Proper pushforward of coherent sheaves Wouter Zomervrucht, March 19, 2014 1. Cechˇ cohomology Definition 1.1. Let X be a ringed space and F an OX-module. Let U = (Ui)i2I be an open cover of X. Put r(U F) := F( \ \ ) C , ∏ Ui0 ... Uir i0,...,ir2I and define maps r+1 r r+1 j C (U, F) ! C (U, F), (ai ...i )i ,...,i 7! (−1) ai ...ıˆ ...i jU \...\U . 0 r 0 r ∑ 0 j r+1 i0 ir+1 j=0 i0,...,ir+1 The r-th Cechˇ cohomology group of F relative to U, denoted Hˇ r(U, F), is the r-th cohomology • group of the cochain complex C (U, F). The purpose of Cechˇ cohomology is to compute the ‘true’ cohomology. For simplicity one may endow the index set I with a total ordering < and consider the ordered complex r (U F) := F( \ \ ) C< , ∏ Ui0 ... Uir i0<...<ir2I r r+1 • with maps C<(U, F) ! C< (U, F) as before. The cohomology of C<(U, F) is canonically isomorphic to the usual Cechˇ cohomology. U = ( ) \ \ Theorem 1.2. Let X be a scheme and Ui i2I an open cover of X such that Ui0 ... Uir is affine for all r ≥ 0 and i0,..., ir 2 I. Then for all quasi-coherent OX-modules F and all r ≥ 0 we have r ∼ r Hˇ (U, F) = H (X, F) as OX(X)-modules. Proof. Let I• be an injective resolution of F and let Tot• denote the total complex associated to the double complex C•(U, I•). We have morphisms of complexes C•(U, F) C•(U, I0) Tot• C0(U, I•) G(X, I•). For an injective sheaf I and r > 0 we have Hˇ r(U, I) = 0, so the row C•(U, Iq) is a resolution of G(X, Iq) for all q ≥ 0. Hence G(X, I•) ! Tot• is a quasi-isomorphism. On the other hand, if U is affine and r > 0 then Hr(U, F) = 0. So the column Cp(U, I•) is a resolution of Cp(U, F) • • for all p ≥ 0, and C (U, F) ! Tot is a quasi-isomorphism as well. An important application is the computation of the cohomology of projective space. Theorem 1.3. Let A be a ring, n ≥ 0 and d 2 Z. Then 8 A[x ,..., x ] if r = 0, <> 0 n d Hr(Pn , O(d)) = 1 A[ 1 ,..., 1 ] if r = n, A x0···xn x0 xn d :> 0 otherwise. See [Stacks 01ED, 01EO, 01FG, 01X8, 01XS] for more details. 1 2. Proper pushforward of coherent sheaves Theorem 2.1. Let f : X ! S be a proper morphism of locally noetherian schemes. Let F be a coherent r OX-module. Then for all r ≥ 0 the higher direct image R f∗F is a coherent OS-module. Proof. We may assume S = Spec A for some noetherian ring A. n r r n ∼ r n First case: the structure map f : PA ! S. We have R f∗F = H (PA, F) , and H (PA, F) is finitely generated by our computation of the cohomology of projective space. n Second case: f : X ! S projective. Write f = g ◦ i with i : X ! PA is a closed immersion n q and g : PA ! S the structure map. Then i∗F is coherent and R i∗F = 0 for q > 0. So the p p Grothendieck spectral sequence for f∗ = g∗i∗ gives R f∗F = R g∗(i∗F), which is coherent. General case: f : X ! S proper. By Chow’s lemma there is a scheme Y with projective maps p : Y ! X and g : Y ! S such that g = f ◦ p and such that there is a dense open U ⊂ X for which p : p−1U ! U is an isomorphism. We apply noetherian induction. ∗ The kernel and cokernel of F! p∗p F are coherent with support on X n U so have ∗ coherent higher direct images; so it suffices to prove that p∗p F has coherent higher direct p q ∗ p+q ∗ images. Observe that R f∗R p∗(p F) is coherent for p ≥ 0, q > 0, and R g∗(p F) is co- herent for all p, q ≥ 0. From the Grothendieck spectral sequence for g∗ = f∗p∗ it follows that p q ∗ R f∗R p∗(p F) is coherent for q = 0 as well. 3. Serre duality Definition 3.1. Let X be an n-dimensional scheme proper over a field k.A dualizing sheaf for X is a coherent sheaf w on X endowed with a trace map t :Hn(X, w) ! k such that Hom(F, w) × Hn(X, F) Hn(X, w) t k is a perfect pairing for all F 2 Coh X. Dualizing shaeves exist and are uniquely unique. We obtain an isomorphism 0( H (F )) ∼ n( F)_ H X, omOX , w = H X, . for all F 2 Coh X. Theorem 3.2. Let X be an n-equidimensional scheme projective over a field k with dualizing sheaf w. I There are maps i n−i _ ji : Ext (F, w) ! H (X, F) for i ≥ 0 and F 2 Coh X. I If X is Cohen–Macaulay, then all ji are isomorphisms. ∼ n I If X is smooth, then w = WX/k. F i(F ) = i( H (F )) Observe that for locally free of finite rank, Ext , w H X, omOX , w . Complexes of sheaves offer two advantages here: if a dualizing complex exists, there will i automatically be isomorphisms ji as in the theorem; and we have duality between H and Hn−i for all coherent F •, not just the locally free ones. Example 3.3. Let k be a field and R a finite-dimensional k-algebra. Then Spec R is proper over ∼ k and 0-dimensional. Its dualizing sheaf is Homk(R, k) . This is locally free of rank 1 if and only if R is Gorenstein. 2.