dga-lemma-analogue-triangle-independent-splittings In Situation \ref{situation-ABC} let $0 \to x \to y \to z \to 0$ be an admissible short exact sequence in $\text{Comp}(\mathcal{A})$. The triangle $$ \xymatrix{x\ar[r] & y\ar[r] & z\ar[r]^{\delta} & x[1]} $$ with $\delta : z \to x[1]$ as defined in Lemma \ref{lemma-get-triangle} is up to canonical isomorphism in $K(\mathcal{A})$, independent of the choices made in Lemma \ref{lemma-get-triangle}.


spaces-properties-lemma-support-section-closed Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be an abelian sheaf on $X_\etale$. Let $U \in \Ob(X_\etale)$ and $\sigma \in \mathcal{F}(U)$. \begin{enumerate} \item The support of $\sigma$ is closed in $|X|$. \item The support of $\sigma + \sigma'$ is contained in the union of the supports of $\sigma, \sigma' \in \mathcal{F}(X)$. \item If $\varphi : \mathcal{F} \to \mathcal{G}$ is a map of abelian sheaves on $X_\etale$, then the support of $\varphi(\sigma)$ is contained in the support of $\sigma \in \mathcal{F}(U)$. \item The support of $\mathcal{F}$ is the union of the images of the supports of all local sections of $\mathcal{F}$. \item If $\mathcal{F} \to \mathcal{G}$ is surjective then the support of $\mathcal{G}$ is a subset of the support of $\mathcal{F}$. \item If $\mathcal{F} \to \mathcal{G}$ is injective then the support of $\mathcal{F}$ is a subset of the support of $\mathcal{G}$. \end{enumerate}


obsolete-lemma-Rlim-of-system Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(K_n)$ be an inverse system of objects of $D(\mathcal{O})$. Let $\mathcal{B} \subset \Ob(\mathcal{C})$ be a subset. Let $d \in \mathbf{N}$. Assume \begin{enumerate} \item $K_n$ is an object of $D^+(\mathcal{O})$ for all $n$, \item for $q \in \mathbf{Z}$ there exists $n(q)$ such that $H^q(K_{n + 1}) \to H^q(K_n)$ is an isomorphism for $n \geq n(q)$, \item every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$, \item for every $U \in \mathcal{B}$ we have $H^p(U, H^q(K_n)) = 0$ for $p > d$ and all $q$. \end{enumerate} Then we have $H^m(R\lim K_n) = \lim H^m(K_n)$ for all $m \in \mathbf{Z}$.


spaces-divisors-lemma-minimal-support-in-weakly-ass-decent Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $x \in |X|$. If \begin{enumerate} \item $X$ is decent (for example quasi-separated or locally separated), \item $x \in \text{Supp}(\mathcal{F})$ \item $x$ is not a specialization of another point in $\text{Supp}(\mathcal{F})$. \end{enumerate} Then $x \in \text{WeakAss}(\mathcal{F})$.


more-morphisms-lemma-flat-morphism-from-CM-scheme Let $f : X \to Y$ be a flat morphism of locally Noetherian schemes. If $X$ is Cohen-Macaulay, then $f$ is Cohen-Macaulay and $\mathcal{O}_{Y, f(x)}$ is Cohen-Macaulay for all $x \in X$.


divisors-lemma-extension-torsion-free Let $X$ be an integral scheme. Let $0 \to \mathcal{F} \to \mathcal{F}' \to \mathcal{F}'' \to 0$ be a short exact sequence of quasi-coherent $\mathcal{O}_X$-modules. If $\mathcal{F}$ and $\mathcal{F}''$ are torsion free, then $\mathcal{F}'$ is torsion free.


derived-lemma-projective-resolution-ses Let $\mathcal{A}$ be an abelian category. Assume $\mathcal{A}$ has enough projectives. For any short exact sequence $0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0$ of $\text{Comp}^{+}(\mathcal{A})$ there exists a commutative diagram in $\text{Comp}^{+}(\mathcal{A})$ $$ \xymatrix{ 0 \ar[r] & P_1^\bullet \ar[r] \ar[d] & P_2^\bullet \ar[r] \ar[d] & P_3^\bullet \ar[r] \ar[d] & 0 \\ 0 \ar[r] & A^\bullet \ar[r] & B^\bullet \ar[r] & C^\bullet \ar[r] & 0 } $$ where the vertical arrows are projective resolutions and the rows are short exact sequences of complexes. In fact, given any projective resolution $P^\bullet \to C^\bullet$ we may assume $P_3^\bullet = P^\bullet$.


decent-spaces-lemma-finite-etale-cover-dense-open-scheme Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If there exists a finite, \'etale, surjective morphism $U \to X$ where $U$ is a scheme, then there exists a dense open subspace of $X$ which is a scheme.


divisors-lemma-equation-codim-1-in-projective-space Let $R$ be a UFD. Let $Z \subset \mathbf{P}^n_R$ be a closed subscheme which has no embedded points such that every irreducible component of $Z$ has codimension $1$ in $\mathbf{P}^n_R$. Then the ideal $I(Z) \subset R[T_0, \ldots, T_n]$ corresponding to $Z$ is principal.


spaces-divisors-lemma-normal-effective-Cartier-divisor-S1 Let $S$ be a scheme and let $X$ be a locally Noetherian normal algebraic space over $S$. Let $D \subset X$ be an effective Cartier divisor. Then $D$ is $(S_1)$.


more-etale-lemma-lqf-shriek-derived Let $f : X \to Y$ be a locally quasi-finite morphism of schemes. The functors $f_!$ and $f^!$ of Definition \ref{definition-f-shriek-lqf} and Lemma \ref{lemma-lqf-f-upper-shriek} induce adjoint functors $f_! : D(X_\etale) \to D(Y_\etale)$ and $Rf^! : D(Y_\etale) \to D(X_\etale)$ on derived categories.


spaces-simplicial-lemma-fppf-hypercovering-equivalence-bounded Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. Let $\mathcal{A} \subset \textit{Ab}(U_\etale)$ denote the weak Serre subcategory of cartesian abelian sheaves. If $U$ is an fppf hypercovering of $X$, then the functor $a^{-1}$ defines an equivalence $$ D^+(X_\etale) \longrightarrow D_\mathcal{A}^+(U_\etale) $$ with quasi-inverse $Ra_*$. Here $a : \Sh(U_\etale) \to \Sh(X_\etale)$ is as in Section \ref{section-simplicial-algebraic-spaces}.


derived-lemma-derived-compare-triangles-split-case Let $\mathcal{A}$ be an abelian category. Let $$ \xymatrix{ 0 \ar[r] & A^\bullet \ar[r] & B^\bullet \ar[r] & C^\bullet \ar[r] & 0 } $$ be a short exact sequences of complexes. Assume this short exact sequence is termwise split. Let $(A^\bullet, B^\bullet, C^\bullet, \alpha, \beta, \delta)$ be the distinguished triangle of $K(\mathcal{A})$ associated to the sequence. The $\delta$-functor of Lemma \ref{lemma-derived-canonical-delta-functor} above maps the short exact sequences $0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0$ to a triangle isomorphic to the distinguished triangle $$ (A^\bullet, B^\bullet, C^\bullet, \alpha, \beta, \delta). $$


more-morphisms-lemma-slice-smooth-once-separable-residue-field-extension Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point with image $s \in S$. Assume \begin{enumerate} \item $f$ is smooth at $x$, \item the residue field extension $\kappa(s) \subset \kappa(x)$ is separable, and \item $x$ is not a generic point of $X_s$. \end{enumerate} Then there exists an affine open neighbourhood $U \subset X$ of $x$ and an effective Cartier divisor $D \subset U$ containing $x$ such that $D \to S$ is smooth.


morphisms-lemma-two-immersions-smooth Let $$ \xymatrix{ Z \ar[r]_i \ar[rd]_j & X \ar[d] \\ & Y } $$ be a commutative diagram of schemes where $i$ and $j$ are immersions and $X \to Y$ is smooth. Then the canonical exact sequence $$ 0 \to  \mathcal{C}_{Z/Y} \to \mathcal{C}_{Z/X} \to i^*\Omega_{X/Y} \to 0 $$ of Lemma \ref{lemma-two-immersions} is exact.


artin-lemma-naive-obstruction-theory-qis Let $S$ and $\mathcal{X}$ be as in Definition \ref{definition-naive-obstruction-theory} and let $\mathcal{X}$ be endowed with a naive obstruction theory. Let $A \to B$ and $y \to x$ be as in (\ref{item-functoriality}). Let $k$ be a $B$-algebra which is a field. Then the functoriality map $E_x \to E_y$ induces bijections $$ H^i(E_x \otimes_A^{\mathbf{L}} k) \to H^i(E_y \otimes_B^{\mathbf{L}} k) $$ for $i = 0, 1$.


topologies-lemma-verify-site-Zariski Let $S$ be a scheme. Let $\Sch_{Zar}$ be a big Zariski site containing $S$. Both $S_{Zar}$ and $(\textit{Aff}/S)_{Zar}$ are sites.


spaces-divisors-lemma-weakly-associated-finite Let $S$ be a scheme. Let $f : X \to Y$ be a finite morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $\text{WeakAss}(f_*\mathcal{F}) = f(\text{WeakAss}(\mathcal{F}))$.


algebra-lemma-catenary-Noetherian-local Let $(A, \mathfrak m)$ be a Noetherian local ring. The following are equivalent \begin{enumerate} \item $A$ is catenary, and \item $\mathfrak p \mapsto \dim(A/\mathfrak p)$ is a dimension function on $\Spec(A)$. \end{enumerate}


flat-lemma-blowup-pd1 Let $X$ be a scheme. Let $\mathcal{F}$ be a perfect $\mathcal{O}_X$-module of tor dimension $\leq 1$. Let $U \subset X$ be a scheme theoretically dense open such that $\mathcal{F}|_U$ is finite locally free of constant rank $r$. Then there exists a $U$-admissible blowup $b : X' \to X$ such that there is a canonical short exact sequence $$ 0 \to \mathcal{K} \to b^*\mathcal{F} \to \mathcal{Q} \to 0 $$ where $\mathcal{Q}$ is finite locally free of rank $r$ and $\mathcal{K}$ is a perfect $\mathcal{O}_X$-module of tor dimension $\leq 1$ whose restriction to $U$ is zero.


spaces-resolve-lemma-dominate-by-blowing-up-in-points Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $T \subset |X|$ be a finite set of closed points $x$ such that (1) $X$ is regular at $x$ and (2) the local ring of $X$ at $x$ has dimension $2$. Let $f : Y \to X$ be a proper morphism of algebraic spaces which is an isomorphism over $U = X \setminus T$. Then there exists a sequence $$ X_n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X $$ where $X_{i + 1} \to X_i$ is the blowing up of $X_i$ at a closed point $x_i$ lying above a point of $T$ and a factorization $X_n \to Y \to X$ of the composition.


obsolete-lemma-finite-type-flat-over-integral-algebra Let $A \to B$ be a finite type, flat ring map with $A$ an integral domain. Then $B$ is a finitely presented $A$-algebra.


spaces-more-morphisms-lemma-flat-finite-presentation-perfect Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent \begin{enumerate} \item $f$ is flat and perfect, and \item $f$ is flat and locally of finite presentation. \end{enumerate}


derived-lemma-additive-exact-homotopy-category Let $\mathcal{A}$, $\mathcal{B}$ be additive categories. Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor. The induced functors $$ \begin{matrix} F : K(\mathcal{A}) \longrightarrow K(\mathcal{B}) \\ F : K^{+}(\mathcal{A}) \longrightarrow K^{+}(\mathcal{B}) \\ F : K^{-}(\mathcal{A}) \longrightarrow K^{-}(\mathcal{B}) \\ F : K^b(\mathcal{A}) \longrightarrow K^b(\mathcal{B}) \end{matrix} $$ are exact functors of triangulated categories.


sites-cohomology-lemma-compare-cohomology-general In Situation \ref{situation-compare}. Let $X$ be in $\mathcal{C}$. \begin{enumerate} \item for $\mathcal{F}'$ in $\mathcal{A}'_X$ we have $H^n_{\tau'}(X, \mathcal{F}') = H^n_\tau(X, \epsilon_X^{-1}\mathcal{F}')$, \item for $K' \in D^+_{\mathcal{A}'_X}(\mathcal{C}_{\tau'}/X)$ we have $H^n_{\tau'}(X, K') = H^n_\tau(X, \epsilon_X^{-1}K')$. \end{enumerate}


trace-lemma-count-points-projective Let $X$ be a smooth, projective, geometrically irreducible curve over a finite field $k$. Then \begin{enumerate} \item the $L$-function $L(X, \mathbf{Q}_\ell)$ is a rational function, \item the eigenvalues $\alpha_1, \ldots, \alpha_{2g}$ of $\pi_X^*$ on $H^1(X_{\bar k}, \mathbf{Q}_\ell)$ are algebraic integers independent of $\ell$, \item the number of rational points of $X$ on $k_n$, where $[k_n : k] = n$, is $$ \# X(k_n) = 1 - \sum\nolimits_{i = 1}^{2g}\alpha_i^n + q^n, $$ \item for each $i$, $|\alpha_i| < q$. \end{enumerate}


more-groupoids-lemma-groupoid-on-field-separated Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. If $U$ is the spectrum of a field, then $R$ is a separated scheme.


more-morphisms-lemma-henselian-relatively-perfect Let $(R, I)$ be a pair consisting of a ring and an ideal $I$ contained in the Jacobson radical. Set $S = \Spec(R)$ and $S_0 = \Spec(R/I)$. Let $f : X \to S$ be proper, flat, and of finite presentation. Denote $X_0 = S_0 \times_S X$. Let $E \in D(\mathcal{O}_X)$ be pseudo-coherent. If the derived restriction $E_0$ of $E$ to $X_0$ is $S_0$-perfect, then $E$ is $S$-perfect.


cotangent-lemma-cotangent-morphism-spaces In the situation above there is a canonical isomorphism $$ L_{X/\Lambda} =  L\pi_!(Li^*\Omega_{\mathcal{O}/\underline{\Lambda}}) = L\pi_!(i^*\Omega_{\mathcal{O}/\underline{\Lambda}}) = L\pi_!(\Omega_{\mathcal{O}/\underline{\Lambda}} \otimes_\mathcal{O} \underline{\mathcal{O}}_X) $$ in $D(\mathcal{O}_X)$.


groupoids-quotients-lemma-pre-equivalence-equivalence-relation-points Let $B \to S$ as in Section \ref{section-conventions-notation}. Let $j : R \to U \times_B U$ be a pre-equivalence relation of algebraic spaces over $B$. Then $$ O_u = \{u' \in |U| \text{ such that } \exists r \in |R|, \ s(r) = u, \ t(r) = u'\}. $$


spaces-more-morphisms-lemma-finite-type-differentials Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type, then $\Omega_{X/Y}$ is a finite type $\mathcal{O}_X$-module.


more-algebra-lemma-K-injective-epimorphism Let $R \to S$ be an epimorphism of rings. Let $I^\bullet$ be a complex of $S$-modules. If $I^\bullet$ is K-injective as a complex of $R$-modules, then $I^\bullet$ is a K-injective complex of $S$-modules.


derham-lemma-cup-product-hodge-graded-commutative Let $p : X \to S$ be a morphism of schemes. The cup product on $H^*_{Hodge}(X/S)$ is associative and graded commutative.


etale-theorem-flat-is-quotient A faithfully flat quasi-compact morphism is a quotient map for the Zariski topology.


spaces-morphisms-lemma-characterize-representable-quasi-compact Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$. The following are equivalent: \begin{enumerate} \item $f$ is quasi-compact (in the sense of Section \ref{section-representable}), and \item for every quasi-compact algebraic space $Z$ and any morphism $Z \to Y$ the algebraic space $Z \times_Y X$ is quasi-compact. \end{enumerate}


properties-lemma-dimension-zero \begin{reference} Email from Ofer Gabber dated June 4, 2016 \end{reference} Let $X$ be a scheme of dimension zero. The following are equivalent \begin{enumerate} \item $X$ is quasi-separated, \item $X$ is separated, \item $X$ is Hausdorff, \item every affine open is closed. \end{enumerate} In this case the connected components of $X$ are points.


spaces-simplicial-lemma-constant-simplicial-space Let $X$ be a topological space. Let $X_\bullet$ be the constant simplicial topological space with value $X$. The functor $$ X_{\bullet, Zar} \longrightarrow X_{Zar},\quad U \longmapsto U $$ is continuous and cocontinuous and defines a morphism of topoi $g : \Sh(X_{\bullet, Zar}) \to \Sh(X)$ as well as a left adjoint $g_!$ to $g^{-1}$. We have \begin{enumerate} \item $g^{-1}$ associates to a sheaf on $X$ the constant cosimplicial sheaf on $X$, \item $g_!$ associates to a sheaf $\mathcal{F}$ on $X_{\bullet, Zar}$ the sheaf $\mathcal{F}_0$, and \item $g_*$ associates to a sheaf $\mathcal{F}$ on $X_{\bullet, Zar}$ the equalizer of the two maps $\mathcal{F}_0 \to \mathcal{F}_1$. \end{enumerate}


modules-lemma-whole-tensor-algebra-permanence Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. \begin{enumerate} \item If $\mathcal{F}$ is quasi-coherent, then so is each $\text{T}(\mathcal{F})$, $\wedge(\mathcal{F})$, and $\text{Sym}(\mathcal{F})$. \item If $\mathcal{F}$ is locally free, then so is each $\text{T}(\mathcal{F})$, $\wedge(\mathcal{F})$, and $\text{Sym}(\mathcal{F})$. \end{enumerate}


more-etale-lemma-derived-lower-shriek-bounded Let $f : X \to Y$ be a finite type separated morphism of quasi-compact and quasi-separated schemes. The functor $Rf_!$ is bounded in the following sense: There exists an integer $N$ such that for $E \in D(X_\etale, \Lambda)$ we have \begin{enumerate} \item $H^i(Rf_!(\tau_{\leq a}E) \to H^i(Rf_!(E))$ is an isomorphism for $i \leq a$, \item $H^i(Rf_!(E)) \to H^i(Rf_!(\tau_{\geq b - N + 1}E))$ 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 $H^i(Rf_!(E)) = 0$ for $i \not \in [a, b + N - 1]$. \end{enumerate}


more-morphisms-lemma-descending-property-ind-quasi-affine The property of being ind-quasi-affine is fpqc local on the base.


etale-cohomology-lemma-cohomology-enlarge-partial-universe Let $\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}$. Let $S$ be a scheme. Let $(\Sch/S)_\tau$ and $(\Sch'/S)_\tau$ be two big $\tau$-sites of $S$, and assume that the first is contained in the second. In this case \begin{enumerate} \item for any abelian sheaf $\mathcal{F}'$ defined on $(\Sch'/S)_\tau$ and any object $U$ of $(\Sch/S)_\tau$ we have $$ H^p_\tau(U, \mathcal{F}'|_{(\Sch/S)_\tau}) = H^p_\tau(U, \mathcal{F}') $$ In words: the cohomology of $\mathcal{F}'$ over $U$ computed in the bigger site agrees with the cohomology of $\mathcal{F}'$ restricted to the smaller site over $U$. \item for any abelian sheaf $\mathcal{F}$ on $(\Sch/S)_\tau$ there is an abelian sheaf $\mathcal{F}'$ on $(\Sch/S)_\tau'$ whose restriction to $(\Sch/S)_\tau$ is isomorphic to $\mathcal{F}$. \end{enumerate}


proetale-lemma-j-shriek-limits Let $j : U \to X$ be a quasi-compact open immersion morphism of schemes. The functor $j_! : \textit{Ab}(U_\proetale) \to \textit{Ab}(X_\proetale)$ commutes with limits.


spaces-divisors-lemma-strict-transform-blowup-fitting-ideal-locally-free Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_B$-module. Let $Z_k \subset S$ be the closed subscheme cut out by $\text{Fit}_k(\mathcal{F})$, see Section \ref{section-fitting-ideals}. Assume that $\mathcal{F}$ is locally free of rank $k$ on $B \setminus Z_k$. Let $B' \to B$ be the blowup of $B$ in $Z_k$ and let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$. Then $\mathcal{F}'$ is locally free of rank $k$.


properties-lemma-finite-locally-free-reduced Let $X$ be a reduced scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then the equivalent conditions of Lemma \ref{lemma-finite-locally-free} are also equivalent to \begin{enumerate} \item[(6)] $\mathcal{F}$ is an $\mathcal{O}_X$-module of finite type and the function $$ \rho_\mathcal{F} : X \to \mathbf{Z}, \quad x \longmapsto \dim_{\kappa(x)} \mathcal{F}_x \otimes_{\mathcal{O}_{X, x}} \kappa(x) $$ is locally constant in the Zariski topology on $X$. \end{enumerate}


derived-lemma-triangle-independent-splittings Let $\mathcal{A}$ be an additive category. Let $0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0$ be termwise split exact sequences as in Definition \ref{definition-split-ses}. Let $(\pi')^n$, $(s')^n$ be a second collection of splittings. Denote $\delta' : C^\bullet \longrightarrow A^\bullet[1]$ the morphism associated to this second set of splittings. Then $$ (1, 1, 1) : (A^\bullet, B^\bullet, C^\bullet, \alpha, \beta, \delta) \longrightarrow (A^\bullet, B^\bullet, C^\bullet, \alpha, \beta, \delta') $$ is an isomorphism of triangles in $K(\mathcal{A})$.


spaces-morphisms-lemma-base-change-syntomic The base change of a syntomic morphism is syntomic.


etale-cohomology-lemma-cohomology-jstar-Gm For all $p \geq 1$, $H_\etale^p(X, j_*\mathbf{G}_{m, \eta}) = 0$.


simplicial-lemma-section Let $f : V \to U$ be a morphism of simplicial sets. Let $n \geq 0$ be an integer. Assume \begin{enumerate} \item The map $f_i : V_i \to U_i$ is a bijection for $i < n$. \item The map $f_n : V_n \to U_n$ is a surjection. \item The canonical morphism $U \to \text{cosk}_n \text{sk}_n U$ is an isomorphism. \item The canonical morphism $V \to \text{cosk}_n \text{sk}_n V$ is an isomorphism. \end{enumerate} Then $f$ is a trivial Kan fibration.


more-morphisms-lemma-normal-goes-up Let $f : X \to Y$ be a morphism of schemes. If $Y$ is a normal scheme and $f$ weakly \'etale, then $X$ is a normal scheme.


chow-lemma-application-herbrand-quotient Let $R$ be a Noetherian local ring with maximal ideal $\mathfrak m$. Let $M$ be a finite $R$-module, and let $\psi : M \to M$ be an $R$-module map. Assume that \begin{enumerate} \item $\Ker(\psi)$ and $\Coker(\psi)$ have finite length, and \item $\dim(\text{Supp}(M)) \leq 1$. \end{enumerate} Write $\text{Supp}(M) = \{\mathfrak m, \mathfrak q_1, \ldots, \mathfrak q_t\}$ and denote $f_i \in \kappa(\mathfrak q_i)^*$ the element such that $\det_{\kappa(\mathfrak q_i)}(\psi_{\mathfrak q_i}) : \det_{\kappa(\mathfrak q_i)}(M_{\mathfrak q_i}) \to \det_{\kappa(\mathfrak q_i)}(M_{\mathfrak q_i})$ is multiplication by $f_i$. Then we have $$ \text{length}_R(\Coker(\psi)) - \text{length}_R(\Ker(\psi)) = \sum\nolimits_{i = 1, \ldots, t} \text{ord}_{R/\mathfrak q_i}(f_i). $$


models-lemma-minimal-model-mapping-property Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_C) = K$ and genus $> 0$. Let $X$ be the minimal model for $C$ (Lemma \ref{lemma-minimal-model-unique}). Let $Y$ be a regular proper model for $C$. Then there is a unique morphism of models $Y \to X$ which is a sequence of contractions of exceptional curves of the first kind.


weil-lemma-decompose-P1 In $M_k$ we have $h(\mathbf{P}^1_k) \cong \mathbf{1} \oplus \mathbf{1}(-1)$.


more-morphisms-lemma-base-change-NL-flat Consider a cartesian diagram of schemes $$ \xymatrix{ X' \ar[r]_{g'} \ar[d] & X \ar[d] \\ Y' \ar[r] & Y } $$ If $X \to Y$ is flat, then the canonical map $(g')^*\NL_{X/Y} \to \NL_{X'/Y'}$ is a quasi-isomorphism. If in addition $\NL_{X/Y}$ has tor-amplitude in $[-1, 0]$ then $L(g')^*\NL_{X/Y} \to \NL_{X'/Y'}$ is a quasi-isomorphism too.


obsolete-lemma-separated-fpqc Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\{f_i : T_i \to T\}_{i \in I}$ be a fpqc covering of schemes over $S$. Then the map $$ \Mor_S(T, X) \longrightarrow \prod\nolimits_{i \in I} \Mor_S(T_i, X) $$ is injective.


moduli-lemma-complexes-qs-lfp The morphism $\Complexesstack_{X/B} \to B$ is quasi-separated and locally of finite presentation.


sites-cohomology-lemma-cohomological-descent-general In Situation \ref{situation-compare}. For any $X$ in $\mathcal{C}$ the category $\mathcal{A}_X \subset \textit{Ab}(\mathcal{C}_\tau/X)$ is a weak Serre subcategory and the functor $$ R\epsilon_{X, *} : D^+_{\mathcal{A}_X}(\mathcal{C}_\tau/X) \longrightarrow D^+_{\mathcal{A}'_X}(\mathcal{C}_{\tau'}/X) $$ is an equivalence with quasi-inverse given by $\epsilon_X^{-1}$.


decent-spaces-lemma-stratify-reasonable Let $S$ be a scheme. Let $X$ be a quasi-compact, reasonable algebraic space over $S$. There exist an integer $n$ and open subspaces $$ \emptyset = U_{n + 1} \subset U_n \subset U_{n - 1} \subset \ldots \subset U_1 = X $$ such that each $T_p = U_p \setminus U_{p + 1}$ (with reduced induced subspace structure) is a scheme.


more-morphisms-lemma-lci-NL Let $f : X \to Y$ be a local complete intersection homomorphism. Then the naive cotangent complex $\NL_{X/Y}$ is a perfect object of $D(\mathcal{O}_X)$ of tor-amplitude in $[-1, 0]$.


etale-cohomology-lemma-base-change-f-star Let $f : X \to S$ be a flat morphism of schemes such that for every geometric point $\overline{x}$ of $X$ the map $$ \mathcal{O}_{S, f(\overline{x})}^{sh} \longrightarrow \mathcal{O}_{X, \overline{x}}^{sh} $$ has geometrically connected fibres. Then for every cartesian diagram of schemes $$ \xymatrix{ X \ar[d]_f & Y \ar[l]^h \ar[d]^e \\ S & T \ar[l]_g } $$ with $g$ quasi-compact and quasi-separated we have $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$ for any sheaf $\mathcal{F}$ of sets on $T_\etale$.


spaces-morphisms-lemma-point-finite-type-monomorphism Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. The following are equivalent: \begin{enumerate} \item $x$ is a finite type point, \item there exists an algebraic space $Z$ whose underlying topological space $|Z|$ is a singleton, and a morphism $f : Z \to X$ which is locally of finite type such that $\{x\} = |f|(|Z|)$, and \item there exists an algebraic space $Z$ and a morphism $f : Z \to X$ with the following properties: \begin{enumerate} \item there is a surjective \'etale morphism $z : \Spec(k) \to Z$ where $k$ is a field, \item $f$ is locally of finite type, \item $f$ is a monomorphism, and \item $x = f(z)$. \end{enumerate} \end{enumerate}


restricted-lemma-fully-faithfulness Let $A$ be a Noetherian G-ring. Let $I \subset A$ be an ideal. Let $B, C$ be finite type $A$-algebras. For any $A$-algebra map $\varphi : B^\wedge \to C^\wedge$ of $I$-adic completions and any $N \geq 1$ there exist \begin{enumerate} \item an \'etale ring map $C \to C'$ which induces an isomorphism $C/IC \to C'/IC'$, \item an $A$-algebra map $\varphi : B \to C'$ \end{enumerate} such that $\varphi$ and $\psi$ agree modulo $I^N$ into $C^\wedge = (C')^\wedge$.


obsolete-lemma-bound-primes Let $A$ be a Noetherian local normal domain of dimension $2$. For $f \in \mathfrak m$ nonzero denote $\text{div}(f) = \sum n_i (\mathfrak p_i)$ the divisor associated to $f$ on the punctured spectrum of $A$. We set $|f| = \sum n_i$. There exist integers $N$ and $M$ such that $|f + g| \leq M$ for all $g \in \mathfrak m^N$.


intersection-lemma-rational-equivalence Let $X$ be a variety. Let $W \subset X$ be a subvariety of dimension $k + 1$. Let $f \in \mathbf{C}(W)^*$ be a nonzero rational function on $W$. Then $\text{div}_W(f)$ is rationally equivalent to zero on $X$. Conversely, these principal divisors generate the abelian group of cycles rationally equivalent to zero on $X$.


etale-cohomology-theorem-enough-injectives The category of abelian sheaves on a site is an abelian category which has enough injectives.


moduli-lemma-pic-functor-qs-lfp The morphism $\Picardfunctor_{X/B} \to B$ is quasi-separated and locally of finite presentation.


stacks-morphisms-lemma-composition-gerbe Let $\mathcal{X} \to \mathcal{Y}$ and $\mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks. If $\mathcal{X}$ is a gerbe over $\mathcal{Y}$ and $\mathcal{Y}$ is a gerbe over $\mathcal{Z}$, then $\mathcal{X}$ is a gerbe over $\mathcal{Z}$.


local-cohomology-lemma-torsion-tensor-product Let $A$ be a Noetherian ring. Let $T \subset \Spec(A)$ be a subset stable under specialization. Assume $A$ has finite dimension. Then $$ R\Gamma_T(K) = R\Gamma_T(A) \otimes_A^\mathbf{L} K $$ for $K \in D(A)$. For $K, L \in D(A)$ we have $$ R\Gamma_T(K \otimes_A^\mathbf{L} L) = K \otimes_A^\mathbf{L} R\Gamma_T(L) = R\Gamma_T(K) \otimes_A^\mathbf{L} L = R\Gamma_T(K) \otimes_A^\mathbf{L} R\Gamma_T(L) $$ If $K$ or $L$ is in $D_T(A)$ then so is $K \otimes_A^\mathbf{L} L$.


restricted-lemma-composition-rig-smooth Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of locally Noetherian formal algebraic spaces over $S$. If $f$ and $g$ are rig-smooth, then so is $g \circ f$.


stacks-properties-lemma-open-image-substack Let $\mathcal X$ be an algebraic stack. Let $U$ be an algebraic space and $U \to \mathcal X$ a surjective smooth morphism. For an open immersion $V \hookrightarrow U$, there exists an algebraic stack $\mathcal Y$, an open immersion $\mathcal Y \to \mathcal X$, and a surjective smooth morphism $V \to \mathcal Y$.


quot-lemma-coherent-fibred-in-groupoids In Situation \ref{situation-coherent} the functor $p : \Cohstack_{X/B} \longrightarrow (\Sch/S)_{fppf}$ is fibred in groupoids.


spaces-more-cohomology-lemma-compare-cohomology Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Then \begin{enumerate} \item For $K$ in $D(X_\etale)$ we have $H^n_\etale(X, \pi_X^{-1}K) = H^n(X_\etale, K)$. \item For $K$ in $D(X_\etale, \mathcal{O}_X)$ we have $H^n_\etale(X, L\pi_X^*K) = H^n(X_\etale, K)$. \item For $K$ in $D(X_\etale)$ we have $H^n_\etale(Y, \pi_X^{-1}K) = H^n(Y_\etale, f_{small}^{-1}K)$. \item For $K$ in $D(X_\etale, \mathcal{O}_X)$ we have $H^n_\etale(Y, L\pi_X^*K) = H^n(Y_\etale, Lf_{small}^*K)$. \item For $M$ in $D((\textit{Spaces}/X)_\etale)$ we have $H^n_\etale(Y, M) = H^n(Y_\etale, i_f^{-1}M)$. \item For $M$ in $D((\textit{Spaces}/X)_\etale, \mathcal{O})$ we have $H^n_\etale(Y, M) = H^n(Y_\etale, i_f^*M)$. \end{enumerate}


sites-modules-lemma-adjoint-pull-push-modules Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be a morphism of topoi. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. Let $\mathcal{G}$ be a sheaf of $f_*\mathcal{O}$-modules. Then $$ \Mor_{\textit{Mod}(\mathcal{O})}( \mathcal{O} \otimes_{f^{-1}f_*\mathcal{O}} f^{-1}\mathcal{G}, \mathcal{F}) = \Mor_{\textit{Mod}(f_*\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}). $$ Here we use Lemmas \ref{lemma-pullback-module} and \ref{lemma-pushforward-module}, and we use the canonical map $f^{-1}f_*\mathcal{O} \to \mathcal{O}$ in the definition of the tensor product.


sites-cohomology-lemma-inverse-limit-complexes Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{F}_n^\bullet)$ be an inverse system of complexes of $\mathcal{O}$-modules. Let $m \in \mathbf{Z}$. Suppose given $\mathcal{B} \subset \Ob(\mathcal{C})$ and an integer $n_0$ such that \begin{enumerate} \item every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$, \item for every $U \in \mathcal{B}$ \begin{enumerate} \item the systems of abelian groups $\mathcal{F}_n^{m - 2}(U)$ and $\mathcal{F}_n^{m - 1}(U)$ have vanishing $R^1\lim$ (for example these have the Mittag-Leffler property), \item the system of abelian groups $H^{m - 1}(\mathcal{F}_n^\bullet(U))$ has vanishing $R^1\lim$ (for example it has the Mittag-Leffler property), and \item we have $H^m(\mathcal{F}_n^\bullet(U)) = H^m(\mathcal{F}_{n_0}^\bullet(U))$ for all $n \geq n_0$. \end{enumerate} \end{enumerate} Then the maps $H^m(\mathcal{F}^\bullet) \to \lim H^m(\mathcal{F}_n^\bullet) \to H^m(\mathcal{F}_{n_0}^\bullet)$ are isomorphisms of sheaves where $\mathcal{F}^\bullet = \lim \mathcal{F}_n^\bullet$ is the termwise inverse limit.


moduli-curves-lemma-extend-curves-to-spaces Let $T \to B$ be a morphism of algebraic spaces. The category $$ \Mor_B(T, B\text{-}\Curvesstack) = \Mor(T, \Curvesstack) $$ is the category of families of curves over $T$.


spaces-duality-lemma-dualizing-spaces Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $K$ be a dualizing complex on $X$. Then $K$ is an object of $D_{\textit{Coh}}(\mathcal{O}_X)$ and $D = R\SheafHom_{\mathcal{O}_X}(-, K)$ induces an anti-equivalence $$ D : D_{\textit{Coh}}(\mathcal{O}_X) \longrightarrow D_{\textit{Coh}}(\mathcal{O}_X) $$ which comes equipped with a canonical isomorphism $\text{id} \to D \circ D$. If $X$ is quasi-compact, then $D$ exchanges $D^+_{\textit{Coh}}(\mathcal{O}_X)$ and $D^-_{\textit{Coh}}(\mathcal{O}_X)$ and induces an equivalence $D^b_{\textit{Coh}}(\mathcal{O}_X) \to D^b_{\textit{Coh}}(\mathcal{O}_X)$.


restricted-lemma-formal-modifications-locally-algebraic Let $S$ be a scheme. Let $\mathfrak X' \to \mathfrak X$ be a formal modification (Definition \ref{definition-formal-modification}) of locally Noetherian formal algebraic spaces over $S$. Given \begin{enumerate} \item any adic Noetherian topological ring $A$, \item any adic morphism $\text{Spf}(A) \longrightarrow \mathfrak X$ \end{enumerate} there exists a proper morphism $X \to \Spec(A)$ of algebraic spaces and an isomorphism $$ \text{Spf}(A) \times_{\mathfrak X} \mathfrak X' \longrightarrow X_{/Z} $$ over $\text{Spf}(A)$ of the base change of $\mathfrak X$ with the formal completion of $X$ along the ``closed fibre'' $Z = X \times_{\Spec(A)} \text{Spf}(A)_{red}$ of $X$ over $A$.


topology-lemma-topological-ring-limits The category of topological rings has limits and limits commute with the forgetful functors to (a) the category of topological spaces and (b) the category of rings.


dualizing-lemma-CM Let $A$ be a Noetherian ring. If $A$ has a dualizing complex $\omega_A^\bullet$, then $\{\mathfrak p \in \Spec(A) \mid A_\mathfrak p\text{ is Cohen-Macaulay}\}$ is a dense open subset of $\Spec(A)$.


more-morphisms-lemma-quasi-finite-Noetherian-universally-open Let $f : X \to Y$ be a morphism of schemes. If \begin{enumerate} \item $f$ is locally quasi-finite, \item $Y$ is unibranch and locally Noetherian, and \item every irreducible component of $X$ dominates an irreducible component of $Y$, \end{enumerate} then $f$ is universally open.


more-morphisms-lemma-weakly-etale-formally-unramified Let $f : X \to Y$ be a weakly \'etale morphism of schemes. Then $f$ is formally unramified, i.e., $\Omega_{X/Y} = 0$.


obsolete-lemma-make-integral-not-in-ideal Let $\varphi : R \to S$ be a ring map. Suppose $t \in S$ satisfies the relation $\varphi(a_0) + \varphi(a_1)t + \ldots + \varphi(a_n) t^n = 0$. Let $J \subset S$ be an ideal such that for at least one $i$ we have $\varphi(a_i) \not \in J$. Then there exists a $u \in S$, $u \not\in J$ such that both $u$ and $ut$ are integral over $R$.


pione-lemma-extend-covering-general Let $S$ be a quasi-compact and quasi-separated integral normal scheme with generic point $\eta$. Let $f : X \to S$ be a quasi-compact and quasi-separated smooth morphism with geometrically connected fibres. Let $\sigma : S \to X$ be a section of $f$. Let $Z \to X_\eta$ be a finite \'etale Galois cover (Section \ref{section-finite-etale-under-galois}) with group $G$ of order invertible on $S$ such that $Z$ has a $\kappa(\eta)$-rational point mapping to $\sigma(\eta)$. Then there exists a finite \'etale Galois cover $Y \to X$ with group $G$ whose restriction to $X_\eta$ is $Z$.


groupoids-quotients-lemma-orbit-space-locally-finite-type-over-base Let $B \to S$ as in Section \ref{section-conventions-notation}. Let $j = (t, s) : R \to U \times_B U$ be a pre-relation. Assume $R, U$ are locally of finite type over $B$. Let $\phi : U \to X$ be an $R$-invariant morphism of algebraic spaces over $B$. Then $\phi$ is an orbit space for $R$ if and only if the natural map $$ U(k)/\big(\text{equivalence relation generated by }j(R(k))\big) \longrightarrow X(k) $$ is bijective for all algebraically closed fields $k$ over $B$.


sites-lemma-localize-cocontinuous-downstairs Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a cocontinuous functor. Let $V$ be an object of $\mathcal{D}$. Let ${}^u_V\mathcal{I}$ be the category introduced in Section \ref{section-more-functoriality-PSh}. We have a commutative diagram $$ \vcenter{ \xymatrix{ \,_V^u\mathcal{I} \ar[r]_j \ar[d]_{u'} & \mathcal{C} \ar[d]^u \\ \mathcal{D}/V \ar[r]^-{j_V} & \mathcal{D} } } \quad\text{where}\quad \begin{matrix} j : (U, \psi) \mapsto U \\ u' : (U, \psi) \mapsto (\psi : u(U) \to V) \end{matrix} $$ Declare a family of morphisms $\{(U_i, \psi_i) \to (U, \psi)\}$ of ${}^u_V\mathcal{I}$ to be a covering if and only if $\{U_i \to U\}$ is a covering in $\mathcal{C}$. Then \begin{enumerate} \item ${}^u_V\mathcal{I}$ is a site, \item $j$ is continuous and cocontinuous, \item $u'$ is cocontinuous, \item we get a commutative diagram of topoi $$ \xymatrix{ \Sh({}^u_V\mathcal{I}) \ar[r]_j \ar[d]_{f'} & \Sh(\mathcal{C}) \ar[d]^f \\ \Sh(\mathcal{D}/V) \ar[r]^-{j_V} & \Sh(\mathcal{D}) } $$ where $f$ (resp.\ $f'$) corresponds to $u$ (resp.\ $u'$), and \item we have $f'_*j^{-1} = j_V^{-1}f_*$. \end{enumerate}


examples-lemma-perfect-closure-polynomial-ring Let $A = \mathbb{F}_p[T]$ be the polynomial ring in one variable over $\mathbb{F}_p$. Let $A_{perf}$  denote the perfect closure of $A$. Then $A \rightarrow A_{perf}$ is flat and formally unramified, but not formally \'etale.


etale-cohomology-lemma-cohomology-istar-Z For all $q \geq 1$, $H_\etale^q(X, \bigoplus_{x \in X_0} {i_x}_* \underline{\mathbf{Z}}) = 0$.


spaces-more-morphisms-lemma-check-universally-injective Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume \begin{enumerate} \item $f$ is locally of finite type, \item for every \'etale morphism $V \to Y$ the map $|X \times_Y V| \to |V|$ is injective. \end{enumerate} Then $f$ is universally injective.


algebra-lemma-smooth-independent-presentation Let $R \to S$ be a ring map of finite presentation. If for some presentation $\alpha$ of $S$ over $R$ the naive cotangent complex $\NL(\alpha)$ is quasi-isomorphic to a finite projective $S$-module placed in degree $0$, then this holds for any presentation.


perfect-lemma-finite-push-pseudo-coherent Let $f : X \to Y$ be a finite morphism of schemes such that $f_*\mathcal{O}_X$ is pseudo-coherent as an $\mathcal{O}_Y$-module\footnote{This means that $f$ is pseudo-coherent, see More on Morphisms, Lemma \ref{more-morphisms-lemma-finite-pseudo-coherent}.}. Let $E \in D_\QCoh(\mathcal{O}_X)$. Then $E$ is $m$-pseudo-coherent if and only if $Rf_*E$ is $m$-pseudo-coherent.


sites-modules-lemma-skyscraper-modules-exact Let $(\Sh(\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $p$ be a point of the topos $\Sh(\mathcal{C})$. \begin{enumerate} \item The functor $p_* : \textit{Mod}(\mathcal{O}_p) \to \textit{Mod}(\mathcal{O})$, $M \mapsto p_*M$ is exact. \item The canonical surjection $p^{-1}p_*M \to M$ is $\mathcal{O}_p$-linear. \item The functorial direct sum decomposition $p^{-1}p_*M = M \oplus I(M)$ of Lemma \ref{lemma-skyscraper-exact} is {\bf not} $\mathcal{O}_p$-linear in general. \end{enumerate}


spaces-descent-lemma-descending-fppf-property-locally-separated The property $\mathcal{P}(f) =$``$f$ is locally separated'' is fppf local on the base.


defos-lemma-inf-obs-map-rel-ringed-topoi Let $(f, f')$ be a morphism of first order thickenings as in Situation \ref{situation-morphism-thickenings-ringed-topoi}. Let $\mathcal{F}'$, $\mathcal{G}'$ be $\mathcal{O}'$-modules and set $\mathcal{F} = i^*\mathcal{F}'$ and $\mathcal{G} = i^*\mathcal{G}'$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be an $\mathcal{O}$-linear map. Assume that $\mathcal{F}'$ and $\mathcal{G}'$ are flat over $\mathcal{O}_{\mathcal{B}'}$ and that $(f, f')$ is a strict morphism of thickenings. There exists an element $$ o(\varphi) \in \Ext^1_\mathcal{O}(\mathcal{F}, \mathcal{G} \otimes_\mathcal{O} f^*\mathcal{J}) $$ whose vanishing is a necessary and sufficient condition for the existence of a lift of $\varphi$ to an $\mathcal{O}'$-linear map $\varphi' : \mathcal{F}' \to \mathcal{G}'$.


spaces-over-fields-lemma-numerical-intersection-effective-Cartier-divisor Let $k$ be a field. Let $X$ be a proper algebraic space over $k$. Let $Z \subset X$ be a closed subspace of dimension $d$. Let $\mathcal{L}_1, \ldots, \mathcal{L}_d$ be invertible $\mathcal{O}_X$-modules. Assume there exists an effective Cartier divisor $D \subset Z$ such that $\mathcal{L}_1|_Z \cong \mathcal{O}_Z(D)$. Then $$ (\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z) = (\mathcal{L}_2 \cdots \mathcal{L}_d \cdot D) $$


more-morphisms-lemma-smooth-etale-permanence Let $X \to Y \to Z$ be morphisms of schemes. Assume $X \to Z$ smooth and $Y \to Z$ \'etale. Then $X \to Y$ is smooth.


stacks-geometry-lemma-catenary-covers If $\mathcal{X}$ is a pseudo-catenary locally Noetherian algebraic stack, and if $\mathcal{Y} \to \mathcal{X}$ is a locally of finite type morphism, then there exists a smooth surjective morphism $V \to \mathcal{Y}$ whose source is a universally catenary scheme; thus $\mathcal{Y}$ is again pseudo-catenary.


artin-lemma-naive-obstruction-theory-gives-openness Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\Sch/S)_{fppf}^{opp}$ be a category fibred in groupoids. Assume that $\mathcal{X}$ satisfies (RS*) and that $\mathcal{X}$ has a naive obstruction theory. Then openness of versality holds for $\mathcal{X}$ provided the complexes $E_x$ of Definition \ref{definition-naive-obstruction-theory} have finitely generated cohomology groups for pairs $(A, x)$ where $A$ is of finite type over $S$.


topologies-lemma-composition-fppf Given schemes $X$, $Y$, $Y$ in $(\Sch/S)_{fppf}$ and morphisms $f : X \to Y$, $g : Y \to Z$ we have $g_{big} \circ f_{big} = (g \circ f)_{big}$.


etale-cohomology-lemma-compare-cohomology Let $f : T \to S$ be a morphism of schemes. Then \begin{enumerate} \item For $K$ in $D(S_\etale)$ we have $H^n_\etale(S, \pi_S^{-1}K) = H^n(S_\etale, K)$. \item For $K$ in $D(S_\etale, \mathcal{O}_S)$ we have $H^n_\etale(S, L\pi_S^*K) = H^n(S_\etale, K)$. \item For $K$ in $D(S_\etale)$ we have $H^n_\etale(T, \pi_S^{-1}K) = H^n(T_\etale, f_{small}^{-1}K)$. \item For $K$ in $D(S_\etale, \mathcal{O}_S)$ we have $H^n_\etale(T, L\pi_S^*K) = H^n(T_\etale, Lf_{small}^*K)$. \item For $M$ in $D((\Sch/S)_\etale)$ we have $H^n_\etale(T, M) = H^n(T_\etale, i_f^{-1}M)$. \item For $M$ in $D((\Sch/S)_\etale, \mathcal{O})$ we have $H^n_\etale(T, M) = H^n(T_\etale, i_f^*M)$. \end{enumerate}


more-morphisms-lemma-nr-branches-fibre Let $X \to S$ be a morphism of schemes and $x \in X$ a point with image $s$. Then \begin{enumerate} \item the number of branches of the fibre $X_s$ at $x$ is equal to the supremum of the number of irreducible components of the fibre $U_s$ passing through $u$ taken over elementary \'etale neighbourhoods $(U, u) \to (X, x)$, \item the number of geometric branches of the fibre $X_s$ at $x$ is equal to the supremum of the number of irreducible components of the fibre $U_s$ passing through $u$ taken over \'etale neighbourhoods $(U, u) \to (X, x)$, \item the fibre $X_s$ is unibranch at $x$ if and only if for every elementary \'etale neighbourhood $(U, u) \to (X, x)$ there is exactly one irreducible component of the fibre $U_s$ passing through $u$, and \item $X$ is geometrically unibranch at $x$ if and only if for every \'etale neighbourhood $(U, u) \to (X, x)$ there is exactly one irreducible component of $U_s$ passing through $u$. \end{enumerate}


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


spaces-resolve-lemma-iso-completions Let $(A, \mathfrak m, \kappa)$ be a local ring with finitely generated maximal ideal $\mathfrak m$. Let $X$ be a decent algebraic space over $A$. Let $Y = X \times_{\Spec(A)} \Spec(A^\wedge)$ where $A^\wedge$ is the $\mathfrak m$-adic completion of $A$. For a point $q \in |Y|$ with image $p \in |X|$ lying over the closed point of $\Spec(A)$ the map $\mathcal{O}_{X, p}^h \to \mathcal{O}_{Y, q}^h$ of henselian local rings induces an isomorphism on completions.


adequate-lemma-pure-injective-resolutions Let $A$ be a ring. \begin{enumerate} \item Any $A$-module has a pure injective resolution. \end{enumerate} Let $M \to N$ be a map of $A$-modules. Let $M \to M^\bullet$ be a universally exact resolution and let $N \to I^\bullet$ be a pure injective resolution. \begin{enumerate} \item[(2)] There exists a map of complexes $M^\bullet \to I^\bullet$ inducing the given map $$ M = \Ker(M^0 \to M^1) \to \Ker(I^0 \to I^1) = N $$ \item[(3)] two maps $\alpha, \beta : M^\bullet \to I^\bullet$ inducing the same map $M \to N$ are homotopic. \end{enumerate}


etale-cohomology-lemma-pullback-ctf Let $\Lambda$ be a Noetherian ring. Let $f : X \to Y$ be a morphism of schemes. If $K \in D_{ctf}(Y_\etale, \Lambda)$ then $Lf^*K \in D_{ctf}(X_\etale, \Lambda)$.


spaces-flat-lemma-flat In Situation \ref{situation-flat}. \begin{enumerate} \item The functor $F_{flat}$ satisfies the sheaf property for the fpqc topology. \item If $f$ is quasi-compact and locally of finite presentation and $\mathcal{F}$ is of finite presentation, then the functor $F_{flat}$ is limit preserving. \end{enumerate}


algebraization-lemma-well-defined-chi-triple The quantity $\chi(\mathcal{F}, \mathcal{F}_0, \alpha)$ in (\ref{equation-chi-triple}) does not depend on the choice of $\mathcal{F}', \alpha', \alpha'_0$ as in Lemma \ref{lemma-prepare-chi-triple}.


groupoids-lemma-immersion-group-schemes-closed-over-field Let $i : G' \to G$ be an immersion of group schemes over a field $k$. Then $i$ is a closed immersion, i.e., $i(G')$ is a closed subgroup scheme of $G$.


etale-cohomology-lemma-describe-h1-mun Let $S$ be a scheme. There is a canonical identification $$ H_\etale^1(S, \mu_n) = \text{group of pairs }(\mathcal{L}, \alpha)\text{ up to isomorphism as above} $$ if $n$ is invertible on $S$. In general we have $$ H_{fppf}^1(S, \mu_n) = \text{group of pairs }(\mathcal{L}, \alpha)\text{ up to isomorphism as above}. $$ The same result holds with fppf replaced by syntomic.


etale-cohomology-lemma-exactness-lower-shriek Let $\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}$. Let $f : X \to Y$ be a morphism of schemes. Let $$ f_{big} : \Sh((\Sch/X)_\tau) \longrightarrow \Sh((\Sch/Y)_\tau) $$ be the corresponding morphism of topoi as in Topologies, Lemma \ref{topologies-lemma-morphism-big}, \ref{topologies-lemma-morphism-big-etale}, \ref{topologies-lemma-morphism-big-smooth}, \ref{topologies-lemma-morphism-big-syntomic}, or \ref{topologies-lemma-morphism-big-fppf}. \begin{enumerate} \item The functor $f_{big}^{-1} : \textit{Ab}((\Sch/Y)_\tau) \to \textit{Ab}((\Sch/X)_\tau)$ has a left adjoint $$ f_{big!} : \textit{Ab}((\Sch/X)_\tau) \to \textit{Ab}((\Sch/Y)_\tau) $$ which is exact. \item The functor $f_{big}^* : \textit{Mod}((\Sch/Y)_\tau, \mathcal{O}) \to \textit{Mod}((\Sch/X)_\tau, \mathcal{O})$ has a left adjoint $$ f_{big!} : \textit{Mod}((\Sch/X)_\tau, \mathcal{O}) \to \textit{Mod}((\Sch/Y)_\tau, \mathcal{O}) $$ which is exact. \end{enumerate} Moreover, the two functors $f_{big!}$ agree on underlying sheaves of abelian groups.


spaces-divisors-lemma-base-change-relative-assassin-quasi-finite With notation and assumptions as in Lemma \ref{lemma-base-change-relative-assassin}. Assume $g$ is locally quasi-finite, or more generally that for every $y' \in |Y'|$ the transcendence degree of $y'/g(y')$ is $0$. Then $\text{Ass}_{X'/Y'}(\mathcal{F}')$ is the inverse image of $\text{Ass}_{X/Y}(\mathcal{F})$.


categories-lemma-spell-out-representable-map-stack-in-groupoids Let $\mathcal{C}$ be a category. Let $\mathcal{X}$, $\mathcal{Y}$ be categories fibred in groupoids over $\mathcal{C}$. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism. If $F$ is representable then every one of the functors $$ F_U : \mathcal{X}_U \longrightarrow \mathcal{Y}_U $$ between fibre categories is faithful.


sheaves-lemma-adjoint-pull-push-presheaves-modules Let $f : X \to Y$ be a continuous map of topological spaces. Let $\mathcal{O}$ be a presheaf of rings on $X$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}$-modules. Let $\mathcal{G}$ be a presheaf of $f_*\mathcal{O}$-modules. Then $$ \Mor_{\textit{PMod}(\mathcal{O})}( \mathcal{O} \otimes_{p, f_pf_*\mathcal{O}} f_p\mathcal{G}, \mathcal{F}) = \Mor_{\textit{PMod}(f_*\mathcal{O})}(\mathcal{G}, f_*\mathcal{F}). $$ Here we use Lemmas \ref{lemma-pullback-presheaf-module} and \ref{lemma-pushforward-presheaf-module}, and we use the map $c_\mathcal{O} : f_pf_*\mathcal{O} \to \mathcal{O}$ in the definition of the tensor product.


topologies-lemma-composition-etale Given schemes $X$, $Y$, $Y$ in $\Sch_\etale$ and morphisms $f : X \to Y$, $g : Y \to Z$ we have $g_{big} \circ f_{big} = (g \circ f)_{big}$ and $g_{small} \circ f_{small} = (g \circ f)_{small}$.


spaces-resolve-lemma-equivalence-properties Let $X, x_i, U_i \to X, u_i$ be as in (\ref{equation-equivalence}). If $f : Y \to X$ corresponds to $g_i : Y_i \to U_i$ under $F$, then $f$ is quasi-compact, quasi-separated, separated, locally of finite presentation, of finite presentation, locally of finite type, of finite type, proper, integral, finite, if and only if $g_i$ is so for $i = 1, \ldots, n$.


modules-lemma-determinant-as-socle Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a flat and finitely presented $\mathcal{O}_X$-module. Denote $$ \det(\mathcal{F}) \subset \wedge^*_{\mathcal{O}_X}(\mathcal{F}) $$ the annihilator of $\mathcal{F} \subset \wedge^*_{\mathcal{O}_X}(\mathcal{F})$. Then $\det(\mathcal{F})$ is an invertible $\mathcal{O}_X$-module.


derham-lemma-multiplication-log Let $p : X \to S$ be a morphism of schemes. Let $Y \subset X$ be an effective Cartier divisor. Assume the de Rham complex of log poles is defined for $Y \subset X$ over $S$. \begin{enumerate} \item The maps $\wedge : \Omega^p_{X/S} \times \Omega^q_{X/S} \to \Omega^{p + q}_{X/S}$ extend uniquely to $\mathcal{O}_X$-bilinear maps $$ \wedge : \Omega^p_{X/S}(\log Y) \times \Omega^q_{X/S}(\log Y) \to \Omega^{p + q}_{X/S}(\log Y) $$ satisfying the Leibniz rule $ \text{d}(\omega \wedge \eta) = \text{d}(\omega) \wedge \eta + (-1)^{\deg(\omega)} \omega \wedge \text{d}(\eta)$, \item with multiplication as in (1) the map $\Omega^\bullet_{X/S} \to \Omega^\bullet_{X/S}(\log(Y)$ is a homomorphism of differential graded $\mathcal{O}_S$-algebras, \item via the maps in (1) we have $\Omega^p_{X/S}(\log Y) = \wedge^p(\Omega^1_{X/S}(\log Y))$, and \item the map $\text{Res} : \Omega^\bullet_{X/S}(\log Y) \to \Omega^\bullet_{Y/S}[-1]$ satisfies $$ \text{Res}(\omega \wedge \eta) = \text{Res}(\omega) \wedge \eta|_Y + (-1)^{\deg(\omega)} \omega|_Y \wedge \text{Res}(\eta) $$ \end{enumerate}


more-morphisms-lemma-relative-finite-presentation-characterize 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. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is of finite presentation relative to $S$, \item for every affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ the $\mathcal{O}_X(U)$-module $\mathcal{F}(U)$ is finitely presented relative to $\mathcal{O}_S(V)$. \end{enumerate} Moreover, if this is true, then for every open subschemes $U \subset X$ and $V \subset S$ with $f(U) \subset V$ the restriction $\mathcal{F}|_U$ is of finite presentation relative to $V$.


divisors-lemma-zero-fitting-ideal-omega-unramified Let $f : X \to S$ be a morphism of schemes which is locally of finite type. The closed subscheme $Z \subset X$ cut out by the $0$th fitting ideal of $\Omega_{X/S}$ is exactly the set of points where $f$ is not unramified.


etale-cohomology-lemma-restrict-and-shriek-from-etale-c Let $\Lambda$ be a Noetherian ring. If $j : U \to X$ is an \'etale morphism of schemes, then \begin{enumerate} \item $K|_U \in D_c(U_\etale, \Lambda)$ if $K \in D_c(X_\etale, \Lambda)$, and \item $j_!M \in D_c(X_\etale, \Lambda)$ if $M \in D_c(U_\etale, \Lambda)$ and the morphism $j$ is quasi-compact and quasi-separated. \end{enumerate}


more-algebra-lemma-characterize-injective Let $R$ be a ring. Let $J$ be an $R$-module. The following are equivalent \begin{enumerate} \item $J$ is injective, \item $\Ext^1_R(M, J) = 0$ for every $R$-module $M$. \end{enumerate}


flat-lemma-finite-type-flat-along-fibre-variant Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $s \in S$. Assume that \begin{enumerate} \item $f$ is of finite type, \item $\mathcal{F}$ is of finite type, and \item $\mathcal{F}$ is flat over $S$ at every point of the fibre $X_s$. \end{enumerate} 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 fibre $X_s = X \times_S s'$ such that the pullback of $\mathcal{F}$ to $V$ is flat over $\mathcal{O}_{S', s'}$.


equiv-lemma-trace-map \begin{reference} The proof given here follows the argument given in \cite[Remark 3.4]{MS} \end{reference} Let $f : X' \to X$ be a proper birational morphism of integral Noetherian schemes with $X$ regular. The map $\mathcal{O}_X \to Rf_*\mathcal{O}_{X'}$ canonically splits in $D(\mathcal{O}_X)$.


flat-lemma-Noetherian-h-covering Let $X$ be a Noetherian scheme. Let $\{X_i \to X\}_{i \in I}$ be a finite family of finite type morphisms. The following are equivalent \begin{enumerate} \item $\coprod_{i \in I} X_i \to X$ is universally submersive (Morphisms, Definition \ref{morphisms-definition-submersive}), and \item $\{X_i \to X\}_{i \in I}$ is an h covering. \end{enumerate}


varieties-lemma-good-intersection Let $A$ be a Noetherian local ring of dimension $1$. Let $L = \prod A_\mathfrak p$ where the product is over the minimal primes of $A$. Let $K \to L$ be an integral ring map. Then there exist $a \in \mathfrak m_A$ and $x \in K$ which map to the same element of $L$ such that $\mathfrak m_A = \sqrt{(a)}$.


constructions-lemma-structure-morphism-proj Let $S$ be a graded ring. The scheme $\text{Proj}(S)$ has a canonical morphism towards the affine scheme $\Spec(S_0)$, agreeing with the map on topological spaces coming from Algebra, Definition \ref{algebra-definition-proj}.


obsolete-lemma-good-blowing-up Let $b : X' \to X$ be the blowing up of a smooth projective scheme over a field $k$ in a smooth closed subscheme $Z \subset X$. Picture $$ \xymatrix{ E \ar[r]_j \ar[d]_\pi & X' \ar[d]^b \\ Z \ar[r]^i & X } $$ Assume there exists an element of $K_0(X)$ whose restriction to $Z$ is equal to the class of $\mathcal{C}_{Z/X}$ in $K_0(Z)$. Then $[Lb^*\mathcal{O}_Z] = [\mathcal{O}_E] \cdot \alpha''$ in $K_0(X')$ for some $\alpha'' \in K_0(X')$.


chow-lemma-flat-pullback-divisor-invertible-sheaf Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $X$, $Y$ be locally of finite type over $S$. Assume $X$, $Y$ are integral and $n = \dim_\delta(Y)$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_Y$-module. Let $f : X \to Y$ be a flat morphism of relative dimension $r$. Let $\mathcal{L}$ be an invertible sheaf on $Y$. Then $$ f^*(c_1(\mathcal{L}) \cap [Y]) = c_1(f^*\mathcal{L}) \cap [X] $$ in $\CH_{n + r - 1}(X)$.


etale-theorem-flat-map-open Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be a morphism which is flat and locally of finite type. Then $f$ is (universally) open.


chow-lemma-lci-gysin-well-defined Let $(S, \delta)$ be as in Situation \ref{situation-setup}. Let $f : X \to Y$ be a local complete intersection morphism of schemes locally of finite type over $S$. The bivariant class $f^!$ is independent of the choice of the factorization $f = g \circ i$ with $g$ smooth (provided one exists).


obsolete-lemma-tangent-and-inf We have the following canonical $k$-vector space identifications: \begin{enumerate} \item In Deformation Problems, Example \ref{examples-defos-example-finite-projective-modules} if $x_0 = (k, V)$, then $T_{x_0}\mathcal{F} = (0)$ and $\text{Inf}_{x_0}(\mathcal{F}) = \text{End}_k(V)$ are finite dimensional. \item In Deformation Problems, Example \ref{examples-defos-example-representations} if $x_0 = (k, V, \rho_0)$, then $T_{x_0}\mathcal{F} = \Ext^1_{k[\Gamma]}(V, V) = H^1(\Gamma, \text{End}_k(V))$ and $\text{Inf}_{x_0}(\mathcal{F}) = H^0(\Gamma, \text{End}_k(V))$ are finite dimensional if $\Gamma$ is finitely generated. \item In Deformation Problems, Example \ref{examples-defos-example-continuous-representations} if $x_0 = (k, V, \rho_0)$, then $T_{x_0}\mathcal{F} = H^1_{cont}(\Gamma, \text{End}_k(V))$ and $\text{Inf}_{x_0}(\mathcal{F}) = H^0_{cont}(\Gamma, \text{End}_k(V))$ are finite dimensional if $\Gamma$ is topologically finitely generated. \item In Deformation Problems, Example \ref{examples-defos-example-graded-algebras} if $x_0 = (k, P)$, then $T_{x_0}\mathcal{F}$ and $\text{Inf}_{x_0}(\mathcal{F}) = \text{Der}_k(P, P)$ are finite dimensional if $P$ is finitely generated over $k$. \end{enumerate}


stacks-morphisms-lemma-quasi-finite-permanence Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks. If $g \circ f$ is quasi-finite and $g$ is quasi-separated and quasi-DM then $f$ is quasi-finite.


more-morphisms-lemma-base-change-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. Let $S' \to S$ be a morphism of schemes, set $X' = X \times_S S'$ and denote $\mathcal{F}'$ the pullback of $\mathcal{F}$ to $X'$. If $\mathcal{F}$ is of finite presentation relative to $S$, then $\mathcal{F}'$ is of finite presentation relative to $S'$.


spaces-morphisms-proposition-generic-flatness-reduced-quasi-separated 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 sheaf of $\mathcal{O}_X$-modules. Assume \begin{enumerate} \item $Y$ is reduced, \item $f$ is quasi-separated, \item $f$ is of finite type, and \item $\mathcal{F}$ is a finite type $\mathcal{O}_X$-module. \end{enumerate} Then there exists an open dense subspace $W \subset Y$ such that the base change $X_W \to W$ of $f$ is flat and of finite presentation and such that $\mathcal{F}|_{X_W}$ is flat over $W$ and of finite presentation over $\mathcal{O}_{X_W}$.


algebra-lemma-ML-also Let $R$ be a ring and $M$ an $R$-module. Then $M$ is Mittag-Leffler if and only if for every finite free $R$-module $F$ and module map $f: F \to M$, there exists a finitely presented $R$-module $Q$ and a module map $g : F \to Q$ such that $g$ and $f$ dominate each other, i.e., $\Ker(f \otimes_R \text{id}_N) = \Ker(g \otimes_R \text{id}_N)$ for every $R$-module $N$.


more-algebra-lemma-perfect-push-perfect Let $A \to B$ be a ring map. Assume that $B$ is perfect as an $A$-module. Let $K^\bullet$ be a perfect complex of $B$-modules. Then $K^\bullet$ is perfect as a complex of $A$-modules.


flat-lemma-explain-why-pure-ML Let $R$ be a henselian local ring with maximal ideal $\mathfrak m$. Let $R \to S$ be a ring map. Let $N$ be an $S$-module. Assume $N$ is countably generated and Mittag-Leffler as an $R$-module. Then for any $R$-module $M$ and for any prime $\mathfrak q \subset S$ which is an associated prime of $N \otimes_R M$ we have $\mathfrak q + \mathfrak m S \not = S$.


pic-lemma-pic-descends Let $k$ be a field. Let $X$ be a quasi-compact and quasi-separated scheme over $k$ with $H^0(X, \mathcal{O}_X) = k$. If $X$ has a $k$-rational point, then for any Galois extension $k'/k$ we have $$ \Pic(X) = \Pic(X_{k'})^{\text{Gal}(k'/k)} $$ Moreover the action of $\text{Gal}(k'/k)$ on $\Pic(X_{k'})$ is continuous.


stacks-morphisms-lemma-composition-integral Compositions of integral, resp.\ finite morphisms of algebraic stacks are integral, resp.\ finite.


sites-proposition-sheafification-adjoint-topology Let $\mathcal{C}$ be a category endowed with a topology. Let $\mathcal{F}$ be a presheaf of sets on $\mathcal{C}$. The canonical map $\mathcal{F} \to \mathcal{F}^\#$ has the following universal property: For any map $\mathcal{F} \to \mathcal{G}$, where $\mathcal{G}$ is a sheaf of sets, there is a unique map $\mathcal{F}^\# \to \mathcal{G}$ such that $\mathcal{F} \to \mathcal{F}^\# \to \mathcal{G}$ equals the given map.


stacks-lemma-stack-in-groupoids-pushforward Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous functor of sites. Let $p : \mathcal{S} \to \mathcal{D}$ be a stack in groupoids over $\mathcal{D}$. Then $u^p\mathcal{S}$ is a stack in groupoids over $\mathcal{C}$.


more-algebra-lemma-derived-complete-henselian Let $A$ be a ring derived complete with respect to an ideal $I$. Then $(A, I)$ is a henselian pair.


more-groupoids-lemma-groupoid-on-field-open-multiplication Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. If $U$ is the spectrum of a field, then the composition morphism $c : R \times_{s, U, t} R \to R$ is open.


stacks-cohomology-lemma-relative-leray Let $f : \mathcal{X} \to \mathcal{Y}$ and $g : \mathcal{Y} \to \mathcal{Z}$ be quasi-compact and quasi-separated morphisms of algebraic stacks. Let $\mathcal{F}$ be a quasi-coherent sheaf on $\mathcal{X}$. Then there exists a spectral sequence with $E_2$-page $$ E_2^{p, q} = R^pg_{\QCoh, *}(R^qf_{\QCoh, *}\mathcal{F}) $$ converging to $R^{p + q}(g \circ f)_{\QCoh, *}\mathcal{F}$.


divisors-lemma-relative-weak-assassin-assassin-finite-type 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. Then $\text{WeakAss}_{X/S}(\mathcal{F}) = \text{Ass}_{X/S}(\mathcal{F})$.


examples-defos-lemma-local-ring In Example \ref{example-schemes} let $X$ be a scheme over $k$ Let $p \in X$ be a point. With $\Deformationcategory_{\mathcal{O}_{X, p}}$ as in Example \ref{example-rings} there is a natural functor $$ \Deformationcategory_X \longrightarrow \Deformationcategory_{\mathcal{O}_{X, p}} $$ of deformation categories.


sites-cohomology-lemma-cech-cohomology Let $\mathcal{C}$ be a site. Let $\mathcal{U} = \{U_i \to U\}_{i \in I}$ be a covering of $\mathcal{C}$. There is a transformation $$ \check{\mathcal{C}}^\bullet(\mathcal{U}, -) \longrightarrow R\Gamma(U, -) $$ of functors $\textit{Ab}(\mathcal{C}) \to D^{+}(\mathbf{Z})$. In particular this gives a transformation of functors $\check{H}^p(U, \mathcal{F}) \to H^p(U, \mathcal{F})$ for $\mathcal{F}$ ranging over $\textit{Ab}(\mathcal{C})$.


stacks-morphisms-lemma-characterize-etale Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent \begin{enumerate} \item $f$ is \'etale, and \item $f$ is locally of finite presentation, flat, and unramified, \item $f$ is locally of finite presentation, flat, and its diagonal is \'etale. \end{enumerate}


more-morphisms-lemma-category-quasi-projective Let $S$ be a scheme which has an ample invertible sheaf. Let $\text{QP}_S$ be the full subcategory of the category of schemes over $S$ satisfying the equivalent conditions of Lemma \ref{lemma-quasi-projective}. \begin{enumerate} \item if $S' \to S$ is a morphism of schemes and $S'$ has an ample invertible sheaf, then base change determines a functor $\text{QP}_S \to \text{QP}_{S'}$, \item if $X \in \text{QP}_S$ and $Y \in \text{QP}_X$, then $Y \in \text{QP}_S$, \item the category $\text{QP}_S$ is closed under fibre products, \item the category $\text{QP}_S$ is closed under finite disjoint unions, \item if $X \to S$ is projective, then $X \in \text{QP}_S$, \item if $X \to S$ is quasi-affine of finite type, then $X$ is in $\text{QP}_S$, \item if $X \to S$ is quasi-finite and separated, then $X \in \text{QP}_S$, \item if $X \to S$ is a quasi-compact immersion, then $X \in \text{QP}_S$, \item add more here. \end{enumerate}


categories-lemma-preserve-products Let $\mathcal{I}$ be an index category, i.e., a category. Assume that for every pair of objects $x, y$ of $\mathcal{I}$ there exists an object $z$ and morphisms $x \to z$ and $y \to z$. Then \begin{enumerate} \item If $M$ and $N$ are diagrams of sets over $\mathcal{I}$, then $\colim (M_i \times N_i) \to \colim M_i \times \colim N_i$ is surjective, \item in general colimits of diagrams of sets over $\mathcal{I}$ do not commute with finite nonempty products. \end{enumerate}


more-algebra-lemma-pull-relative-pseudo-coherent-module Let $R \to A \to B$ be finite type ring maps. Let $m \in \mathbf{Z}$. Let $M$ be an $A$-module. Assume $B$ is flat over $A$ and $B$ as a $B$-module is pseudo-coherent relative to $A$. If $M$ is $m$-pseudo-coherent (resp.\ pseudo-coherent) relative to $R$, then $M \otimes_A B$ is $m$-pseudo-coherent (resp.\ pseudo-coherent) relative to $R$.


etale-cohomology-lemma-pullback-on-h2-curve Let $\pi : X \to Y$ be a nonconstant morphism of smooth projective curves over an algebraically closed field $k$ and let $n \geq 1$ be invertible in $k$. The map $$ \pi^* : H^2_\etale(Y, \mu_n) \longrightarrow H^2_\etale(X, \mu_n) $$ is given by multiplication by the degree of $\pi$.


etale-cohomology-theorem-quasi-finite-etale-locally Let $A\to B$ be finite type ring map and $\mathfrak p \subset A$ a prime ideal. Then there exist an \'etale ring map $A \to A'$ and a prime $\mathfrak p' \subset A'$ lying over $\mathfrak p$ such that \begin{enumerate} \item $\kappa(\mathfrak p) = \kappa(\mathfrak p')$, \item $ B \otimes_A A' = B_1\times \ldots \times B_r \times C$, \item $ A'\to B_i$ is finite and there exists a unique prime $q_i\subset B_i$ lying over $\mathfrak p'$, and \item all irreducible components of the fibre $\Spec(C \otimes_{A'} \kappa(\mathfrak p'))$ of $C$ over $\mathfrak p'$ have dimension at least 1. \end{enumerate}


decent-spaces-lemma-relative-conditions-local Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{P} \in \{(\beta), decent, reasonable, very\ reasonable\}$. The following are equivalent \begin{enumerate} \item $f$ is $\mathcal{P}$, \item for every affine scheme $Z$ and every morphism $Z \to Y$ the base change $Z \times_Y X \to Z$ of $f$ is $\mathcal{P}$, \item for every affine scheme $Z$ and every morphism $Z \to Y$ the algebraic space $Z \times_Y X$ is $\mathcal{P}$, and \item there exists a Zariski covering $Y = \bigcup Y_i$ such that each morphism $f^{-1}(Y_i) \to Y_i$ has $\mathcal{P}$. \end{enumerate} If $\mathcal{P} \in \{(\beta), decent, reasonable\}$, then this is also equivalent to \begin{enumerate} \item[(5)] there exists a scheme $V$ and a surjective \'etale morphism $V \to Y$ such that the base change $V \times_Y X \to V$ has $\mathcal{P}$. \end{enumerate}


sdga-lemma-extension-by-zero-dg In the situation above we have $$ \Hom_{\text{Mod}_{(\mathcal{A}, \text{d})}^{dg}}( j_!\mathcal{M}, \mathcal{N}) = \Hom_{\text{Mod}_{(\mathcal{A}_U, \text{d})}^{dg}}( \mathcal{M}, j^*\mathcal{N}) $$


equiv-lemma-boundedness Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$. Let $\mathcal{A}$ be an abelian category. Let $H : D_{perf}(\mathcal{O}_X) \to \mathcal{A}$ be a homological functor (Derived Categories, Definition \ref{derived-definition-homological}) such that for all $K$ in $D_{perf}(\mathcal{O}_X)$ the object $H^i(K)$ is nonzero for only a finite number of $i \in \mathbf{Z}$. Then there exists an integer $m \geq 1$ such that $H^i(\mathcal{F}) = 0$ for any coherent $\mathcal{O}_X$-module $\mathcal{F}$ and $i \not \in [-m, m]$. Similarly for cohomological functors.


algebra-lemma-universally-injective-submodule-powerseries Let $R$ be a Noetherian ring and let $M$ be a $R$-module.  Suppose $M$ is a direct sum of countably generated $R$-modules, and suppose there is a universally injective map $M \to R[[t_1, \ldots, t_n]]$ for some $n$. Then $M$ is projective.


divisors-lemma-flat-pullback-reflexive Let $f : X \to Y$ be a flat morphism of integral locally Noetherian schemes. Let $\mathcal{G}$ be a coherent reflexive $\mathcal{O}_Y$-module. Then $f^*\mathcal{G}$ is a coherent reflexive $\mathcal{O}_X$-module.


sites-cohomology-lemma-compose-cup-product Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')$ and $f' : (\Sh(\mathcal{C}'), \mathcal{O}') \to (\Sh(\mathcal{C}''), \mathcal{O}'')$ be morphisms of ringed topoi. The relative cup product of Remark \ref{remark-cup-product} is compatible with compositions in the sense that the diagram $$ \xymatrix{ R(f' \circ f)_*K \otimes_{\mathcal{O}''}^\mathbf{L} R(f' \circ f)_*L \ar@{=}[rr] \ar[d] & & Rf'_*Rf_*K \otimes_{\mathcal{O}''}^\mathbf{L} Rf'_*Rf_*L \ar[d] \\ R(f' \circ f)_*(K \otimes_\mathcal{O}^\mathbf{L} L) \ar@{=}[r] & Rf'_*Rf_*(K \otimes_\mathcal{O}^\mathbf{L} L) & Rf'_*(Rf_*K \otimes_{\mathcal{O}'}^\mathbf{L}  Rf_*L) \ar[l] } $$ is commutative in $D(\mathcal{O}'')$ for all $K, L$ in $D(\mathcal{O})$.


morphisms-lemma-universally-catenary-local-rings-universally-catenary Let $S$ be a locally Noetherian scheme. The following are equivalent: \begin{enumerate} \item $S$ is universally catenary, and \item all local rings $\mathcal{O}_{S, s}$ of $S$ are universally catenary. \end{enumerate}


etale-proposition-etale-regular Let $A$, $B$ be Noetherian local rings. Let $f : A \to B$ be an \'etale homomorphism of local rings. Then $A$ is regular if and only if $B$ is so.


etale-theorem-formally-unramified Let $f : X \to S$ be a morphism of schemes. Assume $S$ is a locally Noetherian scheme, and $f$ is locally of finite type. Then the following are equivalent: \begin{enumerate} \item $f$ is unramified, \item the morphism $f$ is formally unramified: for any affine $S$-scheme $T$ and subscheme $T_0$ of $T$ defined by a square-zero ideal, the natural map $$ \Hom_S(T, X) \longrightarrow \Hom_S(T_0, X) $$ is injective. \end{enumerate}


models-lemma-numerical-type-rational-point In Situation \ref{situation-regular-model} assume $C$ has a $K$-rational point. Then \begin{enumerate} \item $X_k$ has a $k$-rational point $x$ which is a smooth point of $X_k$ over $k$, \item if $x \in C_i$, then $H^0(C_i, \mathcal{O}_{C_i}) = k$ and $m_i = 1$, and \item $H^0(X_k, \mathcal{O}_{X_k}) = k$ and $X_k$ has genus equal to the genus of $C$. \end{enumerate}


schemes-lemma-fibre-products The category of schemes has a final object, products and fibre products. In other words, the category of schemes has finite limits, see Categories, Lemma \ref{categories-lemma-finite-limits-exist}.


restricted-lemma-composition-rig-etale Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of locally Noetherian formal algebraic spaces over $S$. If $f$ and $g$ are rig-\'etale, then so is $g \circ f$.


dpa-lemma-ci-well-defined Let $(A, \mathfrak m)$ be a Noetherian complete local ring. The following are equivalent \begin{enumerate} \item for every surjection of local rings $R \to A$ with $R$ a regular local ring, the kernel of $R \to A$ is generated by a regular sequence, and \item for some surjection of local rings $R \to A$ with $R$ a regular local ring, the kernel of $R \to A$ is generated by a regular sequence. \end{enumerate}


more-algebra-lemma-permanence-tame Let $A$ be a discrete valuation ring with fraction field $K$. \begin{enumerate} \item If $L/K$ is a finite separable extension which is tamely ramified with respect to $A$, then there exists a Galois extension $M/K$ containing $L$ which is tamely ramified with respect to $A$. \item If $L_1/K$, $L_2/K$ are finite separable extensions which are tamely ramified with respect to $A$, then there exists a a finite separable extension $L/K$ which is tamely ramified with respect to $A$ containing $L_1$ and $L_2$. \end{enumerate}


spaces-more-morphisms-lemma-regular-immersion-noetherian Let $S$ be a scheme. Let $i : Z \to X$ be an immersion of algebraic spaces over $S$. Assume $X$ is locally Noetherian. Then $i$ is Koszul-regular $\Leftrightarrow$ $i$ is $H_1$-regular $\Leftrightarrow$ $i$ is quasi-regular.


more-algebra-lemma-factor-through-finite-projective Let $R$ be a ring. Let $\varphi : M \to N$ be an $R$-module map. If $\varphi$ factors through a projective module and $M$ is a finite $R$-module, then $\varphi$ factors through a finite projective module.


bootstrap-lemma-morphism-spaces-is-representable-by-spaces Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Then $f$ is representable by algebraic spaces.


pione-lemma-faithful In Situation \ref{situation-local-lefschetz}. Assume one of the following holds \begin{enumerate} \item $\dim(A/\mathfrak p) \geq 2$ for every minimal prime $\mathfrak p \subset A$ with $f \not \in \mathfrak p$, or \item every connected component of $U$ meets $U_0$. \end{enumerate} Then $$ \textit{F\'Et}_U \longrightarrow \textit{F\'Et}_{U_0},\quad V \longmapsto V_0 = V \times_U U_0 $$ is a faithful functor.


quot-lemma-complexes-fibred-in-groupoids In Situation \ref{situation-complexes} the functor $p : \Complexesstack_{X/B} \longrightarrow (\Sch/S)_{fppf}$ is fibred in groupoids.


fields-lemma-field-semi-simple Every exact sequence of modules over a field splits.


stacks-cohomology-lemma-leray Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. Let $\mathcal{F}$ be a quasi-coherent sheaf on $\mathcal{X}$. Then there exists a spectral sequence with $E_2$-page $$ E_2^{p, q} = H^p(\mathcal{Y}, R^qf_{\QCoh, *}\mathcal{F}) $$ converging to $H^{p + q}(\mathcal{X}, \mathcal{F})$.


curves-lemma-globally-generated-curve In Situation \ref{situation-Cohen-Macaulay-curve} assume that $X$ is integral. Let $0 \to \omega_X \to \mathcal{F} \to \mathcal{Q} \to 0$ be a short exact sequence of coherent $\mathcal{O}_X$-modules with $\mathcal{F}$ torsion free, $\dim(\text{Supp}(\mathcal{Q})) = 0$, and $\dim_k H^0(X, \mathcal{Q}) \geq 2$. Then $\mathcal{F}$ is globally generated.


obsolete-lemma-lqf-f-shriek-composition Let $f : X \to Y$ and $g : Y \to Z$ be composable locally quasi-finite morphisms of schemes. Then $g_! \circ f_! = (g \circ f)_!$ and $f^! \circ g^! = (g \circ f)^!$.


spaces-pushouts-lemma-reverse-commutes-with-flat-base-change In Situation \ref{situation-formal-glueing}. Let $X' \to X$ be a flat morphism of algebraic spaces. Set $Z' = X' \times_X Z$ and $Y' = X' \times_X Y$. The pullbacks $\QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_{X'})$ and $\QCoh(Y \to X, Z) \to \QCoh(Y' \to X', Z')$ are compatible with the functors (\ref{equation-reverse}) and \ref{equation-formal-glueing-modules}).


etale-cohomology-lemma-restrict-and-shriek-from-etale-ctf Let $\Lambda$ be a Noetherian ring. If $j : U \to X$ is an \'etale morphism of schemes, then \begin{enumerate} \item $K|_U \in D_{ctf}(U_\etale, \Lambda)$ if $K \in D_{ctf}(X_\etale, \Lambda)$, and \item $j_!M \in D_{ctf}(X_\etale, \Lambda)$ if $M \in D_{ctf}(U_\etale, \Lambda)$ and the morphism $j$ is quasi-compact and quasi-separated. \end{enumerate}


obsolete-lemma-change-equation-multiply Let $R$ be a ring and let $\varphi : R[x] \to S$ be a ring map. Let $t \in S$. If $t$ is integral over $R[x]$, then there exists an $\ell \geq 0$ such that for every $a \in R$ the element $\varphi(a)^\ell t$ is integral over $\varphi_a : R[y] \to S$, defined by $y \mapsto \varphi(ax)$ and $r \mapsto \varphi(r)$ for $r\in R$.


spaces-resolve-theorem-resolve Let $S$ be a scheme. Let $Y$ be a two dimensional integral Noetherian algebraic space over $S$. The following are equivalent \begin{enumerate} \item there exists an alteration $X \to Y$ with $X$ regular, \item there exists a resolution of singularities of $Y$, \item $Y$ has a resolution of singularities by normalized blowups, \item the normalization $Y^\nu \to Y$ is finite and $Y^\nu$ has finitely many singular points $y_1, \ldots, y_m \in |Y|$ such that the completions of the henselian local rings $\mathcal{O}_{Y^\nu, y_i}^h$ are normal. \end{enumerate}


chow-lemma-tame-symbol The tame symbol (\ref{equation-tame-symbol}) satisfies (\ref{item-bilinear-better}), (\ref{item-skew-better}), (\ref{item-normalization}), (\ref{item-1-x-better}) and hence gives a map $\partial_A : Q(A)^* \times Q(A)^* \to \kappa(\mathfrak m)^*$ satisfying (\ref{item-bilinear}), (\ref{item-skew}), (\ref{item-1-x}).


algebra-lemma-tensor-algebra-localization Let $R$ be a ring and let $S \subset R$ be a multiplicative subset. Then $S^{-1}T_R(M) = T_{S^{-1}R}(S^{-1}M)$ for any $R$-module $M$. Similar for symmetric and exterior algebras.


spaces-more-morphisms-lemma-characterize-formally-etale Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ The following are equivalent: \begin{enumerate} \item $f$ is formally \'etale, \item $f$ is formally unramified and the universal first order thickening of $X$ over $Y$ is equal to $X$, \item $f$ is formally unramified and $\mathcal{C}_{X/Y} = 0$, and \item $\Omega_{X/Y} = 0$ and $\mathcal{C}_{X/Y} = 0$. \end{enumerate}


obsolete-lemma-infinite-sequence Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$ having (RS*). Let $x$ be an object of $\mathcal{X}$ over an affine scheme $U$ of finite type over $S$. Let $u_n \in U$, $n \geq 1$ be pairwise distinct finite type points such that $x$ is not versal at $u_n$ for all $n$. After replacing $u_n$ by a subsequence, there exist morphisms $$ x \to x_1 \to x_2 \to \ldots \quad\text{in }\mathcal{X}\text{ lying over }\quad U \to U_1 \to U_2 \to \ldots $$ over $S$ such that \begin{enumerate} \item for each $n$ the morphism $U \to U_n$ is a first order thickening, \item for each $n$ we have a short exact sequence $$ 0 \to \kappa(u_n) \to \mathcal{O}_{U_n} \to \mathcal{O}_{U_{n - 1}} \to 0 $$ with $U_0 = U$ for $n = 1$, \item for each $n$ there does {\bf not} exist a pair $(W, \alpha)$ consisting of an open neighbourhood $W \subset U_n$ of $u_n$ and a morphism $\alpha : x_n|_W \to x$ such that the composition $$ x|_{U \cap W} \xrightarrow{\text{restriction of }x \to x_n} x_n|_W \xrightarrow{\alpha} x $$ is the canonical morphism $x|_{U \cap W} \to x$. \end{enumerate}


etale-theorem-smooth-etale-over-n-space Let $\varphi : X \to Y$ be a morphism of schemes. Let $x \in X$. If $\varphi$ is smooth at $x$, then there exist an integer $n \geq 0$ and affine opens $V \subset Y$ and $U \subset X$ with $x \in U$ and $\varphi(U) \subset V$ such that there exists a commutative diagram $$ \xymatrix{ X \ar[d] & U \ar[l] \ar[d] \ar[r]_-\pi & \mathbf{A}^n_R \ar[d] \ar@{=}[r] &  \Spec(R[x_1, \ldots, x_n]) \ar[dl] \\ Y & V \ar[l] \ar@{=}[r] & \Spec(R) } $$ where $\pi$ is \'etale.


weil-proposition-weil-cohomology-theory-classical Let $k$ be an algebraically closed field. Let $F$ be a field of characteristic $0$. A classical Weil cohomology theory is the same thing as a $\mathbf{Q}$-linear functor $$ G : M_k \longrightarrow \text{graded }F\text{-vector spaces} $$ of symmetric monoidal categories together with an isomorphism $F[2] \to G(\mathbf{1}(1))$ of graded $F$-vector spaces such that in addition \begin{enumerate} \item $G(h(X))$ lives in nonnegative degrees, and \item $\dim_F G^0(h(X)) = 1$ \end{enumerate} for any smooth projective variety $X$.


pione-lemma-fully-faithful In Situation \ref{situation-local-lefschetz} assume $A$ has depth $\geq 3$ and $A$ is henselian or more generally $(A, (f))$ is a henselian pair. Then the restriction functor $\textit{F\'Et}_U \to \textit{F\'Et}_{U_0}$ is fully faithful.


restricted-lemma-base-change-finite-type Consider the property $P$ on arrows of $\textit{WAdm}^{adic*}$ defined in Lemma \ref{lemma-finite-type}. Then $P$ is stable under base change as defined in Formal Spaces, Remark \ref{formal-spaces-remark-base-change-variant-adic-star}.


stacks-morphisms-lemma-immersion-unramified An immersion is unramified.


examples-lemma-base-change-regular-sequence There exists a local homomorphism of local rings $A \to B$ and a regular sequence $x, y$ in the maximal ideal of $B$ such that $B/(x, y)$ is flat over $A$, but such that the images $\overline{x}, \overline{y}$ of $x, y$ in $B/\mathfrak m_AB$ do not form a regular sequence, nor even a Koszul-regular sequence.


divisors-lemma-minimal-support-in-weakly-ass Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $x \in \text{Supp}(\mathcal{F})$ be a point in the support of $\mathcal{F}$ which is not a specialization of another point of $\text{Supp}(\mathcal{F})$. Then $x \in \text{WeakAss}(\mathcal{F})$. In particular, any generic point of an irreducible component of $X$ is weakly associated to $\mathcal{O}_X$.


crystalline-lemma-flat-base-change-divided-power-envelope Let $(A, I, \gamma)$ be a divided power ring. Let $B \to B'$ be a homomorphism of $A$-algebras. Assume that \begin{enumerate} \item $B/IB \to B'/IB'$ is flat, and \item $\text{Tor}_1^B(B', B/IB) = 0$. \end{enumerate} Then for any ideal $IB \subset J \subset B$ the canonical map $$ D_B(J) \otimes_B B' \longrightarrow D_{B'}(JB') $$ is an isomorphism.


more-etale-lemma-pseudo-functor Let $f : X \to Y$, $g : Y \to Z$, $h : Z \to T$ be separated morphisms of finite type of quasi-compact and quasi-separated schemes. Then the diagram $$ \xymatrix{ Rh_! \circ Rg_! \circ Rf_! \ar[r]_{\gamma_C} \ar[d]^{\gamma_A} & R(h \circ g)_! \circ Rf_! \ar[d]_{\gamma_{A + B}} \\ Rh_! \circ R(g \circ f)_! \ar[r]^{\gamma_{B + C}} & R(h \circ g \circ f)_! } $$ of isomorphisms of Lemma \ref{lemma-shriek-composition} commutes (for the meaning of the $\gamma$'s see proof).


proetale-lemma-geometric-lift-to-cover Let $S$ be a scheme. Let $\overline{s}$ be a geometric point of $S$. Let $\mathcal{U} = \{\varphi_i : S_i \to S\}_{i\in I}$ be a pro-\'etale covering. Then there exist $i \in I$ and geometric point $\overline{s}_i$ of $S_i$ mapping to $\overline{s}$.


spaces-chow-lemma-first-chern-class In Situation \ref{situation-setup} let $X/B$ be good. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. The first Chern class of $\mathcal{L}$ on $X$ of Definition \ref{definition-chern-classes} is equal to the bivariant class of Lemma \ref{lemma-cap-c1-bivariant}.


spaces-more-cohomology-lemma-proper-push-pull-fppf-etale In Lemma \ref{lemma-push-pull-fppf-etale} if $f$ is proper, then we have \begin{enumerate} \item $a_Y^{-1} \circ f_{small, *} = f_{big, fppf, *} \circ a_X^{-1}$, and \item $a_Y^{-1}(Rf_{small, *}K) = Rf_{big, fppf, *}(a_X^{-1}K)$ for $K$ in $D^+(X_\etale)$ with torsion cohomology sheaves. \end{enumerate}


more-algebra-lemma-characterize-pseudo-coherent-colimit-ext Let $R$ be a ring. Let $K \in D^-(R)$. Let $m \in \mathbf{Z}$. Then $K$ is $m$-pseudo-coherent if and only if for any filtered colimit $M = \colim M_i$ of $R$-modules we have $\colim \Ext^n_R(K, M_i) = \Ext^n_R(K, M)$ for $n < -m$ and $\colim \Ext^{-m}_R(K, M_i) \to \Ext^{-m}_R(K, M)$ is injective.


algebraization-lemma-formal-functions Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. Let $X$ be a Noetherian scheme over $A$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Assume that $H^p(X, \mathcal{F})$ is a finite $A$-module for all $p$. Then there are short exact sequences $$ 0 \to R^1\lim H^{p - 1}(X, \mathcal{F}/I^n\mathcal{F}) \to H^p(X, \mathcal{F})^\wedge \to \lim H^p(X, \mathcal{F}/I^n\mathcal{F}) \to 0 $$ of $A$-modules where $H^p(X, \mathcal{F})^\wedge$ is the usual $I$-adic completion. If $f$ is proper, then the $R^1\lim$ term is zero.


spaces-morphisms-lemma-i-upper-shriek Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$. 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}.)


limits-lemma-glueing-near-multiple-closed-points Let $S$ be a scheme. Let $s_1, \ldots, s_n \in S$ be pairwise distinct closed points such that $U = S \setminus \{s_1, \ldots, s_n\} \to S$ is quasi-compact. With $S_i = \Spec(\mathcal{O}_{S, s_i})$ and $U_i = S_i \setminus \{s_i\}$ there is an equivalence of categories $$ FP_S \longrightarrow FP_U \times_{(FP_{U_1} \times \ldots \times FP_{U_n})} (FP_{S_1} \times \ldots \times FP_{S_n}) $$ where $FP_T$ is the category of schemes of finite presentation over the scheme $T$.


divisors-lemma-make-maps-regular-section Suppose given \begin{enumerate} \item $X$ a locally Noetherian scheme, \item $\mathcal{L}$ an invertible $\mathcal{O}_X$-module, \item $s$ a regular meromorphic section of $\mathcal{L}$, and \item $\mathcal{F}$ coherent on $X$ without embedded associated points and $\text{Supp}(\mathcal{F}) = X$. \end{enumerate} Let $\mathcal{I} \subset \mathcal{O}_X$ be the ideal of denominators of $s$. Let $T \subset X$ be the union of the supports of $\mathcal{O}_X/\mathcal{I}$ and $\mathcal{L}/s(\mathcal{I})$ which is a nowhere dense closed subset $T \subset X$ according to Lemma \ref{lemma-regular-meromorphic-ideal-denominators}. Then there are canonical injective maps $$ 1 : \mathcal{I}\mathcal{F} \to \mathcal{F}, \quad s : \mathcal{I}\mathcal{F} \to \mathcal{F} \otimes_{\mathcal{O}_X}\mathcal{L} $$ whose cokernels are supported on $T$.


criteria-lemma-etale-base-change-restriction-of-scalars Let $S$ be a scheme. Let $X \to Z \to B$ and $B' \to B$ be morphisms of algebraic spaces over $S$. Set $Z' = B' \times_B Z$ and $X' = B' \times_B X$. Then $$ \text{Res}_{Z'/B'}(X') = B' \times_B \text{Res}_{Z/B}(X) $$ in $\Sh((\Sch/S)_{fppf})$.


more-algebra-lemma-construct-absolute-integral-closure For any ring $A$ there exists an extension $A \subset B$ such that \begin{enumerate} \item $B$ is a filtered colimit of finite free $A$-algebras, \item $B$ is free as an $A$-module, and \item $B$ is absolutely integrally closed. \end{enumerate}


restricted-lemma-flat-rig-surjective Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Let $B$ be an $I$-adically complete $A$-algebra. Assume that \begin{enumerate} \item the $I$-torsion in $A$ is $0$, \item $A/I^n \to B/I^nB$ is flat and of finite type for all $n$. \end{enumerate} Then $\text{Spf}(B) \to \text{Spf}(A)$ is rig-surjective if and only if $A/I \to B/IB$ is faithfully flat.


derived-lemma-j-is-exact Let $\mathcal{A}$ be an abelian category with enough injectives. Any resolution functor $j : K^{+}(\mathcal{A}) \to K^{+}(\mathcal{I})$ is exact.


sdga-lemma-H0-over-D In Definition \ref{definition-derived-category} the functor $H^0 : K(\text{Mod}_{(\mathcal{A}, \text{d})}) \to \textit{Mod}(\mathcal{O})$ factors through a homological functor $H^0 : D(\mathcal{A}, \text{d}) \to \textit{Mod}(\mathcal{O})$.


sites-cohomology-lemma-square-triangle-general Let $\mathcal{C}$ be a site. Consider a commutative diagram $$ \xymatrix{ \mathcal{D} \ar[r] \ar[d] & \mathcal{F} \ar[d] \\ \mathcal{E} \ar[r] & \mathcal{G} } $$ of presheaves of sets on $\mathcal{C}$ and assume that \begin{enumerate} \item $\mathcal{G}^\# = \mathcal{E}^\# \amalg_{\mathcal{D}^\#} \mathcal{F}^\#$, and \item $\mathcal{D}^\# \to \mathcal{F}^\#$ is injective. \end{enumerate} Then there is a canonical distinguished triangle $$ R\Gamma(\mathcal{G}, K) \to R\Gamma(\mathcal{E}, K) \oplus R\Gamma(\mathcal{F}, K) \to R\Gamma(\mathcal{D}, K) \to R\Gamma(\mathcal{G}, K)[1] $$ functorial in $K \in D(\mathcal{C})$ where $R\Gamma(\mathcal{G}, -)$ is the cohomology discussed in Section \ref{section-limp}.


perfect-lemma-triangulated Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. The full subcategory of $D(\mathcal{O}_X)$ consisting of $S$-perfect objects is a saturated\footnote{Derived Categories, Definition \ref{derived-definition-saturated}.} triangulated subcategory.


spaces-cohomology-lemma-Noetherian-h1-zero Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Assume that for every coherent $\mathcal{O}_X$-module $\mathcal{F}$ we have $H^1(X, \mathcal{F}) = 0$. Then $X$ is an affine scheme.


flat-lemma-Noetherian-base-change Let $f : X \to S$ be a morphism of schemes which is of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_X$-module. Let $s \in S$. If $\mathcal{O}_{S, s}$ is Noetherian then $\mathcal{F}$ is pure along $X_s$ if and only if $\mathcal{F}$ is universally pure along $X_s$.


more-morphisms-lemma-connected-along-section-locally-constructible Let $f : X \to Y$, $s : Y \to X$ be as in Situation \ref{situation-connected-along-section}. If $f$ is of finite presentation then $X^0$ is locally constructible in $X$.


more-algebra-lemma-flat-base-change-perfect Let $A \to B$ be a flat ring map. Let $M$ be a perfect $A$-module. Then $M \otimes_A B$ is a perfect $B$-module.


more-algebra-lemma-perfect-modulo-two-ideals Let $R$ be a ring. Let $I, J \subset R$ be ideals. Let $K$ be an object of $D(R)$. Assume that \begin{enumerate} \item $K \otimes_R^\mathbf{L} R/I$ is perfect in $D(R/I)$, and \item $K \otimes_R^\mathbf{L} R/J$ is perfect in $D(R/J)$. \end{enumerate} Then $K \otimes_R^\mathbf{L} R/IJ$ is perfect in $D(R/IJ)$.


more-morphisms-lemma-base-change-connected-along-section Let $f : X \to Y$, $s : Y \to X$ be as in Situation \ref{situation-connected-along-section}. If $g : Y' \to Y$ is any morphism, consider the base change diagram $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]^{f'} & X \ar[d]_f \\ Y' \ar@/^1pc/[u]^{s'} \ar[r]^g & Y \ar@/_1pc/[u]_s } $$ so that we obtain $(X')^0 \subset X'$. Then $(X')^0 = (g')^{-1}(X^0)$.


varieties-lemma-dominate-valuation-ring-dimension-fibres Let $f : X \to \Spec(R)$ be a morphism from an irreducible scheme to the spectrum of a valuation ring. If $f$ is locally of finite type and surjective, then the special fibre is equidimensional of dimension equal to the dimension of the generic fibre.


more-groupoids-lemma-property-G-invariant Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. Let $G \to U$ be its stabilizer group scheme. Let $\tau \in \{fppf, \linebreak[0] \etale, \linebreak[0] smooth, \linebreak[0] syntomic\}$. Let $\mathcal{P}$ be a property of morphisms which is $\tau$-local on the target. Assume $\{s : R \to U\}$ and $\{t : R \to U\}$ are coverings for the $\tau$-topology. Let $W \subset U$ be the maximal open subscheme such that $G_W \to W$ has property $\mathcal{P}$. Then $W$ is $R$-invariant (see Groupoids, Definition \ref{groupoids-definition-invariant-open}).


spaces-more-morphisms-lemma-finite-after-blowing-up Let $S$ be a scheme. Let $B$ be a quasi-compact and quasi-separated algebraic space over $B$. Let $X$ be an algebraic space over $S$. Let $U \subset B$ be a quasi-compact open subspace. Assume \begin{enumerate} \item $X \to B$ is proper, and \item $X_U \to U$ is finite locally free. \end{enumerate} Then there exists a $U$-admissible blowup $B' \to B$ such that the strict transform of $X$ is finite locally free over $B'$.


schemes-lemma-widetilde-constructions Let $(X, \mathcal{O}_X) = (\Spec(R), \mathcal{O}_{\Spec(R)})$ be an affine scheme. There are canonical isomorphisms \begin{enumerate} \item $ \widetilde{M \otimes_R N} \cong \widetilde M \otimes_{\mathcal{O}_X} \widetilde N $, see Modules, Section \ref{modules-section-tensor-product}. \item $ \widetilde{\text{T}^n(M)} \cong \text{T}^n(\widetilde M) $, $ \widetilde{\text{Sym}^n(M)} \cong \text{Sym}^n(\widetilde M) $, and $ \widetilde{\wedge^n(M)} \cong \wedge^n(\widetilde M) $, see Modules, Section \ref{modules-section-symmetric-exterior}. \item if $M$ is a finitely presented $R$-module, then $ \SheafHom_{\mathcal{O}_X}(\widetilde M, \widetilde N) \cong \widetilde{\Hom_R(M,  N)} $, see Modules, Section \ref{modules-section-internal-hom}. \end{enumerate}


morphisms-lemma-differentials-relative-immersion-smooth Let $i : Z \to X$ be an immersion of schemes over $S$. Assume that $Z$ is smooth over $S$. Then the canonical exact 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 short exact.


more-morphisms-lemma-perfect-conormal-free-lci Let $i : X \to Y$ be an immersion. If \begin{enumerate} \item $i$ is perfect, \item $Y$ is locally Noetherian, and \item the conormal sheaf $\mathcal{C}_{Z/X}$ is finite locally free, \end{enumerate} then $i$ is a regular immersion.


sites-modules-lemma-special-locally-free Any of the properties (1) -- (8) of Definition \ref{definition-site-local} is intrinsic (see discussion in Section \ref{section-intrinsic}).


simplicial-lemma-cosk-up Let $\mathcal{C}$ be a category which has finite limits. \begin{enumerate} \item For every $n$ the functor $\text{sk}_n : \text{Simp}(\mathcal{C}) \to \text{Simp}_n(\mathcal{C})$ has a right adjoint $\text{cosk}_n$. \item For every $n' \geq n$ the functor $\text{sk}_n : \text{Simp}_{n'}(\mathcal{C}) \to \text{Simp}_n(\mathcal{C})$ has a right adjoint, namely $\text{sk}_{n'}\text{cosk}_n$. \item For every $m \geq n \geq 0$ and every $n$-truncated simplicial object $U$ of $\mathcal{C}$ we have $\text{cosk}_m \text{sk}_m \text{cosk}_n U = \text{cosk}_n U$. \item If $U$ is a simplicial object of $\mathcal{C}$ such that the canonical map $U \to \text{cosk}_n \text{sk}_nU$ is an isomorphism for some $n \geq 0$, then the canonical map $U \to \text{cosk}_m \text{sk}_mU$ is an isomorphism for all $m \geq n$. \end{enumerate}


duality-lemma-compactly-supported-triangle In Situation \ref{situation-shriek} let $f : X \to Y$ be a morphism of $\textit{FTS}_S$. Let $$ K \to L \to M \to K[1] $$ be a distinguished triangle of $D^b_{\textit{Coh}}(\mathcal{O}_X)$. Then there exists an inverse system of distinguished triangles $$ K_n \to L_n \to M_n \to K_n[1] $$ in $D^b_{\textit{Coh}}(\mathcal{O}_Y)$ such that the pro-systems $(K_n)$, $(L_n)$, and $(M_n)$ give $Rf_!K$, $Rf_!L$, and $Rf_!M$.


algebra-lemma-tor-welldefined Let $R$ be a ring. Let $M_1, M_2, N$ be $R$-modules. Suppose that $F_\bullet$ is a free resolution of the module $M_1$ and that $G_\bullet$ is a free resolution of the module $M_2$. Let $\varphi : M_1 \to M_2$ be a module map. Let $\alpha : F_\bullet \to G_\bullet$ be a map of complexes inducing $\varphi$ on $M_1 = \Coker(d_{F, 1}) \to M_2 = \Coker(d_{G, 1})$, see Lemma \ref{lemma-compare-resolutions}. Then the induced maps $$ H_i(\alpha) : H_i(F_\bullet \otimes_R N) \longrightarrow H_i(G_\bullet \otimes_R N) $$ are independent of the choice of $\alpha$. If $\varphi$ is an isomorphism, so are all the maps $H_i(\alpha)$. If $M_1 = M_2$, $F_\bullet = G_\bullet$, and $\varphi$ is the identity, so are all the maps $H_i(\alpha)$.


groupoids-quotients-lemma-set-theoretically-invariant-invariant-when-reduced In the situation of Definition \ref{definition-set-theoretically-invariant}. Let $\phi : U \to X$ be a morphism of algebraic spaces over $B$. Assume \begin{enumerate} \item $\phi$ is set-theoretically $R$-invariant, \item $R$ is reduced, and \item $X$ is locally separated over $B$. \end{enumerate} Then $\phi$ is $R$-invariant.


spaces-simplicial-lemma-restriction-injective-to-component-site-module In Situation \ref{situation-simplicial-site}. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$ such that $f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m$ is flat for all $\varphi : [n] \to [m]$. If $\mathcal{I}$ is injective in $\textit{Mod}(\mathcal{O})$, then $\mathcal{I}_n$ is injective in $\textit{Mod}(\mathcal{O}_n)$.


flat-lemma-pure-along-X-s In Situation \ref{situation-pre-pure} the following are equivalent \begin{enumerate} \item there exists an impurity $(S^h \to S, s' \leadsto s, \xi)$ of $\mathcal{F}$ above $s$ where $S^h$ is the henselization of $S$ at $s$, \item there exists an impurity $(T \to S, t' \leadsto t, \xi)$ of $\mathcal{F}$ above $s$ such that $(T, t) \to (S, s)$ is an elementary \'etale neighbourhood, and \item there exists an impurity $(T \to S, t' \leadsto t, \xi)$ of $\mathcal{F}$ above $s$ such that $T \to S$ is quasi-finite at $t$. \end{enumerate}


spaces-resolve-lemma-closed-immersion-on-fibre Let $(A, \mathfrak m, \kappa)$ be a Noetherian local ring. Let $g : X \to Y$ be a morphism in the category (\ref{equation-modification}). If the induced morphism $X_\kappa \to Y_\kappa$ of special fibres is a closed immersion, then $g$ is a closed immersion.


schemes-lemma-kernel-cokernel-quasi-coherent Let $X = \Spec(R)$ be an affine scheme. Kernels and cokernels of maps of quasi-coherent $\mathcal{O}_X$-modules are quasi-coherent.


spaces-perfect-proposition-Noetherian Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Then the functor (\ref{equation-compare}) $$ D(\QCoh(\mathcal{O}_X)) \longrightarrow D_\QCoh(\mathcal{O}_X) $$ is an equivalence with quasi-inverse given by $RQ_X$.


spaces-more-morphisms-lemma-formally-etale-not-affine Let $S$ be a scheme. Let $f : X \to Y$ be a formally \'etale morphism of algebraic spaces over $S$. Then given any solid commutative diagram $$ \xymatrix{ X \ar[d]_f & T \ar[d]^i \ar[l]_a \\ Y & T' \ar[l] \ar@{-->}[lu] } $$ where $T \subset T'$ is a first order thickening of algebraic spaces over $Y$ there exists exactly one dotted arrow making the diagram commute. In other words, in Definition \ref{definition-formally-etale} the condition that $T$ be affine may be dropped.


adequate-lemma-quotient-easy Let $S$ be a scheme. Let $\mathcal{C} \subset \textit{Adeq}(\mathcal{O})$ denote the full subcategory consisting of parasitic adequate modules. Then $$ D(\textit{Adeq}(\mathcal{O}))/D_\mathcal{C}(\textit{Adeq}(\mathcal{O})) = D(\QCoh(\mathcal{O}_S)) $$ and similarly for the bounded versions.


curves-lemma-point-over-separable-extension Let $k$ be a field. Let $X$ be a smooth proper curve over $k$ with $H^0(X, \mathcal{O}_X) = k$ and genus $g \geq 2$. Then there exists a closed point $x \in X$ with $\kappa(x)/k$ separable of degree $\leq 2g - 2$.


pione-proposition-purity-smooth-over-depth2 Let $A \to B$ be a local homomorphism of local Noetherian rings. Assume $A$ has depth $\geq 2$, $A \to B$ is formally smooth for the $\mathfrak m_B$-adic topology, and $\dim(B) > \dim(A)$. For any open $V \subset Y = \Spec(B)$ which contains \begin{enumerate} \item any prime $\mathfrak q \subset B$ such that $\mathfrak q \cap A \not = \mathfrak m_A$, \item the prime $\mathfrak m_A B$ \end{enumerate} the functor $\textit{F\'Et}_Y \to \textit{F\'Et}_V$ is an equivalence. In particular purity holds for $B$.


spaces-divisors-lemma-check-injective-on-weakass Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. 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 $\mathcal{F}_{\overline{x}} \to \mathcal{G}_{\overline{x}}$ is injective, or \item $x \not \in \text{WeakAss}(\mathcal{F})$. \end{enumerate} Then $\varphi$ is injective.


derived-lemma-hocolim-subsequence Let $\mathcal{D}$ be a triangulated category. Let $(K_n, f_n)$ be a system of objects of $\mathcal{D}$. Let $n_1 < n_2 < n_3 < \ldots$ be a sequence of integers. Assume $\bigoplus K_n$ and $\bigoplus K_{n_i}$ exist. Then there exists an isomorphism $\text{hocolim} K_{n_i} \to \text{hocolim} K_n$ such that $$ \xymatrix{ K_{n_i} \ar[r] \ar[d]_{\text{id}} & \text{hocolim} K_{n_i} \ar[d] \\ K_{n_i} \ar[r] & \text{hocolim} K_n } $$ commutes for all $i$.


hypercovering-lemma-product-hypercoverings Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. If $K, L$ are hypercoverings of $X$, then $K \times L$ is a hypercovering of $X$.


spaces-more-morphisms-lemma-differentials-formally-unramified-formally-smooth Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $h : Z \to X$ be a formally unramified morphism of algebraic spaces over $B$. Assume that $Z$ is formally smooth over $B$. Then the canonical exact sequence $$ 0 \to \mathcal{C}_{Z/X} \to i^*\Omega_{X/B} \to \Omega_{Z/B} \to 0 $$ of Lemma \ref{lemma-universally-unramified-differentials-sequence} is short exact.


etale-cohomology-lemma-pullback-c Let $\Lambda$ be a Noetherian ring. Let $f : X \to Y$ be a morphism of schemes. If $K \in D_c(Y_\etale, \Lambda)$ then $Lf^*K \in D_c(X_\etale, \Lambda)$.


obsolete-lemma-P1-localize Let $R$ be a ring. Let $F(X, Y) \in R[X, Y]$ be homogeneous of degree $d$. Let $S = R[X, Y]/(F)$ as a graded ring. Let $\mathfrak p \subset R$ be a prime such that some coefficient of $F$ is not in $\mathfrak p$. There exists an $f \in R$ $f \not\in \mathfrak p$, an integer $e$, and a $G \in R[X, Y]_e$ such that multiplication by $G$ induces isomorphisms $(S_n)_f \to (S_{n + e})_f$ for all $n \geq d$.


topology-lemma-spectral-if-continuous-wrt-constructible-top Let $X$ and $Y$ be spectral spaces. Let $f : X \to Y$ be a continuous map. Then $f$ is spectral if and only if $f$ is continuous in the constructible topology.


constructions-lemma-localization-map-proj With hypotheses and notation as in Lemma \ref{lemma-morphism-proj} above. Assume there exists a $g \in A_0$ such that $\psi$ induces an isomorphism $A_g \to B$. Then $U(\psi) = Y$, $r_\psi : Y \to X$ is an open immersion which induces an isomorphism of $Y$ with the inverse image of $D(g) \subset \Spec(A_0)$. Moreover the map $\theta$ is an isomorphism.