12 January 2006 1. Contents 1.1 Hilbert schemes 0. Contents. 1. Sheaf cohomology for schemes 2. Cohomology of projective spaces 3. Flat maps 4. Base change 5. Hilbert polynomials 6. Castelnuovo-Mumford regularity 7. Fitting ideals 8. Flattening strati(cid:12)cations 9. Representation of functors 10.The Quotient functor \hilball.tex 12 January 2006 1. Cohomology of sheaves on schemes cohom 1.1 1. Cohomology of sheaves on schemes. (1.1) Setup. Given a noetherian scheme S and a f:X ! S separated morphism of (cid:12)nite type. Moreover, given a quasi{coherent O {module F. Let g:T ! S X be a morphism from a scheme T. We write X = T (cid:2) X and the maps of the T S resulting cartesian diagram we denote as follows: X (cid:0)(cid:0)g(cid:0)X(cid:0)! X T fT f ? ? T? (cid:0)(cid:0)(cid:0)(cid:0)! S?: y g y Moreover, we write F = g(cid:3) F. T X We choose an a(cid:14)ne open covering U = fU ;:::;U g of X. 0 r (1.2) De(cid:12)nition. Assume that S = SpecA is a(cid:14)ne. We have a sequence of A{modules d0 d1 F :0 ! F(U ) (cid:0)! F(U \U ) (cid:0)! U i0 i0 i1 0(cid:20)Mi0(cid:20)r 0(cid:20)iM0 r and i < 0. (1.4) Note. Assume that S = SpecA. The map which sends a quasi{coherent O {module F to the A{module Hi(X;F) is a covariant functor from quasi{ X coherent O {modules to A{modules. Indeed, given a homomorphism F ! G X of quasi{coherent O {modules. We obtain a map X F(U \(cid:1)(cid:1)(cid:1)\U ) ! G(U \(cid:1)(cid:1)(cid:1)\U ); i0 ip i0 ip \hilball.tex 12 January 2006 1. Cohomology of sheaves on schemes cohom 1.2 for each i ;:::;i , and consequently a map 0 p F ! G U U of complexes of A{modules. Thus there is an A{linear map Hi(X;F)! Hi(X;G) of cohomology modules, for each i. It is clear from the construction of the latter map that the map from quasi{coherent O {modules to A{modules that sends F X to Hi(X;F) is a functor. (1.5) Note. Assume that S = SpecA. From a short exact sequence 0 ! F0 ! F ! F00 ! 0 of quasi{coherent O {modules, we obtain a long exact sequence X (cid:1)(cid:1)(cid:1) ! Hi(X;F0) ! Hi(X;F) ! Hi(X;F00) ! Hi+1(X;F0) ! (cid:1)(cid:1)(cid:1) : Indeed, we have an exact sequence 0 ! F0(U \(cid:1)(cid:1)(cid:1)\U ) ! F(U \(cid:1)(cid:1)(cid:1)\U ) ! F00(U \(cid:1)(cid:1)(cid:1)\U ) ! 0; i0 ip i0 ip i0 ip for each 0 (cid:20) i < (cid:1)(cid:1)(cid:1) < i (cid:20) r. Hence we obtain a short exact sequence 0 p 0 ! F0 ! F ! F00 ! 0 U U U of complexes that gives rise to the long exact sequence. (1.6) Note. Assume that S = SpecA. Let (cid:19):Y (cid:18) X be a closed immersion of schemes, and let G be a quasi{coherent O {module. The map i induces an Y equality Hi(Y;G) = Hi(X;(cid:19) G) (cid:3) of A{modules. Indeed, let V = U \ Y = i(cid:0)1(U ). Then V = fV ;:::;V g is an i i i 0 r a(cid:14)ne open coveringof Y and we have that(i G)(U \(cid:1)(cid:1)(cid:1)\U ) = G(V \(cid:1)(cid:1)(cid:1)\V ). (cid:3) i0 ip i0 ip Consequently (i G) = G and we obtain the equality. (cid:3) U V (1.7) De(cid:12)nition. Given a morphism g:T ! S from a noetherian scheme T. Given an open a(cid:14)ne subset SpecA of S and let U = fU ;:::;U g be an a(cid:14)ne 0 r open a(cid:14)ne covering of f(cid:0)1(SpecA). Moreover, let SpecB be an open a(cid:14)ne subset of T that maps to SpecA. For every open a(cid:14)ne subset U of X that maps into 12 January 2006 1. Cohomology of sheaves on schemes cohom 1.3 SpecA we have that V = X \ g(cid:0)1U = SpecB (cid:2) U is an a(cid:14)ne open SpecB X SpecA subset of X and we have that SpecB B (cid:10) F(U) = (F (cid:10) O )(V) = g(cid:3) F(V) = F (V): A OSpecA SpecB X SpecB Hence, if we let V = X \ g(cid:0)1U , we obtain an open a(cid:14)ne covering V = i SpecB X i fV ;:::;V g of X , and we have an isomorphism 0 r SpecB B (cid:10) F(U \(cid:1)(cid:1)(cid:1)\U ) ! F (V \(cid:1)(cid:1)(cid:1)\V ) A i0 ip SpecB i0 ip for each 0 (cid:20) i < (cid:1)(cid:1)(cid:1) < i (cid:20) r of B{modules. Consequently we obtain an 0 p isomorphism B (cid:10) F ! (F ) (1.7.1) A U SpecB V of complexes of B{modules. Thus we obtain an A{B{linear map F ! B (cid:10) F ! (F ) (1.7.2) U A U SpecB V where the left map sends f to 1(cid:10) f. A We obtain a restriction map Hi(X ;F ) ! Hi(X ;F ) (1.7.3) SpecA SpecA SpecB SpecB of H0(X ;O ){H0(SpecB;O ){modules. SpecA XSpecA SpecB In particular, when we associate to each open a(cid:14)ne subscheme SpecA of S the A{module Hi(X ;F ), we obtain a pre{sheaf of O {modules. The SpecA SpecA S associated O {module we denote by Rif F . We have that S (cid:3) ^ Rif FjSpecA = Hi(X ;F ); (1.7.4) (cid:3) SpecA SpecA ! for all open a(cid:14)ne subsets SpecA of S ([H], (III x8, Proposition 8.6)). ! (1.8) Note. From (1.7.4) it follows that the sheaves Rif F are quasi{coherent (cid:3) O {modules. Moreover, it follows from the Notes (1.3){(1.6), applied to an a(cid:14)ne S open covering of S, that: (1) We have Rif F = 0 for i > r and i < 0, when X, and thus all a(cid:14)ne open (cid:3) subsets of X, can be covered by r+1 open a(cid:14)ne subsets. (2) The correspondence that sends a quasi{coherent O {module F to the X quasi{coherent O {module Rif F is functorial in F. X (cid:3) (3) Given a short exact sequence 0 ! F0 ! F ! F00 ! 0 of quasi{coherent O {modules, we obtain a long exact sequence X (cid:1)(cid:1)(cid:1) ! Rif F0 ! Rif F ! Rif F00 ! Ri+1f F ! (cid:1)(cid:1)(cid:1) (cid:3) (cid:3) (cid:3) (cid:3) of O {modules. S (4) Given a closed immersion (cid:19):Y (cid:18) X and a quasi{coherent sheaf G on Y we have that (Rif )(cid:19) G = Ri(f (cid:19) )G = Ri((f(cid:19)) )G. (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) 12 January 2006 1. Cohomology of sheaves on schemes cohom 1.4 (1.9) De(cid:12)nition. Given a complex F:0 ! F0 (cid:0)d!0 F1 (cid:0)d!1 (cid:1)(cid:1)(cid:1) (cid:0)d(cid:0)r(cid:0)(cid:0)!1 Fr ! 0 of A{modules. We write Zi = Zi(F) = Kerdi and Bi = Bi(F) = Imdi(cid:0)1. Then Hi(F) = Zi(F)=Bi(F) is the cohomology of the sequence F. There are exact sequences 0 ! Zi(F) ! Fi ! Bi+1(F) ! 0; (1.9.1) and 0 ! Bi(F) ! Zi(F) ! Hi(F) ! 0 (1.9.2) of A{modules for i = 0;:::;r. Given an A{algebra B. We obtain a complex B (cid:10) F:0 ! B (cid:10) F0 (cid:0)i(cid:0)d(cid:0)B(cid:0)(cid:10)(cid:0)A(cid:0)d!0 B (cid:10) F1 (cid:0)i(cid:0)d(cid:0)(cid:10)(cid:0)A(cid:0)d!1 (cid:1)(cid:1)(cid:1) (cid:0)id(cid:0)(cid:0)(cid:10)(cid:0)A(cid:0)dr(cid:0)(cid:0)!1 B (cid:10) Fr ! 0 A A A A of B{modules, and a map of complexes F ! B (cid:10) F; A which sends an element m in Fi to 1 (cid:10) m in B (cid:10) Fi. For each i we get a A A map Hi(F) ! Hi(B (cid:10) F) of cohomology, which is a map of A{B{modules. We A extend this map to a map B (cid:10) Hi(F) ! Hi(B (cid:10) F) (1.9.3) A A of B{modules which is called the map obtained by changing the base from A to B , or simply the base change map . (1.10) Note. The natural map B (cid:10) Bi(F) ! Bi(B (cid:10) F) of B{modules is A A a surjection because B (cid:10) Fi = Fi(B (cid:10) F) for all i, and di (b (cid:10) m) = A A B(cid:10)AF A b(cid:10) di (m) where b 2 B and m 2 Fi(cid:0)1. A F (1.11) De(cid:12)nition. Given a morphism g:T ! S from a noetherian scheme T. Let SpecA of S be an a(cid:14)ne subscheme and SpecB an open a(cid:14)ne subscheme of T !! which maps to SpecA. We obtain from the maps (1.7.1) and (1.9.3) a base change map B (cid:10) Hi(F ) ! Hi(B (cid:10) F ) = Hi((F ) ), that is, a B{linear (base A U A U SpecB V ! change map) B (cid:10) Hi(X ;F ) ! Hi(X ;F ): (1.11.1) A SpecA SpecA SpecB XSpecB We apply this map to each member S of an a(cid:14)ne open cover of S, and to each i ! member of an a(cid:14)ne open cover of g(cid:0)1(S ). It follows from the De(cid:12)nitions of (1.7) i that we obtain a base change map O (cid:10) Rif F = g(cid:3)Rif F ! Rif (g(cid:3) F) = Rif F : (1.11.2) T OS (cid:3) (cid:3) T(cid:3) X T(cid:3) T ! When S = SpecA we obtain a (base change) map ^ O (cid:10) Hi(X;F)! Rif F : (1.11.3) T OSpecA T(cid:3) T 12 January 2006 2. Cohomology of sheaves on projective spaces projc 2.1 2. Cohomology of sheaves on projective spaces. (2.1) Setup. Given a noetherian ring A and a free A{module E of rank r +1. We choose an A{basis e ;e ;:::;e of E. Denote by R = Sym (E) the symmetric 0 1 r A algebra of E over A and write P(E) = Proj(R) for the r{dimensional projective space over SpecA . The choice of basis e ;:::;e de(cid:12)nes an isomorphism between 0 r R and the polynomial ring A[x ;x ;:::;x ] in the variables x ;:::;x with coef- 0 1 r 0 r (cid:12)cients in the ring A. In this way we obtain an isomorphism P(E) (cid:24)= Pr . The A r+1 open a(cid:14)ne sets D (e ) cover P(E). + i Denote by p:P(E) ! SpecA the structure map of the projective space, and by O (1)the tautological invertible sheaf on P(E). There is a canonical surjection P(E) p(cid:3)E ! O (1) of O {modules. P(E) P(E) ! A standard calculation ([H], (III, Theorem 5.1)) gives: (1) The canonical map R ! H0(P(E);O (m)) is an isomorphism. m P(E) (2) We have that Hi(P(E);O (m)) = 0 for i > 0 and m (cid:21) 0. P(E) Given an ideal I in R. Let X = Proj(R=I), and let (cid:19):X ! P(E) be the corresponding closed immersion. The r+1 open a(cid:14)ne sets U = X\D (e ) cover i + i X. Given a coherent O {module F on X. For each integer n we write F(n) = X F (cid:10) i(cid:3)O (n). Then we have that i (F(n)) = i (F (cid:10) i(cid:3)O (n)) = OX P(E) (cid:3) (cid:3) OX P(E) (i F)(n), and i(cid:3)i F(n) ! F(n) is an isomorphism for all n. (cid:3) (cid:3) Write K = (cid:0)(X;F(m)). Then we have a canonical isomorphism ([H], m2Z ! (II x5, Proposition 5.15)) L (cid:12):K ! F: Hence F is the sheaf associated to a graded R=I{module K. We can take this e R=I{module to be (cid:12)nitely generated. Indeed, we can choose a (cid:12)nite number of homogeneous elements m of K of degree d such that the elements m=yd, where i y is the class of e in R=I, generate F(U ), for i = 0;:::;r. The submodule of i i i K generated by these elements for i = 0;1;:::;r de(cid:12)nes F. We choose a (cid:12)nitely generated R=I{module M such that F = M . F F (2.2) Theorem. (Serre) There is an m such that for m (cid:21) m we have: 0 0 f (1) The canonical map (M ) ! H0(X;F(m)) F m is an isomorphism. (2) There is an equality Hi(X;F(m)) = 0 for i > 0 ^ (3) The canonical map O (cid:10) H0(X;F(m)) = f(cid:3)f F(m) ! F(m) of X OSpecA (cid:3) O {modules is surjective. X \hilball.tex 12 January 2006 2. Cohomology of sheaves on projective spaces projc 2.2 Proof. To simplify the notation we (cid:12)rst show that it su(cid:14)ces to prove the Theorem ! when X = P(E). ItfollowsfromNote(1.6)andtheequalityi (F(m)) = (i F)(m) (cid:3) (cid:3) ! of Setup (2.1) that Hi(X;F(m)) = Hi(P(E);((cid:19) F)(m)). Let M be the R=I{ (cid:3) F ! submodule of (cid:8) Hi(X;F(m)) chosen in Setup (2.1). Denote by M the module m2Z ^ M considered as a R{submodule of (cid:8) Hi(P(E);((cid:19) F)(m)). Since (M ) = F F m2Z (cid:3) F on X, we obtain that (M) = i F on P(E). Hence we can choose the module M (cid:3) ! for the module M of Setup (2.1). It follows that it su(cid:14)ces to prove assertions i(cid:3)F (1) and (2) of the Theogrem in the case when X = P(E). Since i(cid:3)i F ! F is an (cid:3) isomorphism it also follows that it su(cid:14)ces to prove assertion (3) in this case. When M = R(d)is R with gradind translated by d we have that F = O (d), P(E) ! and, as we noted in (2.1), we have M = R (cid:24)= H0(P(E);O (d+m)); and Hi(P(E);O (d+m)) = 0 m d+m P(E) P(E) for i > 0 and d+m (cid:21) 0. Hence assertions (1) and (2) of the Theorem hold for the modules O (d). P(E) In general, choose a short exact sequence of graded R{modules 0 ! K ! L ! M ! 0; (2.2.1) where L is the direct sum of (cid:12)nitely many modules of the form R(d). Since A is noetherian we have that K is a (cid:12)nitely generated A{module. We shall prove, by descending induction on i, that the second assertion of the Theorem holds. ! Since P(E) can be covered by r+1 open a(cid:14)nes it follows from Note (1.3) that the assertionholdsfori > r. AssumethatwehaveprovedthatHi+1(P(E);F(m)) = 0 for all coherent O {modules F for su(cid:14)ciently big m depending on F. From P(E) ! the short exact sequence sequence (2.2.1) we obtain a long exact sequence (cid:1)(cid:1)(cid:1) ! Hi(P(E);K(m)) ! Hi(P(E);L(m)) ! Hi(P(E);F(m)) ! Hi+1(P(E);K(m)) ! (cid:1)(cid:1)(cid:1) : e e ! As we already observed assertion (2) of the Theorem holds for L by Note (2.1), e and by the induction assumption Hi+1(P(E);K(m)) = 0 for big m. Consequently we have that Hi(P(E);F(m)) = 0 for big m. Hence we have preoved the second part of the Theorem. In particular we have theat H1(P(E);K~(m)) = 0 Thus the map H0(P(E);L(m)) ! H0(P(E);F(m)) is surjective when m is su(cid:14)ciently big. We obtain a commutative diagram of A{modules e 0 (cid:0)! K (cid:0)! L (cid:0)! M (cid:0)! 0 m m m ? ? ? 0 (cid:0)! H0(P(E?);K(m)) (cid:0)! H0(P(E?);L(m)) (cid:0)! H0(P(E?);F(m)) (cid:0)! 0; y y y e e 12 January 2006 2. Cohomology of sheaves on projective spaces projc 2.3 with exact rows, where the middle vertical map is an isomorphism since we observed that assertion (1) of the Theorem holds for L. Consequently the right vertical map is surjective for big m. Since this holds for all (cid:12)nitely generated R{ modules the left vertical map is also surjective for big m. Consequently we have that the right vertical map is an isomorphism for big m, and we have proved the (cid:12)rst part of the Theorem. The third part of the Theorem holds for the modules O (d) because of the P(E) surjection f(cid:3)Sm+d(E) = Sm+d(E)(cid:10) O ! O (m + d), and the isomor- A P(E) P(E) phism Sm+d(E) ! R ! H0(P(E);O (m + d)). Hence the left vertical m+d P(E) map of the commutative diagram O (cid:10) H0(P(E);L~(m)) (cid:0)! O (cid:10) H0(P(E);F(m)) P(E) OSpecA P(E) OSpecA L~(??m) (cid:0)! F(??m) (cid:0)! 0 y y is surjective for big m. It follows that the right vertical map is surjective, and we have proved the third part of the Theorem. (2.3) Note. There is an m such that for each m (cid:21) m there is a surjection 0 0 On ! F(m) X of O {modules, where n depends on m. Indeed, it follows from the (cid:12)rst part of X ! Theorem (2.2) that we can (cid:12)nd a surjection An ! H0(X;F(m)), for (cid:12)xed big m, ! and from the third part of Theorem (2.2) that we have a surjection O (cid:10) X SpecA ^ H0(X;F(m)) ! F(m) for big m. (2.4) Note. For every integer m we have a map (cid:12) :f F(m)(cid:10) f O (1) ! f F(m+1) (2.4.1) m (cid:3) OSpecA (cid:3) X (cid:3) of O {modules induced by the isomorphism F(m)(cid:10) O (1) ! F(m+1). SpecA OX X Equivalently we have a map (cid:12) (SpecA):H0(X;F(m))(cid:10) H0(X;O (1)) ! H0(X;F(m+1)); (2.4.2) m A X of A{modules. There is an m such that for m (cid:21) m this map is surjective. This 0 0 can be seen from the commutative diagram (M ) (cid:10) (R=I) (cid:0)(cid:0)(cid:0)(cid:0)! (M ) F m A 1 F m+1 ? ? H0(X;F(m))(cid:10)?A H0(X;OX(1)) (cid:0)(cid:0)(cid:0)(cid:0)! H0(X;F?(m+1)); y y 12 January 2006 2. Cohomology of sheaves on projective spaces projc 2.4 where the upper row is multiplication. Since M is a (cid:12)nitely generated (R=I){ F module the multiplication map is surjective for big m. It follows from Theorem ! (2.2) that the right vertical map is an isomorphism for big m. Thus there is an m such that the bottom row is surjective for m (cid:21) m . That is, the map (cid:12) is 0 0 m surjective for big m. ! We also note that if (2.4.1) is surjective for m (cid:21) m , then 0 (cid:11) :f(cid:3)f F(m) ! F(m) m (cid:3) is surjective. To see this we note that from the maps (cid:12) we obtain maps m (cid:12) :f F(m)(cid:10) f O (d) ! f F(m+d) m;d (cid:3) OA (cid:3) X (cid:3) for each integer d. If (cid:12) is surjective for n (cid:21) m we have that (cid:12) is surjective. m m;d We obtain a commutative diagram (cid:3) f(cid:3)f F(m)(cid:10) f(cid:3)f O (d) (cid:0)f(cid:0)(cid:0)(cid:12)m(cid:0)!;d f(cid:3)f F(m+d) (cid:3) OSpecA (cid:3) X (cid:3) (cid:11)m(cid:10)(cid:13)d ? ? F(m)(cid:10)OS?ypecA OX(d) (cid:0)(cid:0)(cid:0)(cid:0)! F(m?y+d) ! for each d, where f(cid:3)(cid:12) is surjective. It follows from Theorem (2.2) that the right m;d vertical map is surjective for d su(cid:14)ciently big. Since the bottom horizontal map is an isomorphism we have that (cid:11) (cid:10)(cid:13) is surjective for big d. However we have m d that (cid:13) :f(cid:3)f O (d) = f(cid:3)Symd(E) ! O (d) is surjective for d (cid:21) 0. Hence (cid:11) is d (cid:3) X X m surjective, as asserted. (2.5) De(cid:12)nition. Let A be a noetherian ring. A graded A{algebra S = (cid:8)1 S i=0 i is called standard if S = A and S is generated, as an A{algebra, by the elements 0 S of degree 1. 1 (2.6) Lemma. Let S be a standard A{algebra and N a (cid:12)nitely generated graded S{module such that N 6= 0 for big m. Then N has a (cid:12)ltration 0 = N (cid:26) m 0 N (cid:26) (cid:1)(cid:1)(cid:1) (cid:26) N = N by graded submodules such that N =N is isomorphic to 1 n i i(cid:0)1 (S=P )(m ), where P is a prime ideal of S, and m is an integer. In particular i i i i the support of N on Proj(S) consists of the homogeneous prime ideals in S that contain one of the ideals P . i e ! Proof. See [H] (I x7 Proposition 7.4). 12 January 2006 2. Cohomology of sheaves on projective spaces projc 2.5 (2.7) Theorem. The A{module Hi(X;F) is (cid:12)nitely generated for all i. Proof. To simplify the notation we note that from the equality H0(X;F) = H0(P(E);(cid:19) F)itfollowsthatweonlyhavetoprovetheTheoremwhenX = P(E). (cid:3) We shall prove the Theorem when X = P(E) by induction on the dimension s of the support SuppF of F = M. When s < 0 we have that F = 0 and the ! statement is true. Assume that s (cid:21) 0. It follows from Lemma (2.6) that M has a (cid:12)nite (cid:12)ltration whose quotients afre isomorphic to (R=P)(d), where P is a prime ideal in R. Since s (cid:21) 0 we have that P does not contain the ideal (e ;:::;e ), 0 r and the support of F is the union of the irreducible varieties Z(P) in P(E). Consequently we can assume that F is the sheaf associated to L = (R=P)(d). Choose a homogeneous element f of degree m in R not contained in P. We have an exact sequence f 0 ! L (cid:0)! L(m) ! N ! 0: (2.7.1) The dimension of SuppN is strictly less than s because SuppF = Z(P) and f is ! an isomorphism at the generic point of Z(P). It follows from Theorem (2.2) that we can choose m so big that H0(P(E);F(m)) is a (cid:12)nitely generated A{module, ! and Hi(P(E);F(m)) = 0 for i > 0. From the short exact sequence (2.7.1) we obtain a long exact sequence, (cid:1)(cid:1)(cid:1) ! Hi(cid:0)1(P(E);N) ! Hi(P(E);F) ! Hi(P(E);F(m)) ! Hi(P(E);N) ! (cid:1)(cid:1)(cid:1) : e Since the A{module Hi(P(E);N) is (cid:12)nitely generated for all i, by thee induction assumption, it follows that Hi(P(E);F) is a (cid:12)nitely generated A{module. e

Hilbert schemes

Similar Hilbert schemes