamma(X, K) \otimes_A^\mathbf{L} - $ applied to the distinguished triangle of Lemma \ref{lemma-unbounded-mayer-vietoris} for $M$. \end{enumerate}


derham-proposition-de-rham-is-weil Let $k$ be a field of characteristic zero. The functor that sends a smooth projective scheme $X$ over $k$ to $H_{dR}^*(X/k)$ is a Weil cohomology theory in the sense of Weil Cohomology Theories, Definition \ref{weil-definition-weil-cohomology-theory}.


perfect-lemma-perfect-descends-fpqc Let $\{f_i : X_i \to X\}$ be an fpqc covering of schemes. Let $E \in D(\mathcal{O}_X)$. Then $E$ is perfect if and only if each $Lf_i^*E$ is perfect.


more-algebra-lemma-flatten-on-affine-blowup Let $(R, \mathfrak m)$ be a local domain with fraction field $K$. Let $S$ be a finite type $R$-algebra. Let $M$ be a finite $S$-module. For every valuation ring $A \subset K$ dominating $R$ there exists an ideal $I \subset \mathfrak m$ and a nonzero element $a \in I$ such that \begin{enumerate} \item $I$ is finitely generated, \item $A$ has center on $R[\frac{I}{a}]$, \item the fibre ring of $R \to R[\frac{I}{a}]$ at $\mathfrak m$ is not zero, and \item the strict transform $S_{I, a}$ of $S$ along $R \to R[\frac{I}{a}]$ is flat and of finite presentation over $R$, and the strict transform $M_{I, a}$ of $M$ along $R \to R[\frac{I}{a}]$ is flat over $R$ and finitely presented over $S_{I, a}$. \end{enumerate}


crystalline-lemma-describe-omega-small In Situation \ref{situation-global}. Let $(U, T, \delta)$ be an object of $\text{Cris}(X/S)$. Let $$ (U(1), T(1), \delta(1)) = (U, T, \delta) \times (U, T, \delta) $$ in $\text{Cris}(X/S)$. Let $\mathcal{K} \subset \mathcal{O}_{T(1)}$ be the quasi-coherent sheaf of ideals corresponding to the closed immersion $\Delta : T \to T(1)$. Then $\mathcal{K} \subset \mathcal{J}_{T(1)}$ is preserved by the divided structure on $\mathcal{J}_{T(1)}$ and we have $$ (\Omega_{X/S})_T = \mathcal{K}/\mathcal{K}^{[2]} $$


models-lemma-canonical-map-of-pic In Situation \ref{situation-regular-model} let $T$ be the numerical type associated to $X$. There exists a canonical map $$ \Pic(C) \to \Pic(T) $$ whose kernel is exactly those invertible modules on $C$ which are the restriction of invertible modules $\mathcal{L}$ on $X$ with $\deg_{C_i}(\mathcal{L}|_{C_i}) = 0$ for $i = 1, \ldots, n$.


decent-spaces-lemma-residual-space-regular A reduced, locally Noetherian singleton algebraic space $Z$ is regular.


limits-lemma-finite-closed-in-finite-finite-presentation Let $f : X \to S$ be a morphism of schemes. Assume \begin{enumerate} \item $f$ is finite, and \item $S$ is quasi-compact and quasi-separated. \end{enumerate} Then there exists a morphism which is finite and of finite presentation $f' : X' \to S$ and a closed immersion $X \to X'$ of schemes over $S$.


spaces-more-morphisms-lemma-differentials-relative-immersion-section Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $i : Z \to X$ be an immersion of algebraic spaces over $B$, and assume $i$ (\'etale locally) has a left inverse. Then the canonical sequence $$ 0 \to \mathcal{C}_{Z/X} \to i^*\Omega_{X/B} \to \Omega_{Z/B} \to 0 $$ of Lemma \ref{lemma-differentials-relative-immersion} is (\'etale locally) split exact.


etale-theorem-formally-etale Let $f : X \to S$ be a morphism that is locally of finite presentation. The following are equivalent \begin{enumerate} \item $f$ is \'etale, \item for all affine $S$-schemes $Y$, and closed subschemes $Y_0 \subset Y$ defined by square-zero ideals, the natural map $$ \Mor_S(Y, X) \longrightarrow \Mor_S(Y_0, X) $$ is bijective. \end{enumerate}


etale-cohomology-lemma-compare-cohomology-etale-ph For a scheme $X$ and $a_X : \Sh((\Sch/X)_{ph}) \to \Sh(X_\etale)$ as above: \begin{enumerate} \item $H^q(X_\etale, \mathcal{F}) = H^q_{ph}(X, a_X^{-1}\mathcal{F})$ for a torsion abelian sheaf $\mathcal{F}$ on $X_\etale$, \item $H^q(X_\etale, K) = H^q_{ph}(X, a_X^{-1}K)$ for $K \in D^+(X_\etale)$ with torsion cohomology sheaves. \end{enumerate} Example: if $A$ is a torsion abelian group, then $H^q_\etale(X, \underline{A}) = H^q_{ph}(X, \underline{A})$.


resolve-lemma-blowup-still-good Let $(A, \mathfrak m, \kappa)$ be a Nagata local normal domain of dimension $2$. Assume $A$ defines a rational singularity and that the completion $A^\wedge$ of $A$ is normal. Then \begin{enumerate} \item $A^\wedge$ defines a rational singularity, and \item if $X \to \Spec(A)$ is the blowing up in $\mathfrak m$, then for a closed point $x \in X$ the completion $\mathcal{O}_{X, x}$ is normal. \end{enumerate}


formal-spaces-lemma-composition-representable The composition of morphisms representable by algebraic spaces is representable by algebraic spaces. The same holds for representable (by schemes).


injectives-lemma-G-modules Let $G$ be a topological group. Let $R$ be a ring. The category $\text{Mod}_{R, G}$ of $R\text{-}G$-modules, see \'Etale Cohomology, Definition \ref{etale-cohomology-definition-G-module-continuous}, has functorial injective hulls. In particular this holds for the category of discrete $G$-modules.


sdga-lemma-good-dga Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{B}, \text{d})$ be a differential graded $\mathcal{O}$-algebra. There exists a quasi-isomorphism of differential graded $\mathcal{O}$-algebras $(\mathcal{A}, \text{d}) \to (\mathcal{B}, \text{d})$ such that $\mathcal{A}$ is graded flat and K-flat as a complex of $\mathcal{O}$-modules and such that the same is true after pullback by any morphism of ringed topoi.


proetale-lemma-characterize-sheaf-big Let $S$ be a scheme contained in a big pro-\'etale site $\Sch_\proetale$. A sheaf $\mathcal{F}$ on the big pro-\'etale site $(\Sch/S)_\proetale$ is given by the following data: \begin{enumerate} \item for every $T/S \in \Ob((\Sch/S)_\proetale)$ a sheaf $\mathcal{F}_T$ on $T_\proetale$, \item for every $f : T' \to T$ in $(\Sch/S)_\proetale$ a map $c_f : f_{small}^{-1}\mathcal{F}_T \to \mathcal{F}_{T'}$. \end{enumerate} These data are subject to the following conditions: \begin{enumerate} \item[(a)] given any $f : T' \to T$ and $g : T'' \to T'$ in $(\Sch/S)_\proetale$ the composition $g_{small}^{-1}c_f \circ c_g$ is equal to $c_{f \circ g}$, and \item[(b)] if $f : T' \to T$ in $(\Sch/S)_\proetale$ is weakly \'etale then $c_f$ is an isomorphism. \end{enumerate}


more-morphisms-lemma-rel-n-pseudo-module Let $X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Then \begin{enumerate} \item $\mathcal{F}$ is $m$-pseudo-coherent relative to $S$ for all $m > 0$, \item $\mathcal{F}$ is $0$-pseudo-coherent relative to $S$ if and only if $\mathcal{F}$ is a finite type $\mathcal{O}_X$-module, \item $\mathcal{F}$ is $(-1)$-pseudo-coherent relative to $S$ if and only if $\mathcal{F}$ is quasi-coherent and finitely presented relative to $S$. \end{enumerate}


obsolete-lemma-dualizing-components In Semistable Reduction, Situation \ref{models-situation-regular-model} the dualizing module of $C_i$ over $k$ is $$ \omega_{C_i} = \omega_X(C_i)|_{C_i} $$ where $\omega_X$ is as above.


defos-lemma-existence-lci If $A \to B$ is a local complete intersection ring map, then there exists a solution to (\ref{equation-to-solve}).


properties-lemma-characterize-nagata Let $X$ be a locally Noetherian scheme. Then $X$ is Nagata if and only if every integral closed subscheme $Z \subset X$ is Japanese.


spaces-groupoids-lemma-isomorphism Let $S$ be a scheme, let $(U, R, s, t, c)$ be a groupoid scheme over $S$. If $(\mathcal{F}, \alpha)$ is a quasi-coherent module on $(U, R, s, t, c)$ then $\alpha$ is an isomorphism.


more-morphisms-lemma-semicontinuous-w Let $f : X \to Y$ be a locally quasi-finite morphism. Let $w : X \to \mathbf{Z}_{> 0}$ be a positive weighting of $f$. Then $w$ is upper semi-continuous.


pione-lemma-structure-decomposition-separable-closure Let $A$ be a discrete valuation ring with fraction field $K$. Let $K^{sep}$ be a separable closure of $K$. Let $A^{sep}$ be the integral closure of $A$ in $K^{sep}$. Let $\mathfrak m^{sep}$ be a maximal ideal of $A^{sep}$. Let $\mathfrak m = \mathfrak m^{sep} \cap A$, let $\kappa = A/\mathfrak m$, and let $\overline{\kappa} = A^{sep}/\mathfrak m^{sep}$. Then $\overline{\kappa}$ is an algebraic closure of $\kappa$. Let $G = \text{Gal}(K^{sep}/K)$, $D = \{\sigma \in G \mid \sigma(\mathfrak m^{sep}) = \mathfrak m^{sep}\}$, and $I = \{\sigma \in D \mid \sigma \bmod \mathfrak m^{sep} = \text{id}_{\kappa(\mathfrak m^{sep})}\}$. The decomposition group $D$ fits into a canonical exact sequence $$ 1 \to I \to D \to \text{Gal}(\kappa^{sep}/\kappa) \to 1 $$ where $\kappa^{sep} \subset \overline{\kappa}$ is the separable closure of $\kappa$. The inertia group $I$ fits into a canonical exact sequence $$ 1 \to P \to I \to I_t \to 1 $$ such that \begin{enumerate} \item $P$ is a normal subgroup of $D$, \item $P$ is a pro-$p$-group if the characteristic of $\kappa_A$ is $p > 1$ and $P = \{1\}$ if the characteristic of $\kappa_A$ is zero, \item there exists a canonical surjective map $$ \theta_{can} : I \to \lim_{n\text{ prime to }p} \mu_n(\kappa^{sep}) $$ whose kernel is $P$, which satisfies $\theta_{can}(\tau \sigma \tau^{-1}) = \tau(\theta_{can}(\sigma))$ for $\tau \in D$, $\sigma \in I$, and which induces an isomorphism $I_t \to \lim_{n\text{ prime to }p} \mu_n(\kappa^{sep})$. \end{enumerate}


intersection-lemma-compose-flat-pullback Let $f : X \to Y$ and $g : Y \to Z$ be flat morphisms of varieties. Then $g \circ f$ is flat and $f^* \circ g^* = (g \circ f)^*$ as maps $Z_k(Z) \to Z_{k + \dim(X) - \dim(Z)}(X)$.


local-cohomology-lemma-annihilator-frobenius-module Let $p$ be a prime number. Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring with $p = 0$ in $A$. Let $M$ be a finite $A$-module such that $M \otimes_{A, F} A \cong M$. Then $M$ is finite free.


moduli-lemma-pic-existence-part Assume $X \to B$ is smooth in addition to being proper. Then $\Picardstack_{X/B} \to B$ satisfies the existence part of the valuative criterion (Morphisms of Stacks, Definition \ref{stacks-morphisms-definition-existence}).


chow-lemma-decompose-section-formulae In the situation of Lemma \ref{lemma-decompose-section} assume $Y$ is locally of finite type over $(S, \delta)$ as in Situation \ref{situation-setup}. Then we have $i_1^*p^*\alpha = p_1^*i^*\alpha$ in $\CH_k(D_1)$ for all $\alpha \in \CH_k(Y)$.


obsolete-lemma-quasi-separated-very-reasonable An algebraic space which is Zariski locally quasi-separated is very reasonable. In particular any quasi-separated algebraic space is very reasonable.


bootstrap-lemma-bootstrap-locally-quasi-finite Let $S$ be a scheme. Let $F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. Let $X$ be a scheme and let $X \to F$ be representable by algebraic spaces and locally quasi-finite. Then $X \to F$ is representable (by schemes).


trace-lemma-switch-l Switching $l$. Let $E$ be a number field. Start with $$ \rho : \pi_1(X)\to SL_2(E_\lambda) $$ absolutely irreducible continuous, where $\lambda$ is a place of $E$ not lying above $p$. Then for any second place $\lambda'$ of $E$ not lying above $p$ there exists a finite extension $E'_{\lambda'}$ and a absolutely irreducible continuous representation $$ \rho': \pi_1(X)\to SL_2(E'_{\lambda'}) $$ which is compatible with $\rho$ in the sense that the characteristic polynomials of all Frobenii are the same.


more-morphisms-lemma-descend-quasi-finite-universally-open Let $S = \lim S_i$ be a limit of a directed system of schemes with affine transition morphisms. Let $0 \in I$ and let $f_0 : X_0 \to Y_0$ be a morphism of schemes over $S_0$. Assume $S_0$, $X_0$, $Y_0$ are quasi-compact and quasi-separated. Let $f_i : X_i \to Y_i$ be the base change of $f_0$ to $S_i$ and let $f : X \to Y$ be the base change of $f_0$ to $S$. If \begin{enumerate} \item $f$ is locally quasi-finite and universally open, and \item $f_0$ is locally of finite presentation, \end{enumerate} then there exists an $i \geq 0$ such that $f_i$ is locally quasi-finite and universally open.


stacks-geometry-lemma-behaviour-of-dimensions-wrt-smooth-morphisms-stacky If $f: \mathcal{U} \to \mathcal{X}$ is a smooth morphism of locally Noetherian algebraic stacks, and if $u \in |\mathcal{U}|$ with image $x \in |\mathcal{X}|$, then $$ \dim_u (\mathcal{U}) = \dim_x(\mathcal{X}) + \dim_{u} (\mathcal{U}_x). $$


chow-lemma-chern-character-dual In Situation \ref{situation-setup} let $X$ be locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module with dual $\mathcal{E}^\vee$. Then $ch_i(\mathcal{E}^\vee) = (-1)^i ch_i(\mathcal{E})$ in $A^i(X)$.


categories-lemma-initial-essentially-constant Let $\mathcal{C}$ be a category. Let $H : \mathcal{I} \to \mathcal{J}$ be a functor of cofiltered index categories. If $H$ is initial, then any diagram $M : \mathcal{J} \to \mathcal{C}$ is essentially constant if and only if $M \circ H$ is essentially constant.


examples-lemma-counter-Grothendieck-existence Counter examples to algebraization of coherent sheaves. \begin{enumerate} \item Grothendieck's existence theorem as stated in Cohomology of Schemes, Theorem \ref{coherent-theorem-grothendieck-existence} is false if we drop the assumption that $X \to \Spec(A)$ is separated. \item The stack of coherent sheaves $\Cohstack_{X/B}$ of Quot, Theorems \ref{quot-theorem-coherent-algebraic-general} and \ref{quot-theorem-coherent-algebraic} is in general not algebraic if we drop the assumption that $X \to S$ is separated \item The functor $\Quotfunctor_{\mathcal{F}/X/B}$ of Quot, Proposition \ref{quot-proposition-quot} is not an algebraic space in general if we drop the assumption that $X \to B$ is separated. \end{enumerate}


etale-lemma-etale-definition Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be locally of finite type. Let $x \in X$. The morphism $f$ is \'etale at $x$ in the sense of Definition \ref{definition-etale-schemes-1} if and only if it is \'etale at $x$ in the sense of Morphisms, Definition \ref{morphisms-definition-etale}.


examples-proposition-quotient-by-finitely-generated-torsion-modules Let $A$ be a Noetherian integral domain. Let $K$ denote its field of fractions. Let $\text{Mod}_A^{fg}$ denote the category of finitely generated $A$-modules and $\mathcal{T}^{fg}$ its Serre subcategory of finitely generated torsion modules. Then $\text{Mod}_A^{fg}/\mathcal{T}^{fg}$ is canonically equivalent to the category of finite dimensional $K$-vector spaces.


spaces-divisors-lemma-relatively-ample-sanity-check Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Assume $Y$ is a scheme. The following are equivalent \begin{enumerate} \item $\mathcal{L}$ is ample on $X/Y$ in the sense of Definition \ref{definition-relatively-ample}, and \item $X$ is a scheme and $\mathcal{L}$ is ample on $X/Y$ in the sense of Morphisms, Definition \ref{morphisms-definition-relatively-ample}. \end{enumerate}


dga-lemma-bc-homotopy The functor (\ref{equation-bc}) defines an exact functor of triangulated categories $K(\text{Mod}_{(A, \text{d})}) \to K(\text{Mod}_{(B, \text{d})})$.


more-groupoids-lemma-invariant-affine-open-around-generic-point-Noetherian Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$ finite, $U$ is locally Noetherian, and $u_1, \ldots, u_m \in U$ points whose orbits consist of generic points of irreducible components of $U$. Then there exist $R$-invariant subschemes $V' \subset V \subset U$ such that \begin{enumerate} \item $u_1, \ldots, u_m \in V'$, \item $V$ is open in $U$, \item $V'$ and $V$ are affine, \item $V' \subset V$ is a thickening, \item the morphisms $s', t'$ of the restriction $(V', R', s', t', c')$ are finite locally free. \end{enumerate}


stacks-properties-lemma-zariski-open-cover-stack-is-scheme Let $\mathcal X$ be an algebraic stack. Let $\mathcal{X}_i$, $i \in I$ be a set of open substacks of $\mathcal{X}$. Assume \begin{enumerate} \item $\mathcal{X} = \bigcup_{i \in I} \mathcal{X}_i$, and \item each $\mathcal{X}_i$ is a scheme \end{enumerate} Then $\mathcal{X}$ is a scheme.


algebra-lemma-length-localize Let $R$ be a ring. Let $M$ be an $R$-module. Let $S \subset R$ be a multiplicative subset. Then $\text{length}_R(M) \geq \text{length}_{S^{-1}R}(S^{-1}M)$.


derived-lemma-products-sums-shifts-triangles Let $\mathcal{D}$ be a pre-triangulated category. Let $I$ be a set. \begin{enumerate} \item Let $X_i$, $i \in I$ be a family of objects of $\mathcal{D}$. \begin{enumerate} \item If $\prod X_i$ exists, then $(\prod X_i)[1] = \prod X_i[1]$. \item If $\bigoplus X_i$ exists, then $(\bigoplus X_i)[1] = \bigoplus X_i[1]$. \end{enumerate} \item Let $X_i \to Y_i \to Z_i \to X_i[1]$ be a family of distinguished triangles of $\mathcal{D}$. \begin{enumerate} \item If $\prod X_i$, $\prod Y_i$, $\prod Z_i$ exist, then $\prod X_i \to \prod Y_i \to \prod Z_i \to \prod X_i[1]$ is a distinguished triangle. \item If $\bigoplus X_i$, $\bigoplus Y_i$, $\bigoplus Z_i$ exist, then $\bigoplus X_i \to \bigoplus Y_i \to \bigoplus Z_i \to \bigoplus X_i[1]$ is a distinguished triangle. \end{enumerate} \end{enumerate}


divisors-lemma-regular-meromorphic-section-exists-noetherian Let $X$ be a locally Noetherian scheme having no embedded points. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Then $\mathcal{L}$ has a regular meromorphic section.


decent-spaces-lemma-birational-birational Let $S$ be a scheme. Let $f : X \to Y$ be a birational morphism of algebraic spaces over $S$ which are decent and have finitely many irreducible components. Assume one of the following conditions is satisfied \begin{enumerate} \item $f$ is locally of finite type and $Y$ reduced (i.e., integral), \item $f$ is locally of finite presentation. \end{enumerate} Then there exist dense opens $U \subset X$ and $V \subset Y$ such that $f(U) \subset V$ and $f|_U : U \to V$ is an isomorphism.


groupoids-lemma-group-scheme-field-countable-affine Let $G$ be a group scheme over a field. There exists an open and closed subscheme $G' \subset G$ which is a countable union of affines.


etale-cohomology-lemma-h-descent-etale-sheaves Let $S$ be a scheme. Then the category fibred in groupoids $$ p : \mathcal{S} \longrightarrow (\Sch/S)_h $$ whose fibre category over $U$ is the category $\Sh(U_\etale)$ of sheaves on the small \'etale site of $U$ is a stack in groupoids.


more-morphisms-lemma-immersion-universal-thickening Let $i : Z \to X$ be an immersion of schemes. Then \begin{enumerate} \item $i$ is formally unramified, \item the universal first order thickening of $Z$ over $X$ is the first order infinitesimal neighbourhood of $Z$ in $X$ of Definition \ref{definition-first-order-infinitesimal-neighbourhood}, and \item the conormal sheaf of $i$ in the sense of Morphisms, Definition \ref{morphisms-definition-conormal-sheaf} agrees with the conormal sheaf of $i$ in the sense of Definition \ref{definition-universal-thickening}. \end{enumerate}


flat-theorem-pullback-trivial-fibres Let $p$ be a prime number. Let $Y$ be a quasi-compact and quasi-separated scheme over $\mathbf{F}_p$. Let $f : X \to Y$ be a proper, surjective morphism of finite presentation with geometrically connected fibres. Then the functor $$ \colim_F \textit{Vect}(Y) \longrightarrow \colim_F \textit{Vect}(X) $$ is fully faithful with essential image described as follows. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module. Assume for all $y \in Y$ there exists integers $n_y, r_y \geq 0$ such that $$ F^{n_y, *}\mathcal{E}|_{X_{y, red}} \cong \mathcal{O}_{X_{y, red}}^{\oplus r_y} $$ Then for some $n \geq 0$ the $n$th Frobenius power pullback $F^{n, *}\mathcal{E}$ is the pullback of a finite locally free $\mathcal{O}_Y$-module.


decent-spaces-lemma-punctured-spec Let $S$ be a scheme. Let $X$ be a decent locally Noetherian algebraic space over $S$. Let $x \in |X|$. Then $$ W = \{x' \in |X| : x' \leadsto x,\ x' \not = x\} $$ is a Noetherian, spectral, sober, Jacobson topological space.


dpa-lemma-divided-powers-on-tor Let $R$ be a commutative ring. Let $S$ and $T$ be commutative $R$-algebras. Then there is a canonical structure of a strictly graded commutative $R$-algebra with divided powers on $$ \operatorname{Tor}_*^R(S, T). $$


obsolete-lemma-helper-finite-type-flat-finite-presentation Let $R$ be a domain with fraction field $K$. Let $S = R[x_1, \ldots, x_n]$ be a polynomial ring over $R$. Let $M$ be a finite $S$-module. Assume that $M$ is flat over $R$. If for every subring $R \subset R' \subset K$, $R \not = R'$ the module $M \otimes_R R'$ is finitely presented over $S \otimes_R R'$, then $M$ is finitely presented over $S$.


models-lemma-sequence-torsion In Situation \ref{situation-regular-model} let $d = \gcd(m_1, \ldots, m_n)$ and let $T$ be the numerical type associated to $X$. Let $h \geq 1$ be an integer prime to $d$. There exists an exact sequence $$ 0 \to \Pic(X)[h] \to \Pic(C)[h] \to \Pic(T)[h] $$


spaces-divisors-lemma-regular-meromorphic-section-exists Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ satisfying (a), (b), and (c) of Lemma \ref{lemma-compute-meromorphic}. Then every invertible $\mathcal{O}_X$-module $\mathcal{L}$ has a regular meromorphic section.


spaces-lemma-product-spaces Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G$ be algebraic spaces over $S$. Then $F \times G$ is an algebraic space, and is a product in the category of algebraic spaces over $S$.


sites-cohomology-lemma-section-RHom-over-U Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K, L$ be objects of $D(\mathcal{O})$. For every object $U$ of $\mathcal{C}$ we have $$ H^0(U, R\SheafHom(L, M)) = \Hom_{D(\mathcal{O}_U)}(L|_U, M|_U) $$ and we have $H^0(\mathcal{C}, R\SheafHom(L, M)) = \Hom_{D(\mathcal{O})}(L, M)$.


pione-theorem-specialization-map-isomorphism-prime-to-p Let $f : X \to S$ be a smooth proper morphism with geometrically connected fibres. Let $s' \leadsto s$ be a specialization. If the characteristic of $\kappa(s)$ is $p$, then the specialization map $$ sp : \pi_1(X_{\overline{s}'}) \to \pi_1(X_{\overline{s}}) $$ is surjective and induces an isomorphism $$ \pi'_1(X_{\overline{s}'}) \cong \pi'_1(X_{\overline{s}}) $$ of the maximal prime-to-p quotients


morphisms-lemma-differentials-relative-immersion-section Let $i : Z \to X$ be an immersion of schemes over $S$, and assume $i$ (locally) has a left inverse. Then the canonical sequence $$ 0 \to \mathcal{C}_{Z/X} \to i^*\Omega_{X/S} \to \Omega_{Z/S} \to 0 $$ of Lemma \ref{lemma-differentials-relative-immersion} is (locally) split exact. In particular, if $s : S \to X$ is a section of the structure morphism $X \to S$ then the map $\mathcal{C}_{S/X} \to s^*\Omega_{X/S}$ induced by $\text{d}_{X/S}$ is an isomorphism.


spaces-properties-lemma-isomorphism-ringed-topoi Let $X$, $Y$ be algebraic spaces over $\mathbf{Z}$. If $$ (g, g^\sharp) : (\Sh(X_\etale), \mathcal{O}_X) \longrightarrow (\Sh(Y_\etale), \mathcal{O}_Y) $$ is an isomorphism of ringed topoi, then there exists a unique morphism $f : X \to Y$ of algebraic spaces such that $(g, g^\sharp)$ is isomorphic to $(f_{small}, f^\sharp)$ and moreover $f$ is an isomorphism of algebraic spaces.


morphisms-lemma-ubiquity-J-2 The following types of schemes are J-2. \begin{enumerate} \item Any scheme locally of finite type over a field. \item Any scheme locally of finite type over a Noetherian complete local ring. \item Any scheme locally of finite type over $\mathbf{Z}$. \item Any scheme locally of finite type over a Noetherian local ring of dimension $1$. \item Any scheme locally of finite type over a Nagata ring of dimension $1$. \item Any scheme locally of finite type over a Dedekind ring of characteristic zero. \item And so on. \end{enumerate}


spaces-resolve-lemma-formally-unramified Let $(A, \mathfrak m)$ be a local Noetherian ring. Let $X$ be an algebraic space over $A$. Assume \begin{enumerate} \item $A$ is analytically unramified (Algebra, Definition \ref{algebra-definition-analytically-unramified}), \item $X$ is locally of finite type over $A$, \item $X \to \Spec(A)$ is \'etale at every point of codimension $0$ in $X$. \end{enumerate} Then the normalization of $X$ is finite over $X$.


proetale-lemma-local-structure-constructible-complex Let $X$ be a weakly contractible affine scheme. Let $\Lambda$ be a Noetherian ring and let $I \subset \Lambda$ be an ideal. Let $K$ be an object of $D_{cons}(X, \Lambda)$ such that the cohomology sheaves of $K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I}$ are locally constant. Then there exists a finite disjoint open covering $X = \coprod U_i$ and for each $i$ a finite collection of finite projective $\Lambda^\wedge$-modules $M_a, \ldots, M_b$ such that $K|_{U_i}$ is represented by a complex $$ (\underline{M^a})^\wedge \to \ldots \to (\underline{M^b})^\wedge $$ in $D(U_{i, \proetale}, \Lambda)$ for some maps of sheaves of $\Lambda$-modules $(\underline{M^i})^\wedge \to (\underline{M^{i + 1}})^\wedge$.


pione-proposition-purity-complete-intersection Let $(A, \mathfrak m)$ be a Noetherian local ring. If $A$ is a complete intersection of dimension $\geq 3$, then purity holds for $A$ in the sense that any finite \'etale cover of the punctured spectrum extends.


stacks-perfect-proposition-derived-pullback-quasi-coherent Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The exact functor $f^*$ induces a commutative diagram $$ \xymatrix{ D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) \ar[r] & D(\mathcal{O}_\mathcal{X}) \\ D_{\mathcal{M}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y}) \ar[r] \ar[u]^{f^*} & D(\mathcal{O}_\mathcal{Y}) \ar[u]^{f^*} } $$ The composition $$ D_{\mathcal{M}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y}) \xrightarrow{f^*} D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) \xrightarrow{q_\mathcal{X}} D_\QCoh(\mathcal{O}_\mathcal{X}) $$ is left derivable with respect to the localization $D_{\mathcal{M}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y}) \to D_\QCoh(\mathcal{O}_\mathcal{Y})$ and we may define $Lf^*_\QCoh$ as its left derived functor $$ Lf_\QCoh^* : D_\QCoh(\mathcal{O}_\mathcal{Y}) \longrightarrow D_\QCoh(\mathcal{O}_\mathcal{X}) $$ (see Derived Categories, Definitions \ref{derived-definition-right-derived-functor-defined} and \ref{derived-definition-everywhere-defined}). If $f$ is quasi-compact and quasi-separated, then $Lf^*_\QCoh$ and $Rf_{\QCoh, *}$ satisfy the following adjointness: $$ \Hom_{D_\QCoh(\mathcal{O}_\mathcal{X})}(Lf^*_\QCoh A, B) = \Hom_{D_\QCoh(\mathcal{O}_\mathcal{Y})}(A, Rf_{\QCoh, *}B) $$ for $A \in D_\QCoh(\mathcal{O}_\mathcal{Y})$ and $B \in D^{+}_\QCoh(\mathcal{O}_\mathcal{X})$.


more-algebra-lemma-Mittag-Leffler Let $$ 0 \to (A_i) \to (B_i) \to (C_i) \to 0 $$ be a short exact sequence of inverse systems of abelian groups. If $(A_i)$ and $(C_i)$ are ML, then so is $(B_i)$.


formal-spaces-lemma-permanence-property-morphisms Let $S$ be a scheme. Let $P$ be a local property of morphisms of $\textit{WAdm}^{count}$ which has the cancellation property. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of locally countably indexed formal algebraic spaces over $S$. If $g \circ f$ and $g$ satisfies the equivalent conditions of Lemma \ref{lemma-property-defines-property-morphisms} then so does $f : X \to Y$.


spaces-simplicial-lemma-restriction-to-components-functorial Let $f : Y \to X$ be a morphism of simplicial spaces. Then $$ \xymatrix{ \Sh(Y_n) \ar[d] \ar[r]_{f_n} & \Sh(X_n) \ar[d] \\ \Sh(Y_{Zar}) \ar[r]^{f_{Zar}} & \Sh(X_{Zar}) } $$ is a commutative diagram of topoi.


topology-lemma-from-hausdorff Let $f : X \to Y$ be continuous map of topological spaces. If $X$ is Hausdorff, then $f$ is separated.


more-morphisms-lemma-pushout-finite-type In Situation \ref{situation-pushout-along-closed-immersion-and-integral} assume $S$ is a locally Noetherian scheme and $X$, $Y$, and $Z$ are locally of finite type over $S$. Then the pushout $Y \amalg_Z X$ (Proposition \ref{proposition-pushout-along-closed-immersion-and-integral}) is locally of finite type over $S$.


spaces-morphisms-lemma-valuative-criterion-representable Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is representable. The following are equivalent \begin{enumerate} \item $f$ satisfies the existence part of the valuative criterion as in Definition \ref{definition-valuative-criterion}, \item given any commutative solid diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & X \ar[d] \\ \Spec(A) \ar[r] \ar@{-->}[ru] & Y } $$ where $A$ is a valuation ring with field of fractions $K$, there exists a dotted arrow, i.e., $f$ satisfies the existence part of the valuative criterion as in Schemes, Definition \ref{schemes-definition-valuative-criterion}. \end{enumerate}


spaces-divisors-lemma-vanshing-gives-ample Let $R$ be a Noetherian ring. Let $X$ be an algebraic space over $R$ such that the structure morphism $f : X \to \Spec(R)$ is proper. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $\mathcal{L}$ is ample on $X/R$ (Definition \ref{definition-relatively-ample}), \item for every coherent $\mathcal{O}_X$-module $\mathcal{F}$ there exists an $n_0 \geq 0$ such that $H^p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0$ for all $n \geq n_0$ and $p > 0$. \end{enumerate}


spaces-more-morphisms-lemma-lci-permanence Let $S$ be a scheme. Let $$ \xymatrix{ X \ar[rr]_f \ar[rd] & & Y \ar[ld] \\ & Z } $$ be a commutative diagram of morphisms of algebraic spaces over $S$. Assume $Y \to Z$ is smooth and $X \to Z$ is a local complete intersection morphism. Then $f : X \to Y$ is a local complete intersection morphism.


etale-cohomology-lemma-central-simple-algebra-pgln \begin{slogan} Central simple algebras are classified by Galois cohomology of PGL. \end{slogan} Let $K$ be a field and let $K^{sep}$ be a separable algebraic closure. Then the set of isomorphism classes of central simple algebras of degree $d$ over $K$ is in bijection with the non-abelian cohomology $H^1(\text{Gal}(K^{sep}/K), \text{PGL}_d(K^{sep}))$.


modules-lemma-flat-resolution-of-flat Let $(X, \mathcal{O}_X)$ be a ringed space. Let $$ \ldots \to \mathcal{F}_2 \to \mathcal{F}_1 \to \mathcal{F}_0 \to \mathcal{Q} \to 0 $$ be an exact complex of $\mathcal{O}_X$-modules. If $\mathcal{Q}$ and all $\mathcal{F}_i$ are flat $\mathcal{O}_X$-modules, then for any $\mathcal{O}_X$-module $\mathcal{G}$ the complex $$ \ldots \to \mathcal{F}_2 \otimes_{\mathcal{O}_X} \mathcal{G} \to \mathcal{F}_1 \otimes_{\mathcal{O}_X} \mathcal{G} \to \mathcal{F}_0 \otimes_{\mathcal{O}_X} \mathcal{G} \to \mathcal{Q} \otimes_{\mathcal{O}_X} \mathcal{G} \to 0 $$ is exact also.


spaces-pushouts-lemma-glue-etale-space-modification Let $S$ be a scheme. Let $f : Y' \to Y$ be a proper morphism of algebraic spaces over $S$. Let $i : Z \to Y$ be a closed immersion. Set $E = Z \times_Y Y'$. Picture $$ \xymatrix{ E \ar[d]_g \ar[r]_j & Y' \ar[d]^f \\ Z \ar[r]^i & Y } $$ If $f$ is an isomorphism over $Y \setminus Z$, then the functor $$ Y_{spaces, \etale} \longrightarrow Y'_{spaces, \etale} \times_{E_{spaces, \etale}} \Sh(Z_{spaces, \etale}) $$ is an equivalence of categories.


trace-lemma-when-in-bounded An object $E$ of $D(\mathcal{A})$ is contained in $D^+(\mathcal{A})$ if and only if $H^i(E) =0 $ for all $i \ll 0$. Similar statements hold for $D^-$ and $D^+$.


cohomology-lemma-subsheaf-irreducible Let $X$ be an irreducible topological space. Let $\mathcal{H} \subset \underline{\mathbf{Z}}$ be an abelian subsheaf of the constant sheaf. Then there exists a nonempty open $U \subset X$ such that $\mathcal{H}|_U = \underline{d\mathbf{Z}}_U$ for some $d \in \mathbf{Z}$.


spaces-morphisms-lemma-section-immersion Let $S$ be a scheme. Let $f : X \to T$ be a morphism of algebraic spaces over $S$. Let $s : T \to X$ be a section of $f$ (in a formula $f \circ s = \text{id}_T$). Then \begin{enumerate} \item $s$ is representable, locally of finite type, locally quasi-finite, separated and a monomorphism, \item if $f$ is locally separated, then $s$ is an immersion, \item if $f$ is separated, then $s$ is a closed immersion, and \item if $f$ is quasi-separated, then $s$ is quasi-compact. \end{enumerate}


moduli-curves-lemma-smooth-curves-h0-smooth The morphisms $\mathcal{M} \to \Spec(\mathbf{Z})$ and $\mathcal{M}_g \to \Spec(\mathbf{Z})$ are smooth.


spaces-more-morphisms-lemma-inverse-systems-surjective Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. A map $(\mathcal{F}_n) \to (\mathcal{G}_n)$ is surjective in $\textit{Coh}(X, \mathcal{I})$ if and only if $\mathcal{F}_1 \to \mathcal{G}_1$ is surjective.


chow-lemma-degree-vector-bundle Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $\leq 1$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module of constant rank. Then $$ \deg(\mathcal{E}) = \deg(c_1(\mathcal{E}) \cap [X]_1) $$ where the left hand side is defined in Varieties, Definition \ref{varieties-definition-degree-invertible-sheaf}.


sites-lemma-localize-enough Let $\mathcal{C}$ be a site. Let $U$ be an object of $\mathcal{C}$. let $\{(p_i, u_i)\}_{i\in I}$ be a family of points of $\mathcal{C}$. For $x \in u_i(U)$ let $q_{i, x}$ be the point of $\mathcal{C}/U$ constructed in Lemma \ref{lemma-point-localize}. If $\{p_i\}$ is a conservative family of points, then $\{q_{i, x}\}_{i \in I, x \in u_i(U)}$ is a conservative family of points of $\mathcal{C}/U$. In particular, if $\mathcal{C}$ has enough points, then so does every localization $\mathcal{C}/U$.


spaces-chow-lemma-delta-is-dimension In Situation \ref{situation-setup} assume $B$ is Jacobson and that $\delta(b) = 0$ for every closed point $b$ of $|B|$. Let $X/B$ be good. If $Z \subset X$ is an integral closed subspace with generic point $\xi \in |Z|$, then the following integers are the same: \begin{enumerate} \item $\delta(\xi) = \delta_{X/B}(\xi)$, \item $\dim(|Z|)$, \item $\text{codim}(\{z\}, |Z|)$ for $z \in |Z|$ closed, \item the dimension of the local ring of $Z$ at $z$ for $z \in |Z|$ closed, and \item $\dim(\mathcal{O}_{Z, \overline{z}})$ for $z \in |Z|$ closed. \end{enumerate}


algebra-lemma-blowup-dominant Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $a \in I$. If $a$ is not contained in any minimal prime of $R$, then $\Spec(R[\frac{I}{a}]) \to \Spec(R)$ has dense image.


descent-theorem-descend-algebra-properties If $A \otimes_R S$ has one of the following properties as an $S$-algebra \begin{enumerate} \item[(a)] of finite type; \item[(b)] of finite presentation; \item[(c)] formally unramified; \item[(d)] unramified; \item[(e)] \'etale; \end{enumerate} then so does $A$ as an $R$-algebra (and of course conversely).


cohomology-lemma-internal-hom-diagonal Let $(X, \mathcal{O}_X)$ be a ringed space. Given $K, L$ in $D(\mathcal{O}_X)$ there is a canonical morphism $$ K \longrightarrow R\SheafHom(L, K \otimes_{\mathcal{O}_X}^\mathbf{L} L) $$ in $D(\mathcal{O}_X)$ functorial in both $K$ and $L$.


more-etale-lemma-f-shriek-separated-direct-sums Let $f : X \to Y$ be a morphism of schemes which is separated and locally of finite type. Then functor $f_!$ commutes with direct sums.


topologies-lemma-fibre-products-ph Let $S$ be a scheme. Let $\Sch_{ph}$ be a big ph site containing $S$. The underlying categories of the sites $\Sch_{ph}$, $(\Sch/S)_{ph}$, and $(\textit{Aff}/S)_{ph}$ have fibre products. In each case the obvious functor into the category $\Sch$ of all schemes commutes with taking fibre products. The category $(\Sch/S)_{ph}$ has a final object, namely $S/S$.


brauer-lemma-dimension-square Let $A$ be a finite central simple algebra over a field $k$. Then $[A : k]$ is a square.


sites-cohomology-lemma-left-dual-complex Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}^\bullet$ be a complex of $\mathcal{O}$-modules. If $\mathcal{F}^\bullet$ has a left dual in the monoidal category of complexes of $\mathcal{O}$-modules (Categories, Definition \ref{categories-definition-dual}) then for every object $U$ of $\mathcal{C}$ there exists a covering $\{U_i \to U\}$ such that $\mathcal{F}^\bullet|_{U_i}$ is strictly perfect and the left dual is as constructed in Example \ref{example-dual}.


properties-lemma-locally-Cohen-Macaulay Let $X$ be a scheme. The following are equivalent: \begin{enumerate} \item The scheme $X$ is Cohen-Macaulay. \item For every affine open $U \subset X$ the ring $\mathcal{O}_X(U)$ is Noetherian and Cohen-Macaulay. \item There exists an affine open covering $X = \bigcup U_i$ such that each $\mathcal{O}_X(U_i)$ is Noetherian and Cohen-Macaulay. \item There exists an open covering $X = \bigcup X_j$ such that each open subscheme $X_j$ is Cohen-Macaulay. \end{enumerate} Moreover, if $X$ is Cohen-Macaulay then every open subscheme is Cohen-Macaulay.


algebra-lemma-characterize-domain Let $R$ be a ring. \begin{enumerate} \item An ideal maximal among the ideals which do not contain a nonzerodivisor is prime. \item If $R$ is nonzero and every nonzero prime ideal in $R$ contains a nonzerodivisor, then $R$ is a domain. \end{enumerate}


divisors-lemma-check-reflexive Let $X$ be an integral locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is reflexive, \item $\mathcal{F}_x$ is a reflexive $\mathcal{O}_{X, x}$-module for all $x \in X$, \item $\mathcal{F}_x$ is a reflexive $\mathcal{O}_{X, x}$-module for all closed points $x \in X$. \end{enumerate}


dpa-lemma-sub-dp-ideal Let $(A, I, \gamma)$ be a divided power ring. Let $E \subset I$ be a subset. Then the smallest ideal $J \subset I$ preserved by $\gamma$ and containing all $f \in E$ is the ideal $J$ generated by $\gamma_n(f)$, $n \geq 1$, $f \in E$.


coherent-lemma-Cohen-Macaulay-over-regular Let $X$ be a regular scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is Cohen-Macaulay and $\text{Supp}(\mathcal{F}) = X$, \item $\mathcal{F}$ is finite locally free of rank $> 0$. \end{enumerate}


pic-lemma-flat-geometrically-connected-fibres-with-section Let $f : X \to S$ be as in Definition \ref{definition-picard-functor}. Assume $f$ has a section $\sigma$ and that $\mathcal{O}_T \to f_{T, *}\mathcal{O}_{X_T}$ is an isomorphism for all $T \in \Ob((\Sch/S)_{fppf})$. Then $$ 0 \to \Pic(T) \to \Pic(X_T) \to \Picardfunctor_{X/S}(T) \to 0 $$ is a split exact sequence with splitting given by $\sigma_T^* : \Pic(X_T) \to \Pic(T)$.


proetale-lemma-ind-etale-implies Let $A \to B$ be ind-\'etale. Then $A \to B$ is weakly \'etale (More on Algebra, Definition \ref{more-algebra-definition-weakly-etale}).


descent-lemma-with-section-exact Suppose that $R \to A$ has a section. Then for any $R$-module $M$ the extended cochain complex (\ref{equation-extended-complex}) is exact.


obsolete-lemma-kill-local Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset A$ be an ideal. Let $M$ be a finite $A$-module. Let $s$ be an integer. Assume \begin{enumerate} \item $A$ has a dualizing complex, \item if $\mathfrak p \not \in V(I)$ and $V(\mathfrak p) \cap V(I) \not = \{\mathfrak m\}$, then $\text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim(A/\mathfrak p) > s$. \end{enumerate} Then there exists an $n > 0$ and an ideal $J \subset A$ with $V(J) \cap V(I) = \{\mathfrak m\}$ such that $JI^n$ annihilates $H^i_\mathfrak m(M)$ for $i \leq s$.


morphisms-lemma-affine-separated An affine morphism is separated and quasi-compact.


proetale-lemma-compare-truncations Let $X$ be a scheme. Let $\Lambda$ be a Noetherian ring. Let $K$ be an object of $D(X_\proetale, \Lambda)$. Set $K_n = K \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I^n}$. If $K_1$ is \begin{enumerate} \item in the essential image of $\epsilon^{-1} :D(X_\etale, \Lambda/I) \to D(X_\proetale, \Lambda/I)$, and \item has tor amplitude in $[a,\infty)$ for some $a \in \mathbf{Z}$, \end{enumerate} then (1) and (2) hold for $K_n$ as an object of $D(X_\proetale, \Lambda/I^n)$.


topology-lemma-two-points Let $X$ be a spectral space. Let $x, y \in X$. Then either there exists a third point specializing to both $x$ and $y$, or there exist disjoint open neighbourhoods containing $x$ and $y$.


formal-spaces-lemma-functoriality-etale-site Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. \begin{enumerate} \item There is a continuous functor $Y_{spaces, \etale} \to X_{spaces, \etale}$ which induces a morphism of sites $$ f_{spaces, \etale} : X_{spaces, \etale} \to Y_{spaces, \etale}. $$ \item The rule $f \mapsto f_{spaces, \etale}$ is compatible with compositions, in other words $(f \circ g)_{spaces, \etale} = f_{spaces, \etale} \circ g_{spaces, \etale}$ (see Sites, Definition \ref{sites-definition-composition-morphisms-sites}). \item The morphism of topoi associated to $f_{spaces, \etale}$ induces, via (\ref{equation-etale-topos}), a morphism of topoi $f_{small} : \Sh(X_\etale) \to \Sh(Y_\etale)$ whose construction is compatible with compositions. \end{enumerate}


morphisms-lemma-i-upper-shriek Let $i : Z \to X$ be a closed immersion of schemes. There is a functor\footnote{This is likely nonstandard notation.} $i^! : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_Z)$ which is a right adjoint to $i_*$. (Compare Modules, Lemma \ref{modules-lemma-i-star-right-adjoint}.)


more-morphisms-lemma-complex-relative-pseudo-coherent-modules Let $X \to S$ be a morphism of schemes which is locally of finite type. Let $m \in \mathbf{Z}$. Let $\mathcal{F}^\bullet$ be a (locally) bounded above complex of $\mathcal{O}_X$-modules such that $\mathcal{F}^i$ is $(m - i)$-pseudo-coherent relative to $S$ for all $i$. Then $\mathcal{F}^\bullet$ is $m$-pseudo-coherent relative to $S$.


proetale-lemma-classical-point In the situation above the scheme $\Spec(\mathcal{O}_{S, \overline{s}}^{sh})$ is an object of $X_\proetale$ and there is a canonical isomorphism $$ \mathcal{F}(\Spec(\mathcal{O}_{S, \overline{s}}^{sh})) = \mathcal{F}_{\overline{s}} $$ functorial in $\mathcal{F}$.


spaces-limits-lemma-sheafify-finite-presentation-map Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $F, G : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$ be functors. If $a : F \to G$ is a transformation which is limit preserving, then the induced transformation of sheaves $F^\# \to G^\#$ is limit preserving.


derived-lemma-limit-K-injectives \begin{slogan} The limit of a ``split'' tower of K-injective complexes is K-injective. \end{slogan} Let $\mathcal{A}$ be an abelian category. Let $$ \ldots \to I_3^\bullet \to I_2^\bullet \to I_1^\bullet $$ be an inverse system of complexes. Assume \begin{enumerate} \item each $I_n^\bullet$ is $K$-injective, \item each map $I_{n + 1}^m \to I_n^m$ is a split surjection, \item the limits $I^m = \lim I_n^m$ exist. \end{enumerate} Then the complex $I^\bullet$ is K-injective.


spaces-more-morphisms-lemma-base-change-CM Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume the fibres of $f$ are locally Noetherian. Let $Y' \to Y$ be locally of finite type. Let $f' : X' \to Y'$ be the base change of $f$. Let $x' \in |X'|$ be a point with image $x \in |X|$. \begin{enumerate} \item If $f$ is Cohen-Macaulay at $x$, then $f' : X' \to Y'$ is Cohen-Macaulay at $x'$. \item If $f$ is flat at $x$ and $f'$ is Cohen-Macaulay at $x'$, then $f$ is Cohen-Macaulay at $x$. \item If $Y' \to Y$ is flat at $f'(x')$ and $f'$ is Cohen-Macaulay at $x'$, then $f$ is Cohen-Macaulay at $x$. \end{enumerate}


morphisms-lemma-image-nowhere-dense-quasi-finite Let $f : Y \to X$ be a quasi-finite morphism of schemes. Let $T \subset Y$ be a closed nowhere dense subset of $Y$. Then $f(T) \subset X$ is a nowhere dense subset of $X$.


restricted-lemma-rig-flat-naive Let $\varphi : A \to B$ be an arrow of $\textit{WAdm}^{Noeth}$. If $A/I$ is Jacobson for some (equivalently any) ideal of definition $I \subset A$ and $\varphi$ is naively rig-flat, then $\varphi$ is rig-flat.


more-algebra-lemma-map-into-Rlim Let $\mathcal{D}$ be a triangulated category. Let $(K_n)$ be an inverse system of objects of $\mathcal{D}$. Let $K$ be a derived limit of the system $(K_n)$. Then for every $L$ in $\mathcal{D}$ we have a short exact sequence $$ 0 \to R^1\lim \Hom_\mathcal{D}(L, K_n[-1]) \to \Hom_\mathcal{D}(L, K) \to \lim \Hom_\mathcal{D}(L, K_n) \to 0 $$


more-morphisms-lemma-regular-fppf-local-source-and-target The property $\mathcal{P}(f)=$``the fibres of $f$ are locally Noetherian and $f$ is regular'' is local in the fppf topology on the target and local in the smooth topology on the source.


spaces-descent-lemma-finite-over-finite-module Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume $f$ is a finite morphism. Then $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite type if and only if $f_*\mathcal{F}$ is an $\mathcal{O}_Y$-module of finite type.


more-morphisms-lemma-smooth-regular A smooth morphism is regular.


derived-lemma-uniqueness-maps-postnikov-systems Let $\mathcal{D}$ be a triangulated category. Given a map (\ref{equation-map-complexes}) assume we are given Postnikov systems for both complexes. If \begin{enumerate} \item $\Hom(X_i[i], Y'_n[n]) = 0$ for $i = 1, \ldots, n$, or \item $\Hom(Y_n[n], X'_{n - i}[n - i]) = 0$ for $i = 1, \ldots, n$, or \item $\Hom(X_{j - i}[-i + 1], X'_j) = 0$ and $\Hom(X_j, X'_{j - i}[-i]) = 0$ for $j \geq i > 0$, \end{enumerate} then there exists at most one morphism between these Postnikov systems.


derived-lemma-enough-K-injectives-Ab4-star Let $\mathcal{A}$ be an abelian category having enough injectives and exact countable products. Then for every complex there is a quasi-isomorphism to a K-injective complex.


etale-cohomology-lemma-base-change-local Consider a cartesian diagram of schemes $$ \xymatrix{ X \ar[d]_f & Y \ar[l]^h \ar[d]^e \\ S & T \ar[l]_g } $$ and a sheaf $\mathcal{F}$ on $T_\etale$. Let $\{U_i \to X\}$ be an \'etale covering such that $U_i \to S$ factors as $U_i \to V_i \to S$ with $V_i \to S$ \'etale and consider the cartesian diagrams $$ \xymatrix{ U_i \ar[d]_{f_i} & U_i \times_X Y \ar[l]^{h_i} \ar[d]^{e_i} \\ V_i & V_i \times_S T \ar[l]_{g_i} } $$ Set $\mathcal{F}_i = \mathcal{F}|_{V_i \times_S T}$. \begin{enumerate} \item If $f_i^{-1}g_{i, *}\mathcal{F}_i = h_{i, *}e_i^{-1}\mathcal{F}_i$ for all $i$, then $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$. \item If $\mathcal{F}$ is an abelian sheaf and $f_i^{-1}R^qg_{i, *}\mathcal{F}_i = R^qh_{i, *}e_i^{-1}\mathcal{F}_i$ for all $i$, then $f^{-1}R^qg_*\mathcal{F} = R^qh_*e^{-1}\mathcal{F}$. \end{enumerate}


cohomology-lemma-sections-with-support-acyclic Let $i : Z \to X$ be the inclusion of a closed subset. Let $\mathcal{I}$ be an injective abelian sheaf on $X$. Then $\mathcal{H}_Z(\mathcal{I})$ is an injective abelian sheaf on $Z$.


smoothing-lemma-neron-colimit \begin{slogan} Unramified extensions of DVRs are ind-smooth AKA N\'eron desingularization \end{slogan} Let $R \subset \Lambda$ be an extension of discrete valuation rings which has ramification index $1$ and induces a separable extension of residue fields and of fraction fields. Then $\Lambda$ is a filtered colimit of smooth $R$-algebras.


topologies-lemma-verify-site-fppf Let $S$ be a scheme. Let $\Sch_{fppf}$ be a big fppf site containing $S$. Then $(\textit{Aff}/S)_{fppf}$ is a site.


spaces-properties-lemma-2-morphism Let $S$ be a scheme. Let $X$, $Y$ be algebraic spaces over $S$. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $t$ be a $2$-morphism from $(f_{small}, f^\sharp)$ to itself, see Modules on Sites, Definition \ref{sites-modules-definition-2-morphism-ringed-topoi}. Then $t = \text{id}$.


spaces-simplicial-lemma-descent-disjoint-union-Artinian-along-fields Let $X \to S$ be a quasi-compact flat surjective morphism. Let $(V, \varphi)$ be a descent datum relative to $X \to S$. If $V$ is a disjoint union of spectra of Artinian rings, then $(V, \varphi)$ is effective.


weil-lemma-negative-cohomology Assume given (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C). If there exists a smooth projective scheme $Y$ over $k$ such that $H^i(Y)$ is nonzero for some $i < 0$, then there exists an equidimensional smooth projective scheme $X$ over $k$ such that the equivalent conditions of Lemma \ref{lemma-H-0-separable} fail for $X$.


decent-spaces-lemma-fun-property-reasonable Let $S$ be a scheme. Let $X$ be a quasi-compact reasonable algebraic space. Then there exists a directed system of quasi-compact and quasi-separated algebraic spaces $X_i$ such that $X = \colim_i X_i$ (colimit in the category of sheaves).