# 3 Serre duality

General references: [ GW2 ] Ch. 25 and the references given there; [ AK ] , [ H ] III.7.

We now come back to Serre duality. The explicit computation of the cohomology groups \(H^i(\mathbb {P}^n_R, {\mathscr O}(d))\), \(R\) some ring, that we have done in *Algebraic Geometry 2* shows that for every line bundle \({\mathscr L}\) on \(X:= \mathbb {P}^n_R\) we have a perfect pairing

Here \(\omega _X := {\mathscr O}(-n-1) \cong \bigwedge ^n \Omega _{X/R}\). In particular, we obtain isomorphisms \(H^{n-i}(X, {\mathscr L}^\vee \otimes \omega _X)\cong H^i(X,{\mathscr L})^\vee \).

The goal of this section is to understand how this generalizes. We will (mostly) content ourselves with understanding the situation for a proper (or even projective) \(k\)-scheme \(X\) (where \(k\) is some field).

**(3.1) The abstract approach**

Using the machinery of derived categories and a suitable version of the Brown representability theorem for triangulated categories, Neeman has proved that for a morphism \(f\colon X\to S\) of noetherian (or more generally: qcqs) schemes, the derived pushforward functor \(Rf_*\colon D_{\rm qcoh}(X)\to D_{\rm qcoh}(S)\) admits a right adjoint \(f^\times \). For \(X\to \operatorname{Spec}(k)\) proper, we then call \(\omega _X^\bullet := f^\times {\mathscr O}_{\operatorname{Spec}k}\in D_{\rm qcoh}(X)\) the *dualizing complex* of \(X\). Since \(f^\times \) is by definition right adjoint to \(Rf_*\), for every \(F\in D_{\rm qcoh}(X)\) (and in particular for every quasi-coherent \({\mathscr O}_X\)-module \(F\)) we obtain the following very general form of Grothendieck-Serre duality,

The formula simplifies for example if \(F\) is a locally free \({\mathscr O}_X\)-module (because then \(\operatorname{Ext}^{-i}_{{\mathscr O}_X}(F, \omega _X^\bullet ) = H^{-i}(X, F^\vee \otimes _{{\mathscr O}_X}\omega _X^\bullet )\)) and especially if the complex \(\omega _X^\bullet \) is concentrated in a single degree. This is the case if \(X\) is smooth over \(k\), in which case \(\omega _X^\bullet = \left(\bigwedge ^{\dim X}\Omega _{X/k}\right)[\dim X]\).

Now consider a closed immersion \(i\colon X\to Y\) of \(S\)-schemes (where we again assume that all schemes are noetherian). See [ GW2 ] Section (25.8). To describe the functor \(i^\times \), we start with the following elementary result.

**Lemma 3.1**

Globalizing this, for a closed immersion \(i\colon Z\to X\) of schemes and an \({\mathscr O}_X\)-module \({\mathscr F}\), we write \(\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits ^{{\mathscr O}_Z}_{{\mathscr O}_X}({\mathscr O}_Z, {\mathscr F}) := i^*\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits _{{\mathscr O}_X}(i_*{\mathscr O}_Z, {\mathscr F})\).

**Lemma 3.2**

The lemma “formally” implies an analogous adjunction between the derived functors \(R\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits _{{\mathscr O}_X}^{{\mathscr O}_Z}({\mathscr O}, -)\) and \(Li_*\). Since \(i_*\) is exact, we can identify \(Li_* = i_* = Ri_*\). One checks that \(i_*R\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits _{{\mathscr O}_X}^{{\mathscr O}_Z}({\mathscr O}_Z, {\mathscr F}) = R\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits _{{\mathscr O}_X}({\mathscr O}_Z, {\mathscr F})\), i.e., when considered as an \({\mathscr O}_X\)-module, this is just the usual \(R\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits \) functor.

Note however that we do not immediately get a description of \(i^\times \) because \(i^\times \) is the right adjoint of \(Ri_*\colon D_{\rm qcoh}(Z)\to D_{\rm qcoh}(X)\), and in general an object of the form \(R\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits _{{\mathscr O}_X}^{{\mathscr O}_Z}({\mathscr O}_Z, {\mathscr F})\) will not lie in \(D_{\rm qcoh}(Z)\) (cf. Example 3.13 below). This is however true under an additional assumption, namely for \(F\) in \(D^+_{\rm qcoh}(X)\) (or in \(D^+_{\rm coh}(X)\)) the complex \(R\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits _{{\mathscr O}_X}^{{\mathscr O}_Z}({\mathscr O}_Z, F)\) lies in \(D^+_{\rm qcoh}(Z)\) (or in \(D^+_{\rm coh}(Z)\), respectively). Therefore we have

**Proposition 3.3**

This gives us a strategy of constructing a dualizing complex/“dualizing sheaf” for closed subschemes of projective space. Formally, we will not use any of the results above; they only serve as a motivation/explanation of why the definitions below are sensible.

**(3.2) The dualizing sheaf of a projective scheme 1**

We start by slightly generalizing the statement of Serre duality for projective space, as follows.

Nov. 22,

2023

**Proposition 3.4**

Let \(k\) be a field, \(n\ge 1\), \(X=\mathbb {P}^n_k\). Write \(\omega _X = \bigwedge ^n\Omega _{X/k} \cong {\mathscr O}_X(-n-1)\).

We have \(H^n(X, \omega _X) \cong k\) (and we fix one such isomorphism).

For every coherent \({\mathscr O}_X\)-module \({\mathscr F}\), the natural pairing

\[ \operatorname{Hom}({\mathscr F}, \omega _X)\times H^n(X,{\mathscr F})\to H^n(X,\omega _X)\cong k \]is perfect, i.e., it induces an isomorphism \(\operatorname{Hom}({\mathscr F},\omega _X)\cong H^n(X, {\mathscr F})^\vee \).

For every \(i\ge 0\), we have isomorphisms, functorial in \({\mathscr F}\),

\[ \operatorname{Ext}^i({\mathscr F}, \omega _X)\cong H^{n-i}(X, {\mathscr F})^\vee . \]

(See below for a brief reminder on the \(\operatorname{Ext}\) functor.)

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 finite direct 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.)

We use Parts (1) and (2) of the previous proposition to define the notion of *dualizing sheaf* for an arbitrary proper \(k\)-scheme \(X\). (This already characterizes a dualizing sheaf, and in fact we will see below that Part (3) will not hold in general; this is related to the fact that the dualizing sheaf captures only one cohomology object of the dualizing complex of the previous section, so unless that complex is concentrated in a single degree, the dualizing sheaf will not capture the full duality.)

**Definition 3.5**

*dualizing sheaf*on \(X\). In other words, a dualizing sheaf is a coherent \({\mathscr O}_X\)-module \(\omega _X\), together with a homomorphism \(H^n(X, \omega _X)\to k\) (the element of \(H^n(X, \omega _X)^\vee \) corresponding to \(\operatorname{id}_{\omega _X}\), called the

*trace map*), if for every coherent \({\mathscr O}_X\)-module the natural pairing

Since a dualizing sheaf is defined as the object representing a certain functor, it is clear that it is unique up to unique isomorphism (if it exists).

By the above, \(\bigwedge ^n\Omega _{\mathbb {P}^n_k/k}\) is a dualizing sheaf on projective space \(\mathbb {P}^n_k\). We will see below that more generally for every *smooth* projective \(k\)-scheme \(X\) of dimension \(n\), \(\bigwedge ^n\Omega _{X/k}\) is a dualizing sheaf on \(X\).

**Theorem 3.6**

We will not prove the theorem here, but will rather concentrate on the case of projective schemes. (In terms of the abstract theory, writing \(\omega _X^\bullet \) for the dualizing complex of \(X\) defined above, one can show that \(H^i(\omega _X^\bullet ) = 0\) for all \(i\not \in [-\dim (X), 0]\); it follows that \(H^{-\dim (X)}(\omega _X^\bullet )\) is a dualizing sheaf on \(X\).)

**(3.3) \(\mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits \) sheaves**

By and large we follow [ H ] III.6. See also [ GW2 ] Sections (F.52), (21.18), (21.21), (22.17) for more general results in the context of derived categories.

**Definition 3.7**

Let \(\mathcal A\) be an abelian category with enough injectives. We define the \(\operatorname{Ext}\) functor from \(\mathcal A\) to the category of abelian groups (for \({\mathscr F}\) in \(\mathcal A\)) as \(\operatorname{Ext}^i({\mathscr F}, -) = R^i\operatorname{Hom}({\mathscr F}, -)\). (One can show that

\[ \operatorname{Ext}^i({\mathscr F}, {\mathscr G}) = \operatorname{Hom}_{D(\mathcal A)}({\mathscr F}, {\mathscr G}[i]). \]This identity can be used as a definition of \(\operatorname{Ext}\) groups for arbitrary abelian categories.)

Now let \(X\) be a ringed space. For an \({\mathscr O}_X\)-module \({\mathscr F}\) we define the \(\mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits \) sheaf functor from \(({\mathscr O}_X\text{-Mod})\) to \(({\mathscr O}_X\text{-Mod})\) by \(\mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits ^i({\mathscr F}, -) = R^i\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits ({\mathscr F}, -)\).

More explicitly, the definition means that we can compute \(\operatorname{Ext}^i({\mathscr F}, {\mathscr G})\) using an injective resolution of \({\mathscr G}\).

**Proposition 3.8**

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.)

Nov. 27,

2023

Since \(\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits _{{\mathscr O}_X}({\mathscr O}_X, -)\) is the identity functor, we have \(\mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits ^i({\mathscr O}_X, {\mathscr G}) = {\mathscr G}\) if \(i=0\), and \(= 0\) if \(i {\gt} 0\). Similarly, we have \(\operatorname{Ext}^i({\mathscr O}_X, {\mathscr G}) = H^i(X, {\mathscr G})\) (where we use that \(H^i\) which we defined as the derived functor of \(\Gamma \colon {\rm Ab}_X\to {\rm Ab}\) restricts to the derived functor of \(\Gamma \colon ({\mathscr O}_X\text{-Mod} \to {\rm Ab}\)).

**Remark 3.9**

Since the functor \(\operatorname{Hom}(-, {\mathscr I})\) for an injective \({\mathscr O}_X\)-module \({\mathscr I}\) is exact, for an \({\mathscr O}_X\)-module \({\mathscr G}\) the family \((\operatorname{Ext}^i(-, {\mathscr G}))_i\) is a \(\delta \)-functor (in particular, we obtain a long exact cohomology sequence when we plug in a short exact sequence in the first entry). A similar remark holds for \(\mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits \) sheaves.

Similarly, one shows that \(\operatorname{Ext}^i({\mathscr F}, {\mathscr G})\) and \(\mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits ^i({\mathscr F}, {\mathscr G})\) can be computed using a projective resolution of \({\mathscr F}\). This is useful working in the derived category of modules over a ring. But when working with \({\mathscr O}_X\)-modules for a ringed space \(X\) (even a noetherian scheme) in general projective objects are very rare. Note that even the structure sheaf \({\mathscr O}_X\) is not a projective \({\mathscr O}_X\)-module in general.

The previous two remarks show that an object \({\mathscr F}\) of an abelian category \(\mathcal A\) with enough projectives is projective if and only if \(\operatorname{Ext}^1({\mathscr F}, {\mathscr G}) = 0\) for all \({\mathscr G}\). More generally, \({\mathscr F}\) has projective dimension \(n\) (i.e., admits a projective resolution of length \(n\), but not of any smaller length) if and only if \(\operatorname{Ext}^{n+1}({\mathscr F}, {\mathscr G})=0\) for all \({\mathscr G}\). (Take a projective resolution \({\mathscr P}_\bullet \to {\mathscr F}\to 0\), then \(\cdots \to {\mathscr P}_i\to {\rm Im}({\mathscr P}_i\to {\mathscr P}_{i-1}):={\mathscr I}_i\to 0\) is a projective resolution of \({\mathscr I}_i\) and \(\operatorname{Ext}^{n+1-i}({\mathscr I}_i, {\mathscr G})\cong \operatorname{Ext}^{n+1}({\mathscr F}, {\mathscr G})\). Now do induction.)

Therefore it is useful to study whether other types of resolutions (specifically, a resolution by locally free \({\mathscr O}_X\)-modules of finite rank) can be used for computing \(\mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits \) sheaves.

**Proposition 3.10**

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.

*Question.* Do you see why for the \(\operatorname{Ext}\) groups this cannot possibly hold? Asked differently, what goes wrong in the proof, if we replace \(\mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits \) by \(\operatorname{Ext}\) and \(\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits \) by \(\operatorname{Hom}\)? (*Hint.* Where have we used that the \({\mathscr E}_i\) are locally free?)

**Corollary 3.11**

Let \(A\) be a noetherian ring and let \(X=\operatorname{Spec}A\). Let \(M\), \(N\) be \(A\)-modules. Assume that \(M\) is finitely generated. For every \(i\ge 0\) we have

\[ \mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits ^i_{{\mathscr O}_X}(\widetilde{M}, \widetilde{N}) \cong \operatorname{Ext}^i_A(M, N)^\sim . \]Let \(X\) be a noetherian scheme and let \({\mathscr F}\), \({\mathscr G}\) be coherent \({\mathscr O}_X\)-modules. Then for all \(i\ge 0\) the \({\mathscr O}_X\)-module \(\mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits _{{\mathscr O}_X}^i({\mathscr F}, {\mathscr G})\) is coherent.

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.

*It is also true, and can be shown using the local-to-global spectral sequence for Ext (see below) and the vanishing of cohomology of quasi-coherent modules on affine schemes in positive degrees, that in Part (1) of the Corollary we have \(\Gamma (X, \mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits ^i_{{\mathscr O}_X}(\widetilde{M}, \widetilde{N})) = \operatorname{Ext}_{{\mathscr O}_X}(\widetilde{M}, \widetilde{N})\), but it does not seem easier/easy to directly identify that Ext group (which has to be computed in the category of all \({\mathscr O}_X\)-modules) with \(\operatorname{Ext}^i_A(M, N)\) (which we can view as \(\operatorname{Ext}\) of \(\widetilde{M}\) and \(\widetilde{N}\) in the category of all quasi-coherent \({\mathscr O}_X\)-modules). I did not explain this well in class.*

**Corollary 3.12**

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, \(\mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits ^i({\mathscr F}, {\mathscr G}) =H^i(\mathop{{\mathscr H}\mkern -5mu\mathit{om}}\nolimits ({\mathscr P}_\bullet , {\mathscr G}))\), 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).

**Example 3.13**

**Proposition 3.14**

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.

**Excursion: Spectral sequences 3.15**

Nov. 29,

2023

**Proposition 3.16**

Let \(f\colon X\to Y\) be a morphism of ringed spaces and let \({\mathscr F}\) be an \({\mathscr O}_X\)-module. There is a convergent spectral sequence

\[ E_2^{pq} = H^p(Y, R^qf_*{\mathscr F})\Longrightarrow H^{p+q}(X, {\mathscr F}). \]Let \(f\colon X\to Y\), \(g\colon Y\to Z\) be morphisms of ringed spaces. Let \({\mathscr F}\) be an \({\mathscr O}_X\)-module. There is a convergent spectral sequence

\[ E_2^{pq} = R^pg_*(R^qf_*{\mathscr F}) \Longrightarrow R^{p+q}(g\circ f)_*{\mathscr F}. \]

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.

**Proposition 3.17**

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.

**Proposition 3.18**

(The local-to-global spectral sequence for \(\operatorname{Ext}\)) Let \(X\) be a ringed space and let \({\mathscr F}\), \({\mathscr G}\) be \({\mathscr O}_X\)-modules. We have a convergent spectral sequence

\[ E_2^{pq} = H^p(X, \mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits ^q({\mathscr F}, {\mathscr G})) \Longrightarrow \operatorname{Ext}^{p+q}({\mathscr F}, {\mathscr G}). \]Let \(i\colon Z\to X\) be a closed immersion of schemes (or of arbitrary ringed spaces), let \({\mathscr F}\) be an \({\mathscr O}_Z\)-module and \({\mathscr G}\) an \({\mathscr O}_X\)-module. We have a convergent spectral sequence

\[ E_2^{pq} = \operatorname{Ext}_{{\mathscr O}_Z}^p({\mathscr F}, \mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits _{{\mathscr O}_X}^q({\mathscr O}_Z, {\mathscr G})) \Longrightarrow \operatorname{Ext}_{{\mathscr O}_X}^{p+q}(i_*{\mathscr F}, {\mathscr G}). \]

(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.

**Corollary 3.19**

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 \(E_2^{0, n} = \Gamma (X, \mathop{{\mathscr E}\mkern -5mu\mathit{xt}}\nolimits ^n_{{\mathscr O}_X}({\mathscr F}, {\mathscr G}(d)))\).

**(3.4) The dualizing sheaf of a projective scheme 2**

Dec. 4,

2023

We need the following lemma. See below (Corollary 3.31) for a different proof of the lemma.

**Lemma 3.20**

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 \(i{\lt}r\).

**Theorem 3.21**

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.

**(3.5) Cohen-Macaulay schemes**

References: [ GW2 ] Ch. 19; [ M2 ] Ch. 16, 17, [ BH ] 1.1, 1.2

**Definition 3.22**

A sequence \(f_1,\dots , f_r\in A\) is called a

*regular sequence for \(M\)*(or an*\(M\)-regular sequence*) iffor all \(i\) the element \(f_i\) is \(M/(f_1,\dots , f_{i-1})\)-regular, i.e., multiplication by \(f_i\) defines an injective endomorphism of \(M/(f_1,\dots , f_{i-1})\) and

\(M/(f_1,\dots , f_r)\ne 0\).

Let \(I\) be an ideal of \(A\). The

*depth*of \(M\) with respect to \(I\) is the maximum length of an \(M\)-regular sequence of elements contained in \(I\). We denote it by \(\operatorname{depth}_A(I, M)\). If \(I\) is the maximal ideal of a local ring \(A\), we write \(\operatorname{depth}_A(M):=\operatorname{depth}_A(I, M)\), and we write \(\operatorname{depth}(A) = \operatorname{depth}_A(A)\).

We start with the following lemma. In the proof of the lemma (and also afterwards) we use the notion of *associated prime ideal* of a module. See
[
M2
]
§6. (Over non-noetherian rings, the notion of weakly associated prime ideal is sometimes more useful, see
[
Stacks
]
0546.)

**Lemma 3.23**

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 \(A/I\to M\).

**Proposition 3.24**

We have

\[ \operatorname{depth}_A(I, M) \le \dim (M) := \dim (\operatorname{Supp}(M)). \]We have

\[ \operatorname{depth}_A(I, M) = \min \{ i;\ \operatorname{Ext}^i_A(A/I, M) \ne 0 \} . \]Every maximal \(M\)-regular sequence in \(I\) (i.e., a regular sequence that cannot be further extended) consists of \(\operatorname{depth}_A(I, M)\) elements.

If \(A\) is local, then the property of a sequence \(f_1,\dots , f_r\) of being regular is independent of the order of the \(f_i\). (But not for general noetherian rings \(A\).)

(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.

Dec. 6,

2023

For an \(A\)-module \(M\) we denote by \({\rm projdim}_A(M)\) the projective dimension of \(M\), i.e., the minimal length of a resolution \(0\to P_\ell \to \cdots \to P_0\to M\to 0\) of \(M\) by projective \(A\)-modules.

**Theorem 3.25**

**Fact 3.26**

A local noetherian ring \(A\) is regular if and only if every finitely generated \(A\)-module has finite projective dimension ( [ M2 ] Theorem 19.2). More precisely, for \(A\) regular local, the projective dimension of finite \(A\)-modules is bounded by \(\dim A\). (Below we will only use the latter statement, which is the easier direction of the result. The key point is to show that for any local noetherian ring \(R\), the projective dimension of any finite \(R\)-module is at most the projective dimension of the residue class field, considered as an \(R\)-module. This statement follows without much work from a version of the local criterion of flatness, see e.g. [ AK ] III.5. If \(R\) is regular, then the residue class field has finite projective dimension, since the Koszul complex (see below) is a projective resolution of length \(\dim (R)\).)

Since the formation of Ext for finite modules over a noetherian ring commutes with localization, it follows that every finite module over a regular noetherian ring of finite dimension (but not necessarily local) has finite projective dimension.

**Definition 3.27**

A local noetherian ring \(A\) is called a

*Cohen-Macaulay ring*, if \(\operatorname{depth}(A) = \dim (A)\).A noetherian ring \(A\) is called a

*Cohen-Macaulay ring*, if for every \({\mathfrak p}\in \operatorname{Spec}(A)\) the localization \(A_{{\mathfrak p}}\) is Cohen-Macaulay.A locally noetherian scheme \(X\) is called a

*Cohen-Macaulay scheme*, if for every \(x\in X\) the local ring \({\mathscr O}_{X,x}\) is Cohen-Macaulay.

One can show that localizations of Cohen-Macaulay rings are again Cohen-Macaulay. In particular in Part (2) of the definition it is enough to check localizations with respect to maximal ideals.

**Example 3.28**

Every \(0\)-dimensional noetherian ring is Cohen-Macaulay.

Let \(A\) be a \(1\)-dimensional noetherian ring. Then \(A\) is Cohen-Macaulay if and only if \(A\) does not have any “embedded associated prime ideals” (i.e., non-minimal associated prime ideals). In particular, if \(A\) is reduced, then \(A\) is Cohen-Macaulay. (If \(A\) is reduced, then every associated prime ideal is minimal. Indeed, assume there are prime ideals \({\mathfrak p}\subsetneq {\mathfrak q}= \operatorname{ann}(x)\). Then \(x{\mathfrak q}= 0\subseteq {\mathfrak p}\) and hence \(x\in {\mathfrak p}\subset {\mathfrak q}\), but then \(x^2 = 0\).)

For any \(d\ge 2\), there exist noetherian rings of dimension \(d\) which are not Cohen-Macaulay. For example, for \(k\) a field, neither \(k[X, Y, Z]/(XY, XZ)\) (not equidimensional, cf. below), nor \(k[W, X, Y, Z]/(WY, WZ, XY, XZ)\) are Cohen-Macaulay.

**Facts on Cohen-Macaulay rings 3.29**

Every regular noetherian ring is Cohen-Macaulay. (It is not too hard to see that every minimal generating system of the maximal ideal of a regular local ring is a regular sequence.)

Every connected Cohen-Macaulay scheme locally of finite type over a field is equi-dimensional, i.e., all its irreducible components have the same dimension.

If \(A\) is a Cohen-Macaulay ring, then \(A\) does not have embedded associated prime ideals (i.e., all associated prime ideals of the \(A\)-module \(A\) are minimal prime ideals of \(A\)). In particular, if \(X\) is a generically reduced Cohen-Macaulay scheme, then \(X\) is reduced.

If \(A\) is a local Cohen-Macaulay ring with maximal ideal \(\mathfrak m\), then \(f_1, \dots , f_r\in \mathfrak m\) form an \(A\)-regular sequence if and only if \(\dim A/(f_1,\dots , f_r) = \dim (A) - r\). In this case, the quotient \(A/(f_1,\dots , f_r)\) is again Cohen-Macaulay.

We close the section by a local proof of Lemma 3.20. *This was skipped in the class.*

**Proposition 3.30**

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.

**Corollary 3.31**

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.

**(3.6) The Koszul complex**

References: [ GW2 ] Ch. 19; [ M2 ] Ch. 16, [ BH ] 1.6

Let \(A\) be a ring. Let \(e_1,\dots , e_r\) denote the standard basis of the free \(A\)-module \(A^r\). For \(f_\bullet = (f_1,\dots , f_r)\) with \(f_i\in A\) we define the *Koszul complex* as the (chain) complex \(K_\bullet (f_\bullet )\)

(with \(\bigwedge ^p A^r\) in degree \(p\), i.e., the differential decreases the degree) with differentials

One checks that this in fact defines a complex. For an \(A\)-module \(M\) we write \(K_\bullet (f_\bullet , M):=K_\bullet (f_\bullet )\otimes _A M\) and denote by \(H_p(f_\bullet , M)\) the homology groups.

**Proposition 3.32**

We have \(H_0(f_\bullet , M) = M/(f_1,\dots , f_r)M\).

If \(f_1,\dots , f_r\) is an \(M\)-regular sequence, then \(H_p(f_\bullet , M) = 0\) for all \(p {\gt} 0\).

If \(A\) is a noetherian local ring with maximal ideal \({\mathfrak m}\), \(M\ne 0\) is finitely generated, \(f_1,\dots , f_r\in {\mathfrak m}\) and \(H_1(f_\bullet , M)=0\), then \(f_1,\dots , f_r\) is an \(M\)-regular sequence.

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 \(r=2\), 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 \(x = -f_2 x'\).

**(3.7) The dualizing sheaf of a projective scheme 3**

We can now prove that projective Cohen-Macaulay schemes “satisfy Serre duality in all degrees”, and that this property characterizes them. In terms of the dualizing complex, connected Cohen-Macaulay schemes \(X\) are characterized by the property that the dualizing complex is concentrated in a single degree (namely \(-\dim X\)). Since “full duality” always holds for the dualizing complex, from this point of view it is clear that for connected Cohen-Macaulay schemes it holds for the dualizing sheaf.

Dec. 11,

2023

We start with the following strengthening, for finitely generated modules over noetherian rings, of Remark 3.9.

**Lemma 3.33**

If \(\operatorname{Ext}_A^1(M, N) = 0\) for all finitely generated \(A\)-modules \(N\), then \(M\) is projective.

If \(\operatorname{Ext}_A^{d+1}(M, N) = 0\) for all finitely generated \(A\)-modules \(N\), then \({\rm projdim}(M) \le d\).

Now assume that \(A\) is a regular ring and that \(\dim A {\lt} \infty \). If \(\operatorname{Ext}_A^i(M, A) = 0\) for all \(i {\gt} d\), then \({\rm projdim}(M) \le d\).

(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.

**Theorem 3.34**

Let \(k\) be a field, and let \(X\) be a connected projective \(k\)-scheme of dimension \(n\). Let \({\mathscr O}_X(1)\) denote the pullback of the corresponding twisting sheaf under some closed embedding of \(X\) into a projective space over \(k\). Let \(\omega _X\) denote the dualizing sheaf of \(X\).

For all coherent \({\mathscr O}_X\)-modules \({\mathscr F}\) and all \(i\ge 0\), there are \(k\)-vector space homomorphisms, functorial in \({\mathscr F}\),

\[ \operatorname{Ext}^i({\mathscr F}, \omega )\to H^{n-i}(X, {\mathscr F})^\vee . \]The following are equivalent:

The scheme \(X\) is Cohen-Macaulay.

For every locally free \({\mathscr O}_X\)-module \({\mathscr E}\) of finite rank, every \(i {\lt} n\) and \(d\gg 0\), we have

\[ H^i(X, {\mathscr E}(-d)) = 0. \]The morphisms \(\operatorname{Ext}^i({\mathscr F}, \omega )\to H^{n-i}(X, {\mathscr F})^\vee \) of Part (1) are isomorphisms for all \(i\) and all coherent \({\mathscr O}_X\)-modules \({\mathscr F}\).

(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.

**Corollary 3.35**

**(3.8) The dualizing sheaf of a regularly embedded projective scheme**

Dec. 18,

2023

**Definition 3.36**

*regular*if every \(z\in Z\) admits an affine open neighborhood \(U\subseteq X\) such that the ideal \(\Gamma (U, {\mathscr I})\subset \Gamma (U, {\mathscr O}_X)\) is generated by a regular sequence.

In the situation of the definition, the \({\mathscr O}_Z\)-module \({\mathscr I}/{\mathscr I}^2\) is locally free (locally, a regular sequence generating the ideal induces a basis). See [ M2 ] Theorem 16.2 or [ BouA ] Ch. X §9 no. 7 Thm. 1. Conversely, if \(X\) is regular and \({\mathscr I}/{\mathscr I}^2\) is locally free, then the closed immersion is regular, see [ M2 ] Theorem 19.9.

From Remark 3.29 (4) we obtain the following result.

**Proposition 3.37**

**Proposition 3.38**

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.

More generally one can show that for a \(k\)-scheme \(X\) the property of admitting a regular immersion into a smooth \(k\)-scheme is a property of the local rings \({\mathscr O}_{X,x}\) (being *lci rings*, where lci stands for *locally complete intersection*). So if \(X\) admits a regular closed embedding into \(\mathbb {P}^n_k\) (or any smooth \(k\)-scheme) every closed immersion of \(X\) into a projective space (or in fact into any smooth \(k\)-scheme) is regular. See
[
GW2
]
Corollary 19.44, Proposition 19.49, Proposition 19.50.

**Theorem 3.39**

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 \(\det ((a_{ji})_{i,j}) = \det ((a_{ij})_{i,j})\).

**Corollary 3.40**

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 \(\bigwedge \nolimits ^n \Omega _{X/k}\).

**(3.9) Higher direct images**

Dec. 20,

2023

*higher direct image*(or

*derived direct image*) functors.

As a special case, if \(Y\) is a point, then \(f_*\) can be identified with the global section functor, and then \(R^if_*\) is the cohomology functor \(H^i(X, -)\).

**Proposition 3.41**

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.

**Corollary 3.42**

**Lemma 3.43**

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.

Similar as for cohomology groups, the previous lemma together with the fact that injective sheaves are flasque implies that for \({\mathscr O}_X\)-modules we can compute the higher direct images in the category of \({\mathscr O}_Y\)-modules.

**Proposition 3.44**

As a preparation for showing that the higher direct image functor for qcqs morphisms preserves quasi-coherence, we need the following lemma. The case \(i=0\) is a characterizing property of quasi-coherence. The general case can be proved using tools that we have at hand, but it requires several steps; we skip the proof here.

**Lemma 3.45**

**Proposition 3.46**

Assume that \(Y\) is affine. Then

\[ R^if_*{\mathscr F}\cong H^i(X, {\mathscr F})^\sim . \]The \({\mathscr O}_Y\)-module \(R^if_*{\mathscr F}\) is quasi-coherent.

(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 \(Y\).

**(3.10) Relative versions of some previous results**

From the Leray spectral sequence one obtains the following result. We have already proved Part (2) for closed immersions \(f\) in *Algebraic Geometry 2*.

**Corollary 3.47**

We have \(R^if_*{\mathscr F}= 0\) for all \(i {\gt} 0\).

We have natural isomorphisms

\[ H^p(Y, f_*{\mathscr F}) \cong H^p(X, {\mathscr F}). \]For every morphism \(g\colon Y\to Z\) of schemes we have functorial isomorphisms

\[ R^i(g\circ f)_*{\mathscr F}\cong R^ig_*(f_*{\mathscr F}). \]

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).

The finiteness results for cohomology of coherent modules on projective schemes have the following relative version.

**Proposition 3.48**

For \(d\) sufficiently large, the natural homomorphism \(f^*f_*{\mathscr F}(d)\to {\mathscr F}(d)\) is surjective.

For all \(i\ge 0\), the higher direct image \(R^if_*{\mathscr F}\) is coherent.

For \(i{\gt}0\) and \(d\) sufficiently large, \(R^if_*{\mathscr F}(n) = 0\).

Finally, we state a relative version of duality. See [ Kl ] , Theorem 20, Theorem 21, Corollary 25, for an elementary (i.e., not using derived categories) proof and several more general/more precise results, including a version for coherent, not necessarily locally free, modules; see [ GW2 ] Ch. 25, e.g., Prop. 25.26, for a more general result in terms of the dualizing complex.

**Theorem 3.49**

There exists a coherent \({\mathscr O}_X\)-module \(\omega _f\) (the

*relative dualizing sheaf*) such that\[ R^if_*(\omega _f\otimes _{{\mathscr O}_X}{\mathscr E}^\vee )\cong (R^{r-i}f_*{\mathscr E})^\vee \]for every \(i=0,\dots , r\) and every locally free \({\mathscr O}_X\)-module \({\mathscr E}\) of finite rank (and such that \(\omega _f\) gives rise to a similar, but slightly more difficult to state duality for arbitrary coherent \({\mathscr O}_X\)-modules (not necessarily locally free), which also characterizes \(\omega _f\) up to unique isomorphism).

If \(X\) and \(Y\) are smooth and of finite type over some noetherian base scheme \(S\) of relative dimensions \(d_X\) and \(d_Y\), respectively, then

\[ \omega _f = \bigwedge \nolimits ^{d_X}\Omega _{X/S}\otimes f^*\left(\bigwedge \nolimits ^{d_Y}\Omega _{Y/S}\right)^\vee . \]