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checked that these do not depend on the choices of representatives for the equivalence classes, and that we obtain in this way an S 1A-module S 1M D f m s j m 2 M; s 2 S g and a homomorphism m 7! m 1 W M iS! S 1M of A-modules whose kernel is fa 2 M j sa D 0 for some s 2 S g: PROPOSITION 1.17. The elements of S act inve... |
s the product gh of g; h 2 AŒX, then it divides g or h in F ŒX. Suppose the first, and write f q D g with q 2 F ŒX. Then c.q/ D c.f /c.q/ D c.f q/ D c.g/ 2 A, and so q 2 AŒX. Therefore f divides g in AŒX. We have shown that every element of AŒX is a product of irreducible elements and that every irreducible element of ... |
egral over S 1A. Then n b s C a1 s1 b s n1 C C an sn D 0: for some ai 2 A and si 2 S . On multiplying this equation by sns1 sn, we find that s1 snb 2 A0, and therefore that b s 2 S 1A0. D s1snb ss1sn COROLLARY 1.48. Let A B be rings, and let S be a multiplicative subset of A. If A is integrally closed in B, then S 1A i... |
! Homk-alg.kŒX1; : : :; R/ Homk-alg.kŒXmC1; : : :; R/ induced by the inclusions is a bijection. But this map can be identified with the obvious bijection RmCn ! Rm Rn: In terms of the constructive definition of tensor products, the isomorphism is f ˝ g 7! fgW kŒX1; : : : ; Xm ˝k kŒXmC1; : : : ; XmCn ! kŒX1; : : : ; XmC... |
t the finite sets and the whole space, and so the topology is not Hausdorff (in fact, there are no disjoint nonempty open subsets at all). We shall see in 2.68 below that the proper closed subsets of k2 are finite 2 are much coarser unions of points and curves. Note that the Zariski topologies on C and C (have fewer op... |
lized as the intersection of Y D X 2 and Y D X 2, it has “multiplicity 2”. P Q ! P ! and V W be the set of subsets of kn and let 2.23. Let Then I W and (see FT 7.19). It follows that I and V define a one-to-one correspondence between I . and V . ideals, and (by definition) V . 2.17. Q / P / consists exactly of the radi... |
dical ideals, this becomes a simpler decomposition into prime ideals, as in the corollary. For an ideal .f / with f D Q f mi , the primary decomposition is the decomposition .f / D T.f mi / in Example 2.29. i i i. Regular functions; the coordinate ring of an algebraic set Let V be an algebraic subset of A n, and let I.... |
dent over k, and such that A is finite over kŒx1; : : : ; xd . It is not necessary to assume that A is reduced in Theorem 2.45, nor that k is algebraically closed, although the proof we give requires it to be infinite (for the general proof, see CA 8.1). Let A D kŒx1; : : : ; xn. We prove the theorem by induction on n.... |
; xn/ be the field of fractions of kŒx1; : : : ; xn. As f is not the zero polynomial, some Xi , say, Xn, occurs in it. Then Xn occurs in every nonzero multiple of f , and so no nonzero polynomial in X1; : : : ; Xn1 belongs to .f /. This means that x1; : : : ; xn1 are algebraically independent. On the other hand, xn is ... |
h g with gh0Cg 0h hh0 on U 0, and so is regular at P . on U 0, and so f C f 0 is regular at P . Similarly, ff 0 agrees with gg 0 h and g 0 hh0 (b,c) The definition is local. We next determine O V .U / when U is a basic open subset of V . LEMMA 3.10. Let g; h 2 kŒV with h ¤ 0. The function P 7! g.P /= h.P /mW D.h/ ! k i... |
e image of f in A=m D k, i.e., f .m/ D f (mod m/. This allows us to identify elements of A with functions spm.A/ ! k. We attach a ringed space .V; For f 2 A, let O V / to A by letting V be the set of maximal ideals in A. D.f / D fm j f .m/ ¤ 0g D fm j f … mg: Since D.fg/ D D.f / \ D.g/, there is a topology on V for whi... |
is a closed subset of spm.A/, and every closed subset is of this form. Now let a be a radical ideal in A, so that A=a is again reduced. Corresponding to the homomorphism A ! A=a, we get a regular map Spm.A=a/ ! Spm.A/: The image is V .a/, and spm.A=a/ ! V .a/ is a homeomorphism. Thus every closed subset of spm.A/ has ... |
that, after renumbering, zd C1 will be separably algebraic over k.z1; : : : ; zd /, and this implies that fz1; : : : ; zd g is a transcendence basis for ˝ over k (1.63). l. Noether Normalization Theorem DEFINITION 3.39. The dimension of an affine algebraic variety is the dimension of the underlying topological space (... |
1994 (I 6, Exercise 8; see also Hartshorne 1977, I, Exercise 2.17). 3 — see Apparently, it is not known whether two polynomials always suffice to define a curve in A 3 can’t be defined by two polynomials (ibid. Kunz 1985, p136.5 The union of two skew lines in P 3 can be defined by two polynomials. p. 140), but it is u... |
double point is called a node. There are many names for special types of singularities — see any book, especially an old book, on algebraic curves. Examples The following examples are adapted from Walker, Robert J., Algebraic Curves. Princeton Mathematical Series, vol. 13. Princeton University Press, Princeton, N. J., ... |
: When we identify Tb.A regarded as a regular map) with the differential of F (F regarded as a regular function). 1/; f. Tangent spaces to affine algebraic varieties 89 LEMMA 4.24. Let 'W A km into W D V .b/ kn, then .d'/a maps Ta.V / into Tb.W /, b D '.a/. n be as at the start of this subsection. If ' maps V D V .a/ m... |
imension of the geometric tangent cone at P is the same as the dimension of V (because the Krull dimension of a noetherian local ring is equal to P / is a polynomial ring in dim V variables if and that of its graded ring). Moreover, gr. P / is a only if polynomial ring in d variables, in which case CP .V / D TP .V /. P... |
d connected solvable normal subgroup (except feg). Such a group G may have a finite nontrivial centre Z.G/, and we call two semisimple groups G and G0 locally isomorphic if G=Z.G/ G0=Z.G0/. For example, SLn is semisimple, with centre n, the set of diagonal matrices diag.; : : : ; /, n D 1, and SLn =n D PSLn. A Lie alge... |
e subset of Z on which '1 and '2 agree is closed. PROOF. There are regular functions xi on V such that P 7! .x1.P /; : : : ; xn.P // identifies n (take the xi to be any set of generators for kŒV as a k-algebra). V with a closed subset of A Now xi ı '1 and xi ı '2 are regular functions on Z, and the set where '1 and '2 ... |
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