We have already seen Part (1), as well as Part (2) for \({\mathscr F}\) a line bundle. Clearly, then (2) also holds for finite direct sums of line bundles. For a general \({\mathscr F}\), we can find a presentation

where \({\mathscr E}\) and \({\mathscr E}\) are sums of line bundles and the sequence is exact. Since the functors \(\operatorname{Hom}(-, \omega _X)\) and \(H^n(X, -)^\vee \) are left exact and the result holds for \({\mathscr E}\) and \({\mathscr E}'\), it follows for \({\mathscr F}\).

In view of Part (2), Part (3) follows if we can show that the \(\delta \)-functors \((\operatorname{Ext}^i(-, \omega _X))_i\) and \((H^{n-i}(X, -)^\vee ))_i\) are universal. To prove this, it is enough to show they are coeffaceable. But any \({\mathscr F}\) can be written as a quotient of an \({\mathscr O}_X\)-module of the form \({\mathscr O}_X(-d)^{\oplus N}\) with \(d\gg 0\), and both functors vanish on such sheaves for \(i {\gt} 0\). (In fact, \(d {\gt} n+1\) is enough since then \(\omega _X(d)\) has no higher cohomology.)

This follows from the following fact about injective \({\mathscr O}_X\)-modules: If \({\mathscr I}\) is an injective \({\mathscr O}_X\)-module, then the restriction \({\mathscr I}_{|U}\) is an injective \({\mathscr O}_U\)-module. (In fact, the restriction functor admits an exact left adjoint functor, the extension by zero functor, and hence preserves the class of injective objects.)

Both sides are \(\delta \)-functors in \({\mathscr G}\), and they agree for \(i=0\). Both sides vanish for \(i {\gt} 0\) and \({\mathscr G}\) injective (for the left hand side, use that \(\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits (-, {\mathscr G})\) is exact since injectivity is preserved under restriction to opens, cf. the proof of Proposition 3.8) and hence are universal \(\delta \)-functors.

For Part (1), let \(P_\bullet \to M\to 0\) be a resolution of \(M\) by finite projective \(A\)-modules. We can use this resolution to compute \(\operatorname{Ext}^i_A(M, N)\), and can use \(\widetilde{P}_\bullet \) to compute the left hand side. The result follows since \(\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits _{{\mathscr O}_X}(\widetilde{P}, \widetilde{N}) = \operatorname{Hom}_A(P, N)^\sim \) (because \(P\) is of finite presentation). Part (2) follows from Part (1) since for \(N\) finitely generated, \(\operatorname{Ext}^i_A(M,N)\) is finitely generated, as the considerations in the proof of (1) show.

By Proposition 3.8 we can reduce to the case that \(X = \operatorname{Spec}A\) is affine. Choose a resolution \({\mathscr P}_\bullet \to {\mathscr F}\to 0\) of \({\mathscr F}\) by free \({\mathscr O}_X\)-modules \({\mathscr P}_i\). Passing to the stalks at \(x\), we obtain a free resolution of \({\mathscr F}_x\) as an \({\mathscr O}_{X, x}\)-module.

By the above, \(\mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits ^i({\mathscr F}, {\mathscr G}) = \mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits ({\mathscr P}_i, {\mathscr G})\), and \(\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits ({\mathscr P}_i, {\mathscr G})_x = \operatorname{Hom}({\mathscr P}_{i,x}, {\mathscr G}_x)\) since \({\mathscr P}_i\) is of finite presentation (this is enough since the result clearly holds for \({\mathscr O}_X\) in the first entry, thus for finite free modules, and by the five lemma follows for modules of finite presentation).

All the above functors are \(\delta \)-functors in \({\mathscr G}\) which are moreover effaceable and hence universal. Therefore it is enough to show the equalities in the case \(i=0\) where they are clear.

These are the Grothendieck spectral sequences for the compositions \(\Gamma (Y, -)\circ f_* = \Gamma (X, -)\) and \(g_*\circ f_* = (g\circ f)_*\). In fact, the hypotheses are satisfied because for every injective (and even every flasque) \({\mathscr O}_X\)-module \({\mathscr I}\), the direct image \(f_*{\mathscr I}\) is flasque and hence acyclic for \(\Gamma (Y, -)\) and \(g_*\), resp.

This is the Grothendieck spectral sequence for the composition of functors \(\check{H}^0\) from the category of presheaves(!) on \(X\) to the category of abelian groups, and the inclusion \(\iota \) of the category of \({\mathscr O}_X\)-modules into the category of presheaves of \({\mathscr O}_X\)-modules. Note that \(\iota \) preserves the property of being injective because it has an exact left adjoint functor (namely sheafification). The composition is just the global section functor on the category of sheaves. One concludes by noting that \({\mathscr H}^q\) is the \(q\)-th right derived functor of \(\iota \), and that \(\check{H}^p({\mathscr U}, -)\) is the \(p\)-th right derived functor of \(\check{H}^0({\mathscr U}, -)\) (on the category of presheaves of \({\mathscr O}_X\)-modules. For the latter fact, the key is to show that for \({\mathscr I}\) an injective presheaf we have \(\check{H}^p({\mathscr U}, {\mathscr I})=0\) for all \(p {\gt} 0\). This follows by defining a sheaf version of the Čech complex and showing that it is exact in positive degrees; see  [ GW2 ] Lemma 21.76 and Lemma 21.77 and/or  [ H ] Ch. III, Proposition 4.3.

(1). This is the Grothendieck spectral sequence for the composition \(\Gamma (X, -)\circ \mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits _{{\mathscr O}_X}({\mathscr F}, -) = \operatorname{Hom}_{{\mathscr O}_X}({\mathscr F}, -)\). Note that for \({\mathscr I}\) injective, the sheaf \(\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits _{{\mathscr O}_X}({\mathscr F}, {\mathscr I})\) is flasque (given \(j\colon U\to X\) an open subscheme, and a homomorphism \({\mathscr F}_{|U}\to {\mathscr I}_{|U}\), we get \(j_!({\mathscr F}_{|U})\to {\mathscr I}\), and from the injectivity of \({\mathscr I}\) and the injection \(j_!({\mathscr F}_{|U})\to {\mathscr F}\) a map \({\mathscr F}\to {\mathscr I}\) extending the map given on \(U\)).

(2) Again, this is a Grothendieck spectral sequence, now for the composition \(\operatorname{Hom}_{{\mathscr O}_Z}({\mathscr F}, -)\circ \mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits ^{{\mathscr O}_Z}_{{\mathscr O}_X}({\mathscr O}_Z, -) = \operatorname{Hom}_{{\mathscr O}_X}(i_*{\mathscr F}, -)\). Recall that we have checked this equality in Section 3.1, where we expressed it by saing that \(\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits _{{\mathscr O}_X}^{{\mathscr O}_Z}({\mathscr O}_Z, -)\) is the right adjoint of \(i_*\). (Strictly speaking, we should write \(\mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits _{{\mathscr O}_X}^{{\mathscr O}_Z, q}({\mathscr O}_Z, {\mathscr G})\), but we omit the upper \({\mathscr O}_Z\) at this point.) To obtain the Grothendieck spectral sequence in this situation, we need to check that for \({\mathscr I}\) injective, \(\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits ^{{\mathscr O}_Z}_{{\mathscr O}_X}({\mathscr O}_Z, {\mathscr I})\) is injective. In fact, by the adjunction, \(\operatorname{Hom}(-, \mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits ^{{\mathscr O}_Z}_{{\mathscr O}_X}({\mathscr O}_Z, {\mathscr I})) = \operatorname{Hom}(i_* -, {\mathscr I})\) which is exact.

Consider the spectral sequence of Proposition 3.18 (1). For fixed \(n\), only finitely many terms of the \(E_2\) term play a role in computing the values \(E_r^{pq}\) with \(r\ge 2\), \(p+q=n\) (note that the values vanish anyway unless \(p, q\ge 0\)). We choose \(d_0\) so that for all \(d\ge d_0\), all \(0\le q {\lt} n\) and all \(p {\gt} 0\), the terms \(H^p(X, \mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits ^{q}({\mathscr F}, {\mathscr G})(d))\) vanish. (Such a \(d_0\) exists for each \(q\), since \(\mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits ^{q}({\mathscr F}, {\mathscr G})\) is a coherent \({\mathscr O}_X\)-module, see Algebraic Geometry 2, and the maximum of all these gives the desired bound.) But fixing such a \(d\), the \(E_2\) terms for all \(p, q\) with \(p+q=n\) are equal to the \(E_\infty \) terms and are all zero except when \(p=0\), \(q=n\). It follows that the limit term \(\operatorname{Ext}^n_{{\mathscr O}_X}({\mathscr F}, {\mathscr G}(d))\) equals \(E_2^{0, n} = \Gamma (X, \mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits ^n_{{\mathscr O}_X}({\mathscr F}, {\mathscr G}(d)))\).

Fix \(i\) and write \({\mathscr F}= \mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits _{{\mathscr O}_{\mathbb {P}^n_k}}^i({\mathscr O}_X, \omega )\). Since \({\mathscr F}\) is coherent, it is enough to show that \(H^0(\mathbb {P}^n_k, {\mathscr F}(d)) = 0\) for all \(d\gg 0\). By Corollary 3.19 and the duality theorem for \(\mathbb {P}^n_k\), we have

We can compute the cohomology group on the right as a cohomology group on \(X\). This has to vanish in degrees \({\gt}\dim X\) by the Grothendieck vanishing theorem. But \(n-i {\gt} \dim X\) is equivalent to \(i{\lt}r\).

Let \({\mathscr F}\) be a coherent \({\mathscr O}_X\)-module. We need to show that

We use the spectral sequence of Proposition 3.18 (2) to evaluate the term on the left. By the lemma, the \(E_2\) term of bidegree \((0, r)\) equals the \(E_\infty \) term, and all other terms with bidegree \((p, q)\) with \(p+q=r\) vanish, so that we obtain

By duality on \(\mathbb {P}^n\), the right hand term is isomorphic to \(H^{n-r}(\mathbb {P}^n, {\mathscr F})^\vee = H^{\dim (X)}(X, {\mathscr F})^\vee \). The identification is functorial in \({\mathscr F}\), and thus the theorem is proved.

If \(I\) contains an \(M\)-regular element \(x\), then multiplication by \(x\) on \(\operatorname{Hom}_A(A/I, M)\) at the same time is the zero map (since \(x\in I\)), and injective (since \(x\) is \(M\)-regular). Conversely, assume that all elements of \(I\) annihilate some element of \(M\). This is equivalent to saying that \(I\) is contained in the union of all associated prime ideals of \(M\) (i.e., all prime ideals of \(A\) of the form \(\operatorname{ann}(m) := \{ x\in A;\ xm=0\} \)). By prime avoidance, \(I\) is contained in one of them, say \(I\subseteq {\mathfrak p}=\operatorname{ann}(m)\). Then \(x\to xm\) induces an injection \(A/{\mathfrak p}\to M\), hence a non-zero map \(A/I\to M\).

(1). The minimal prime ideals of \(\operatorname{Supp}(M) = \{ {\mathfrak p}\in \operatorname{Spec}(A);\ M_{{\mathfrak p}}\ne 0\} \) (a closed subset of \(\operatorname{Spec}(A)\)) are associated prime ideals of \(M\). We now do induction on \(\operatorname{depth}_A(I, M)\). If \(\operatorname{depth}_A(I, M)=0\), then there is nothing to do. Otherwise, there exists an \(M\)-regular element \(x\in I\), so \(x\) does not lie in any of the minimal prime ideals in \(\operatorname{Supp}(M)\). Thus \(0\le \dim (M/xM) {\lt} \dim (M)\). By induction, if \(x_1, \dots , x_n\) is a regular sequence for \(M\), then \(\dim (M) \ge n\).

(2), (3). Let \(M\) be an \(A\)-module, and let \(x_1,\dots , x_n\in I\) be an \(M\)-regular sequence. We prove by induction on \(n\) that \(\operatorname{depth}_A(I, M) {\gt} n\) if and only if the given sequence can be extended to an \(M\)-regular sequence of length \(n+1\) in \(I\), if and only if \(\operatorname{Ext}^m_A(A/I, M) = 0\) for all \(m\le n\). The case \(n=0\) is clear by the above lemma. For \(n {\gt} 0\) the short exact sequence

induces an exact sequence

By induction, \(\operatorname{Ext}^{n-1}(A/I, M) = 0\). Since \(x_1\in I\), the map on the right of the above sequence also is \(=0\). That means that we have an isomorphism

By induction, we obtain a chain of isomorphisms

The claim now follows from the previous lemma.

We skip the proof of (4) for the moment; it follows from the connection with the Koszul complex – see below or see any of the references given above.

An \(M\)-regular sequence in \(I\) maps to an \(M_{\mathfrak p}\)-regular sequence under the natural homomorphism \(A\to A_{\mathfrak p}\) for any prime ideal \({\mathfrak p}\in V(I)\), so we have \(\le \). To show the other inequality, we can replace \(M\) by the quotient \(M/(x_1,\dots , x_n)M\) where \(x_\bullet \) is a maximal \(M\)-regular sequence in \(I\), und thus suppose that the left hand side is \(=0\). We then need to show that there exists \({\mathfrak p}\in V(I)\) with \(\operatorname{depth}_{A_{\mathfrak p}}(M_{{\mathfrak p}})=0\). As in the proof of Lemma 3.23, the assumption implies that \(I\) is contained in some associated prime ideal \({\mathfrak p}\), and we obtain an injective \(A\)-module homomorphism \(A/{\mathfrak p}\to M\). Tensoring with \(A_{\mathfrak p}\) preserves the injectivity, so we see that \(\operatorname{Hom}_{A_{\mathfrak p}}(\kappa ({\mathfrak p}), M_{\mathfrak p})\neq 0\), and Lemma 3.23 gives the result.

It is enough to prove that all stalks of this (coherent) sheaf at closed points vanish, so it is enough to show that for every closed point \(x\in \mathbb {P}^n_k\) and \(i{\lt}r\), writing \(A:={\mathscr O}_{\mathbb {P}^n_k, x}\),

Denoting by \(I\subseteq A\) the ideal corresponding to the closed subscheme \(X\), and noting that \({\mathscr E}\) is a locally free sheaf of finite rank, so that \({\mathscr E}_x\) is a finite free \(A\)-module, this means precisely that we need to show \(\operatorname{depth}_A(I, A) \ge r\). By Proposition 3.30, it is enough to show that \(\operatorname{depth}(A_{\mathfrak p})\ge r\) for all \({\mathfrak p}\in V(I)\). But \(A\) is regular (it is the localization of the polynomial ring in \(n\) variables over \(k\) at some maximal ideal), so \(\operatorname{depth}(A_{\mathfrak p}) = \dim (A_{\mathfrak p}) = n - \dim (A/{\mathfrak p})\), and the claim follows.

Part (1) follows directly from the definition. For Parts (2) and (3), we check the cases \(r=1, 2\). See the references for the general case, e.g., [ M2 ] Theorem 16.5. For \(r=1\) the assertion is clear immediately. For \(r=1\), the Koszul complex, tensored with \(M\), is the complex

Assume that \(f_1, f_2\) is an \(M\)-regular sequence. Then clearly the map \(M\to M^2\) is injective. Let \(x,y \in M\) with \(f_1x + f_2y = 0\), so \(y\) is mapped to zero under \(M/(f_1)\xrightarrow {f_2\cdot } M/(f_1)\) and hence \(y\in f_1M\), say \(y = f_1y'\). We find that \(f_1(x + f_2y') = 0\), so \(x = -f_2 y'\). Thus \((x,y) = (-f_2 y', f_1 y')\in {\rm Im}(M\to M^2)\).

Conversely, assume \(A\) is local, \(M\) is finitely generated and that \(H_1\) of the above complex vanishes. It is then straighforward to check that \(M/f_1M\xrightarrow {f_2} M/f_1M\) is injective. To prove that multiplication by \(f_1\) is an injection \(M\to M\), by the Lemma of Nakayama it is enough to show that multiplication by \(f_2\) on the (finitely generated, since \(A\) is noetherian) \(A\)-module \(\operatorname{Ker}(f_{1|M})\) is surjective. But if \(f_1x = 0\), then \((x, 0)\in A^2\) maps to zero and hence must have the form \((-f_2 x', f_1 x')\) for some \(x'\in A\), i.e., \(x'\in \operatorname{Ker}(f_{1|M})\) and \(x = -f_2 x'\).

(1). Since \(M\) is finitely generated, there is a surjection \(A^r\to M\). Let \(N\) be its kernel, a finitely generated \(A\)-module since \(A\) is noetherian. From the short exact sequence \(0\to N\to A^r\to M\to 0\), we obtain a long exact sequence

which gives us a splitting of our short exact sequence.

(2). There exists an exact sequence

with all \(P_i\) projective of finite rank and \(R\) a finitely generated \(A\)-module. We obtain that \(\operatorname{Ext}^1(R, N) = \operatorname{Ext}^{d+1}(M, N) = 0\) for all finitely generated \(N\), whence \(R\) is projective.

(3). We show that \(\operatorname{Ext}_A^i(M, N) = 0\) for all \(i {\gt} d\) and all finite \(A\)-modules \(N\) by descending induction on \(i\). The induction start holds since \(M\) has finite projective dimension (here we use that \(A\) is regular, Fact 3.26). For the induction step, consider a finite \(A\)-module \(N\) and a short exact sequence \(0\to K\to A^r\to N\to 0\). We obtain an exact sequence

For \(i {\gt} d\), the term on the left is \(0\) by assumption. By induction, the term on the right is \(0\). Thus the term in the middle vanishes, too.

(1). The functors on each side are \(\delta \)-functors, and we have the homomorphism (even an isomorphism) for \(i=0\) by the definition of a dualizing sheaf. It is therefore sufficient to show that the left hand side is a universal \(\delta \)-functor, which we show by proving that it is coeffaceable. In fact, given \({\mathscr F}\), we can write it as a quotient of some \({\mathscr O}_X(-d)^{\oplus N}\), and \(\operatorname{Ext}^i({\mathscr O}_X(-d), \omega ) = H^i(X, \omega (d))\) vanishes for \(i {\gt}0\) and \(d \gg 0\).

(2). Clearly, (iii) implies (ii) (cf. the following corollary).

To show (ii) \(\Rightarrow \) (iii), in view of Part (1) it is enough that the functor \({\mathscr F}\mapsto H^{N-i}(X, {\mathscr F})^\vee \) is a universal \(\delta \)-functor. But (ii) shows that it is coeffaceable.

To finish the proof, we show the equivalence of (i) and (ii). Recall that we have embedded \(X\subseteq \mathbb {P}^N_k\) in order to define \({\mathscr O}_X(1)\) (or, expressed using the terminology of very ample line bundles: fixing the “very ample” line bundle \({\mathscr O}_X(1)\) on \(X\) defines an embedding into projective space).

Note that for every locally free \({\mathscr O}_X\)-module \({\mathscr E}\), every \(i\) and \(d \gg 0\) we have, using duality on \(\mathbb {P}^N_k\) and Corollary 3.19,

In particular, it follows that \(H^i(X, {\mathscr E}(-d)) = 0\) for all \(d\gg 0\) if and only if \(\mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits ^{N-i}({\mathscr E}, \omega _{\mathbb {P}^N_k}(d)) = 0\) for all \(d\gg 0\).

(i) \(\Rightarrow \) (ii). Let \({\mathscr E}\) be a locally free \({\mathscr O}_X\)-module, and let \(x\in X\) be a closed point. We can consider \({\mathscr E}_x\) as an \({\mathscr O}_{X,x}\)-module, but also as an \({\mathscr O}_{\mathbb {P}^N_k, x}\)-module, and over either of these rings it has the same depth. Now the depth over \({\mathscr O}_{X,x}\) is \(\dim {\mathscr O}_{X,x} = \dim X = n\), since \(X\) by assumption is Cohen-Macaulay and \({\mathscr E}_x\) is free over \({\mathscr O}_{X,x}\). On the other hand, \({\mathscr O}_{\mathbb {P}^N_k, x}\) is regular so that the depth of \({\mathscr E}\), by the Auslander-Buchsbaum formula, equals \(N - {\rm projdim}_{{\mathscr O}_{\mathbb {P}^N_k, x}}({\mathscr E}_x)\). We find that \({\rm projdim}_{{\mathscr O}_{\mathbb {P}^N_k, x}}({\mathscr E}_x)\) equals the codimension of \(X\) inside \(\mathbb {P}^N_k\).

But then \(\mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits ^i_{{\mathscr O}_{\mathbb {P}^N_k}}({\mathscr E}, {\mathscr G})_x = \operatorname{Ext}({\mathscr E}_x, {\mathscr G}_x) = 0\) for all \(i {\gt} N-n\) and all \({\mathscr G}\). We apply this to \({\mathscr G}= \omega _{\mathbb {P}^N_k}(d)\) for \(d\gg 0\) and get, by 2, that \(H^i(X, {\mathscr E}(-d)) = 0\) for \(i {\lt} n\), as desired.

(ii) \(\Rightarrow \) (i). Conversely, applying (ii) to \({\mathscr E}= {\mathscr O}_X\), we obtain from 2 that \(\mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits ^{N-i}({\mathscr O}_X, \omega _{\mathbb {P}^N_k})(d) = 0\) for all \(d \gg 0\) and all \(i {\lt} n\). Thus all stalks of these sheaves vanish. But the stalk of the second entry is \(\cong {\mathscr O}_{\mathbb {P}^N_k} =: A\), so we get \(\operatorname{Ext}_A^{i}(A/I, A)=0\) for all \(i {\gt} N - n\), where \(I\) denotes the stalk at \(x\) of the ideal sheaf defining \(X\). This implies \({\rm projdim}_A(A/I) \le N-n\) (cf. Lemma 3.33), and via the Auslander-Buchsbaum formula, that \(\operatorname{depth}(A/I) \ge n\). But \(A/I = {\mathscr O}_{X,x}\) has dimension \(n\), so it follows that \({\mathscr O}_{X,x}\) is Cohen-Macaulay.

See  [ GW2 ] Theorem 19.30. The key fact from commutative algebra is the following: For a local noetherian ring \(A\) and an ideal \(I\) such that the quotient \(A/I\) is regular, the ideal \(I\) is generated by a regular sequence if (and only if) \(A\) is regular. See  [ BouAC ] Ch. X §2 no. 2, Cor. de Prop. 2.

If \(S = \operatorname{Spec}k\) is a field (or more generally, whenever \(X\) is a regular schemes), we can also argue as follows. Let \({\mathscr I}\) denote the ideal sheaf attached to \(i\). By Proposition 2.42 and Proposition 2.50 we know that \({\mathscr I}/{\mathscr I}^2\) is a locally free \({\mathscr O}_Z\)-module. Therefore \(i\) is regular by the remark following the definition of regular closed immersion.

We first consider the local situation, i.e., we fix \(x\in X\) and consider an affine open neighborhood \(\operatorname{Spec}A\) of \(x\), where \(Z\) is given by an ideal \(I\subseteq A\) that can be generated by a regular sequence \(f_1,\dots , f_r\). Let \(K_\bullet \) denote the Koszul complex for this sequence. In view of Proposition 3.32, our assumptions ensure that (after shrinking the neighborhood further, if necessary) \(K_\bullet \) is a projective resolution of \(A/I\). So we can use it to compute the Ext groups we are interested in. Let \(M\) be an \(A\)-module. Then we have

The second equality is an easy computation. (In fact, the Koszul complex is “self-dual” (shifting appropriately), [ GW2 ] Section (22.4).)

The final identification of this chain is obtained from the isomorphism

In view of Corollary 3.11, this gives the result in the affine case.

To conclude, we check that the local isomorphisms are independent of the choice of regular sequence generating \(I\). It is then clear that they glue to give the desired natural isomorphism. Given another regular sequence \(g_1,\dots , g_r\) generating \(I\), we can write \(g_j = \sum _i a_{ij}f_i\) for elements \(a_{ij}\in A\) and the residue classes of the \(g_i\) will be another basis of \(I/I^2\). We then have \(g_1\wedge \cdots \wedge g_r = \det ((a_{ij})_{i,j}) f_1\wedge \cdots \wedge f_r\).

On the other hand, one checks that the map \(A^r\to A^r\) given by the matrix \((a_{ij})_{i,j}\) induces a quasi-isomorphism between the Koszul complexes for the two regular sequences. The induced automorphism of \(M/IM\) is multiplication by \(\det ((a_{ji})_{i,j}) = \det ((a_{ij})_{i,j})\).

Fix a closed immersion \(i\colon X\to \mathbb {P}^N_k\), denote by \({\mathscr I}\) the corresponding ideal sheaf, and let \(\omega _{\mathbb {P}^N_k}\) be the dualizing sheaf of \(\mathbb {P}^N_k\). By Proposition 3.38, \(i\) is a regular immersion. Thus by Theorem 3.21 and Theorem 3.39, we have

By Proposition 2.42 the right hand side is isomorphic to \(\bigwedge \nolimits ^n \Omega _{X/k}\).

Consider the commutative diagram

where in the right hand column we have the categories of presheaves of abelian groups on \(X\) and \(Y\), respectively, \(f_*^p\) is the “presheaf direct image” functor (defined in the same way as \(f_*\) for sheaves), and where \(\iota \) is the inclusion and \(-^\sharp \) is the sheafification functor.

Since \(f_*^p\) and sheafification are exact functors, the Grothendieck spectral sequence for the composition of functors shows that \(R^if_*{\mathscr F}\) is the sheafification of the presheaf \(f^p_*R^i\iota {\mathscr F}\). Computing this presheaf (using an injective resolution of \({\mathscr F}\), and using that restriction to open subsets preserves the property of being injective) we obtain the result.

This follows from Proposition 3.41, the fact that flasque sheaves are acyclic for the global section functor and that the restriction of a flasque sheaf to an open is again flasque.

(1). We use the previous lemma. We denote by \({\mathscr H}^i\) the presheaf \(U\mapsto H^i(U, {\mathscr F})\) on \(X\), so that \(R^if_*{\mathscr F}\) is the sheafification of \(f_*{\mathscr H}^i\), as we have already seen. Let \(\varphi \colon A\to \Gamma (X, {\mathscr O}_X)\) be the ring homomorphism obtained from \(f\). For \(s\in A\) we obtain

One checks that these maps are compatible with restriction along inclusions \(D(t)\subseteq D(s)\). Thus the presheaf \(f_*{\mathscr H}^i\) and the sheaf \(H^i(X, {\mathscr F})^\sim \) agree on the basis of the topology given by principal open subsets, and that implies the claim.

If \(X\) is noetherian, one can alternatively argue as follows. The equality holds for \(i=0\), so it is enough to show that both sides are universal \(\delta \)-functors from the category of quasi-coherent \({\mathscr O}_X\)-modules to the category of quasi-coherent \({\mathscr O}_Y\)-modules. (It is important to restrict to quasi-coherent modules here since obviously the claim does not hold in degree \(0\) in general.) For the left hand side, this is clear. The right hand side is a \(\delta \)-functor, since this is true for the cohomology on \(X\) and since the \(\sim \)-operation is exact.

Now since \(X\) is noetherian, every quasi-coherent \({\mathscr O}_X\)-module can be embedded into a flasque quasi-coherent \({\mathscr O}_X\)-module (see  [ H ] Ch. III Cor. 3.6; also see  [ GW2 ] Section (22.18) for stronger results), and then it follows from Lemma 3.43 that the \(\delta \)-functor on the right hand side is effaceable.

Part (2) follows from Part (1) since we can compute the higher direct images locally on \(Y\).

Part (1) follows directly from the vanishing of cohomology of quasi-coherent modules on affine schemes in positive degrees. Parts (2) and (3) then follow from the Leray spectral sequences (Proposition 3.16).