Lecture topic
This post goes through the statement and proof of Serre duality for arbitrary projective schemes, as presented in Chapter III.7 of Hartshorne. Only the necessary tools and definitions to prove the statement are introduced.
Recall a scheme is a topological space $X$ and a sheaf of rings $\mathcal O_X$ such that for every open set $U\subset X$, $\mathcal O_X(U)\cong \Spec(R)$ for some ring $R$. Its dimension is its dimension as a topological space. A projective scheme is a scheme where $X\subset \P^n$. A sheaf (or scheme) over a scheme $X$ is a sheaf (or scheme) $Y$ and a morphism $Y\to X$. Recall also the sheafification $\widetilde {\mathcal F}$ of a presheaf $\mathcal F$.
Definition 1: Let $\mathcal F$ be a sheaf over a projective scheme $X$. Then $\mathcal F$ is
Recall a scheme is a topological space $X$ and a sheaf of rings $\mathcal O_X$ such that for every open set $U\subset X$, $\mathcal O_X(U)\cong \Spec(R)$ for some ring $R$. Its dimension is its dimension as a topological space. A projective scheme is a scheme where $X\subset \P^n$. A sheaf (or scheme) over a scheme $X$ is a sheaf (or scheme) $Y$ and a morphism $Y\to X$. Recall also the sheafification $\widetilde {\mathcal F}$ of a presheaf $\mathcal F$.
Definition 1: Let $\mathcal F$ be a sheaf over a projective scheme $X$. Then $\mathcal F$ is
- proper if it is the image of a proper morphism (separated, finite type, universally closed),
- quasi-coherent if there exists a cover $\{U_i=\Spec(A_i)\}$ of $X$ such that $\mathcal F|_{U_i} = \widetilde{M_i}$ for some $A_i$-module $M_i$,
- coherent if it is quasi-coherent and each $M_i$ is finitely-generated as an $A_i$-module,
- locally free if for every $x\in X$, there exists $U\owns x$ open such that $\mathcal F|_U = \bigoplus_{i\in I} \mathcal O_X|_U$,
- very ample if there is an immersion $i:X\to \P^n$ for some $n$ such that $i^*\mathcal O(1) \cong \mathcal F$.
Often we say $\mathcal F$ is very ample if it has "enough sections," as $\P^n$ has many sections.
Remark 1: Recall some basic definitions of the $\Ext$ functor. Let $\mathcal F,\mathcal G$ be sheaves of $\mathcal O_X$-modules, and $\mathcal L$ a locally free sheaf of finite rank. Then:
Remark 1: Recall some basic definitions of the $\Ext$ functor. Let $\mathcal F,\mathcal G$ be sheaves of $\mathcal O_X$-modules, and $\mathcal L$ a locally free sheaf of finite rank. Then:
- $\Ext^i(\mathcal O_X,\mathcal F) \cong H^i(X,\mathcal F)$ for all $i\>0$ (Proposition III.6.3)
- $\Ext^i(\mathcal F\otimes \mathcal L,\mathcal G) \cong \Ext^i(F,\mathcal L^\vee\otimes \mathcal G)$ (Proposition III.6.7)
- $\Ext^i_{\mathcal O_X}(\mathcal F_x,\mathcal G_x) \cong \mathcal Ext (\mathcal F,\mathcal G)_x $ (Proposition III.6.8)
- $\Ext^i(\mathcal F,\mathcal G(q)) \cong \Gamma(X,\mathcal Ext^i(\mathcal F,\mathcal G(q))$ (Proposition III.6.9)
- $\mathcal Ext^i(\mathcal F\otimes \mathcal L,\mathcal G) \cong \mathcal Ext^i(F,\mathcal L^\vee\otimes \mathcal G) \cong \mathcal Ext^i(\mathcal F,\mathcal G)\otimes \mathcal L^\vee$
- $\mathcal Ext^0(\mathcal O_X,\mathcal F) \cong \mathcal F$
- $\mathcal Ext^i(\mathcal O_X,\mathcal F) \cong 0$ for all $i>0$
Recall that a local ring of a scheme $X$ is $\mathcal O_{X,x}$ for $x\in X$. It is equivalently a ring with a unique maximal left or right ideal. A regular local ring is a local ring $R$ whose maximal ideal is generated by $\dim(R)$ elements.
Preliminary definitions and lemmas
Let $A,B$ be abelian categories (recall this means kernels and cokernels exist).
Definition 2: A $\delta$-functor between $A$ and $B$ is a collection of functors $T^i:A\to B$ that generalize derived functors, in the sense that $R^i\mathcal F = T^i$. A $\delta$-functor is universal if for any other $\delta$-functor $U$, there is a natural transformation $f:T^0\to U^0$ that induces a unique collection of morphisms $f^{i\geqslant 0}:T^i\to U^i$ that extend $f$.
See Weibel for a more thorough definition (and Grothendieck for the original setting). These functors may be covariant or contravariant, homological or cohomological. Note that $\delta$-functors are unique up to isomorphism.
Definition 3: Let $F:A\to B$ be a functor. $F$ is effaceable if for every $X\in A$ there exists a monomorphism $u\in \Hom_A(X,Y)$ such that $F(u)=0$. Similarly, $F$ is coeffaceable if for every $X\in A$ there exists an epimorphism $v\in \Hom_A(Y,X)$ such that $F(v)=0$.
Lemma 1: If a covariant (or contravariant) cohomological $\delta$-functor is effaceable for every $i>0$, then it is universal. Similarly, if a covariant (or contravariant) homological $\delta$-functor is coeffaceable for every $i>0$, then it is universal.
This appears as Proposition II.2.2.1 in Grothendieck and Exercise 2.4.5 in Weibel. Now let $\mathcal F$ be a sheaf over a projective scheme $X$.
Definition 2: A $\delta$-functor between $A$ and $B$ is a collection of functors $T^i:A\to B$ that generalize derived functors, in the sense that $R^i\mathcal F = T^i$. A $\delta$-functor is universal if for any other $\delta$-functor $U$, there is a natural transformation $f:T^0\to U^0$ that induces a unique collection of morphisms $f^{i\geqslant 0}:T^i\to U^i$ that extend $f$.
See Weibel for a more thorough definition (and Grothendieck for the original setting). These functors may be covariant or contravariant, homological or cohomological. Note that $\delta$-functors are unique up to isomorphism.
Definition 3: Let $F:A\to B$ be a functor. $F$ is effaceable if for every $X\in A$ there exists a monomorphism $u\in \Hom_A(X,Y)$ such that $F(u)=0$. Similarly, $F$ is coeffaceable if for every $X\in A$ there exists an epimorphism $v\in \Hom_A(Y,X)$ such that $F(v)=0$.
Lemma 1: If a covariant (or contravariant) cohomological $\delta$-functor is effaceable for every $i>0$, then it is universal. Similarly, if a covariant (or contravariant) homological $\delta$-functor is coeffaceable for every $i>0$, then it is universal.
This appears as Proposition II.2.2.1 in Grothendieck and Exercise 2.4.5 in Weibel. Now let $\mathcal F$ be a sheaf over a projective scheme $X$.
Lemma 2: (Theorem III.5.2 in Hartshorne) If $\mathcal F$ is coherent, there is $q\gg 0$ such that $H^i(X,\mathcal F(q))=0$ all $i>0$.
Definition 4: The dualizing sheaf of $X$ is a coherent sheaf $\omega_X^\circ$ and a trace map $t:H^n(X,\omega_X^\circ) \to k$ such that the isomorphism $\Hom(\mathcal F,\omega_X^\circ)\to H^n(X,\mathcal F)^\vee$ is induced by the natural pairing
\[\Hom(\mathcal F,\omega_X^\circ)\times H^n(X,\mathcal F) \to H^n(X,\omega_X^\circ)\]
composed with $t$.
Lemma 3: (Corollary II.5.18 in Hartshorne) If $\mathcal F$ is coherent, then it is a quotient of $\bigoplus_{i=1}^N \mathcal O_X(-q)$ for $q\gg 0$.
Next we recall some ring theory. Let $A$ be a ring and $M$ an $A$-module.
Definition 5: A sequence $a_1,\dots,a_n\in M$ is $M$-regular if $a_i$ is not a zero divisor of $M/(a_1,\dots,a_{i-1})M$ and $M\neq (a_1,\dots,a_i)M$ for all $i$. The depth of $M$ is the maximal length of an $M$-regular sequence of elements in some maximal ideal $\mathfrak m\leqslant M$. A local Noetherian ring is Cohen-Macaulay if $\text{depth}(A)=\dim(A)$, where dimension is Krull dimension (maximal length of prime ideal chains). A scheme $X$ is Cohen-Macaulay if every point $x\in X$ has a neighborhood $U$ such that the local ring $\mathcal O_X(U)$ is Cohen--Macaulay.
Lemma 4: Let $A$ be a regular local ring of dimension $n$ and $M,N$ be $A$-modules. Then:
Definition 4: The dualizing sheaf of $X$ is a coherent sheaf $\omega_X^\circ$ and a trace map $t:H^n(X,\omega_X^\circ) \to k$ such that the isomorphism $\Hom(\mathcal F,\omega_X^\circ)\to H^n(X,\mathcal F)^\vee$ is induced by the natural pairing
\[\Hom(\mathcal F,\omega_X^\circ)\times H^n(X,\mathcal F) \to H^n(X,\omega_X^\circ)\]
composed with $t$.
Lemma 3: (Corollary II.5.18 in Hartshorne) If $\mathcal F$ is coherent, then it is a quotient of $\bigoplus_{i=1}^N \mathcal O_X(-q)$ for $q\gg 0$.
Next we recall some ring theory. Let $A$ be a ring and $M$ an $A$-module.
Definition 5: A sequence $a_1,\dots,a_n\in M$ is $M$-regular if $a_i$ is not a zero divisor of $M/(a_1,\dots,a_{i-1})M$ and $M\neq (a_1,\dots,a_i)M$ for all $i$. The depth of $M$ is the maximal length of an $M$-regular sequence of elements in some maximal ideal $\mathfrak m\leqslant M$. A local Noetherian ring is Cohen-Macaulay if $\text{depth}(A)=\dim(A)$, where dimension is Krull dimension (maximal length of prime ideal chains). A scheme $X$ is Cohen-Macaulay if every point $x\in X$ has a neighborhood $U$ such that the local ring $\mathcal O_X(U)$ is Cohen--Macaulay.
Lemma 4: Let $A$ be a regular local ring of dimension $n$ and $M,N$ be $A$-modules. Then:
- $\text{pd}(M)\leqslant n$ iff $\Ext^i(M,N)=0$ for all $i>n$ (Proposition III.6.10)
- $\text{pd}(M)+\text{depth}(M)=n$ if $M$ is f.g. (Proposition III.6.12A)
Main theorem and proof
First we state the duality theorem for $X=\P^n$, without proof. Let $\omega_X$ be the canonical sheaf of $X$.
Theorem 1: (Theorem III.7.1 in Hartshorne) For $\mathcal F$ coherent over $\P^n$, for $i\geqslant 0$ there are natural isomorphisms
\[ \Hom(\mathcal F,\omega_X) \cong H^n(X,\mathcal F)^\vee, \hspace{2cm} \Ext^i(\mathcal F,\omega) \cong H^{n-i}(X,\mathcal F)^\vee.\]
Now we give the duality theorem for an arbitrary projective scheme, going through the proof as in Hartshorne.
Theorem 2: (Theorem III.7.6 in Hartshorne) Let $X$ be a projective scheme of dimension $n$ such that $\mathcal O(1)$ is very ample. For $\mathcal F$ coherent,
\begin{align*}
\Ext^i(\mathcal F, \omega_X^\circ) \cong H^{n-i}(X,\mathcal F)^\vee &\ \iff\ H^i(X,\mathcal F(-q))=0 \text{ for all $\mathcal F$ locally free, }i<n,q\gg0,\\
&\ \iff\ X\text{ is CM and equidimensional.}
\end{align*}
Proof: Natural maps $\Ext^i(\mathcal F,\omega^\circ_X)\to H^{n-i}(X,\mathcal F)^\vee$ exist, as $\Ext^i(-,\omega_X^\circ):\text{Coh}(X)\to \text{Mod}$ is a coeffaceable $\delta$-functor for every $i>0$, hence universal by Lemma 1. Indeed, by Lemma 3, we have a surjection
Theorem 1: (Theorem III.7.1 in Hartshorne) For $\mathcal F$ coherent over $\P^n$, for $i\geqslant 0$ there are natural isomorphisms
\[ \Hom(\mathcal F,\omega_X) \cong H^n(X,\mathcal F)^\vee, \hspace{2cm} \Ext^i(\mathcal F,\omega) \cong H^{n-i}(X,\mathcal F)^\vee.\]
Now we give the duality theorem for an arbitrary projective scheme, going through the proof as in Hartshorne.
Theorem 2: (Theorem III.7.6 in Hartshorne) Let $X$ be a projective scheme of dimension $n$ such that $\mathcal O(1)$ is very ample. For $\mathcal F$ coherent,
\begin{align*}
\Ext^i(\mathcal F, \omega_X^\circ) \cong H^{n-i}(X,\mathcal F)^\vee &\ \iff\ H^i(X,\mathcal F(-q))=0 \text{ for all $\mathcal F$ locally free, }i<n,q\gg0,\\
&\ \iff\ X\text{ is CM and equidimensional.}
\end{align*}
Proof: Natural maps $\Ext^i(\mathcal F,\omega^\circ_X)\to H^{n-i}(X,\mathcal F)^\vee$ exist, as $\Ext^i(-,\omega_X^\circ):\text{Coh}(X)\to \text{Mod}$ is a coeffaceable $\delta$-functor for every $i>0$, hence universal by Lemma 1. Indeed, by Lemma 3, we have a surjection
\[\underbrace{\bigoplus_{j=1}^N \mathcal O_X(-q)}_{\mathcal E}\tov u \mathcal F\to 0,\]
for which
\[\Ext^i(\mathcal E,\omega_X^\circ) = \bigoplus_{j=1}^N \Ext^i(\mathcal O_X(-q),\omega_X^\circ)= \bigoplus_{j=1}^N \Ext^i(\mathcal O_X,\omega_X^\circ (q)) = 0\]
for $i>0$. The first equality was distributing $\Ext^i$ over the sum and the second was by applying Remark 1.2. Hence $\Ext^i(-,\omega_X^\circ)(u)=0$ for $i>0$, so the functor is coeffeaceable for $i>0$, and so universal. By Definition 2 there exist maps generalizing the map $\Ext^0$ from Definition 4.
First iff $\Leftarrow$: Since universal $\delta$-functors are unique (up to isomorphism), we show $H^{n-i}(X,-)^\vee:\text{Coh}(X)\to \text{Mod}$ is also universal contravariant, which follows as it is coeffaceable for $i>0$. Using the same sequence and sheaf as in the equation above, we have that
\[H^{n-i}(X,\mathcal E) = \bigoplus_{j=1}^N H^{n-i}(X,\mathcal O_X(-q)) = 0\]
whenever $n-i<n$ by hypothesis, or equivalently, when $i>0$. The dual module is then also zero for $i>0$, so we are done.
First iff $\Rightarrow$: Assume the hypothesis with index $n-i$ and a locally free sheaf $\mathcal F(-q)$ for $q\gg 0$, for which
\begin{align*}
H^i(X,\mathcal F(-q))^\vee & \cong \Ext^{n-i}(\mathcal F(-q),\omega_X^\circ) & (\text{hypothesis}) \\
& \cong \Ext^{n-i}(\mathcal O_X,\mathcal F^\vee \otimes \mathcal O_X(q) \otimes\omega_X^\circ) & (\text{Remark 1.2}) \\
& \cong H^{n-i}(X,(\mathcal F^\vee \otimes\omega_X^\circ)\otimes \mathcal O_X(q)). & (\text{Remark 1.1})
\end{align*}
Tensoring with $\mathcal O_X(q)$ is twisting by $q$, and Lemma 2 says that $H^{n-i}(X,\mathcal G(q))=0$ for $\mathcal G$ coherent, for all $n-i>0$, for $q$ large enough. So for $i<n$ and $q$ large enough $H^i(X,\mathcal F(-q))^\vee=0$, and so its dual, the original cohomology group, is also trivial.
Second iff $\Leftarrow$: Embed $X\hookrightarrow \P^N$. As $X$ is Cohen--Macaulay and equidimensional of dimension $n$, for $\mathcal F$ locally free on $X$, a stalk $\mathcal F_x$ of a closed point $x\in X$ has depth $n$. Also, $\mathcal F_x\subset \mathcal O_{\P^N,x}$, and $\mathcal O_{\P^n,x}$ is regular as $\P^N$ is smooth over $k$. By Lemma 4.2, we have that
\[\text{pd}(\mathcal F_x) +n \leqslant \text{pd}(\mathcal O_{\P^N,x})+n=N, \]
so Lemma 4.1 and Remark 1.3 gives us that, for $i>N-n$,
\[\Ext^i(\mathcal F_x,-)=0
\ \ \implies\ \
\mathcal Ext^i(\mathcal F_x,-)=0
\ \ \implies\ \
\mathcal Ext^i(\mathcal F,-)=0.\]
Applying Theorem 1, Remark 1.4, and letting the functor $\mathcal Ext^i(\mathcal F,-)$ act on $\omega_{\P^N}(q)$, we have
\[H^i(X,\mathcal F(-q))^\vee \cong \Ext_{\P^n}^{N-i}(\mathcal F,\omega_{\P^N}(q)) \cong \Gamma(\P^N,\mathcal Ext_{\P^N}^{N-i}(\mathcal F, \omega_{\P^n}(q))) \cong \Gamma(\P^N, 0) = 0\]
for $q\gg0$ and $N-i>N-n$, or $i<n$. Since the dual is trivial, the cohomology group $H^i(X,\mathcal F(-q))$ is also trivial.
Second iff $\Rightarrow$: Omitted (techniques are similar to previous step, but use many others not used elsewhere). $\square$
for which
\[\Ext^i(\mathcal E,\omega_X^\circ) = \bigoplus_{j=1}^N \Ext^i(\mathcal O_X(-q),\omega_X^\circ)= \bigoplus_{j=1}^N \Ext^i(\mathcal O_X,\omega_X^\circ (q)) = 0\]
for $i>0$. The first equality was distributing $\Ext^i$ over the sum and the second was by applying Remark 1.2. Hence $\Ext^i(-,\omega_X^\circ)(u)=0$ for $i>0$, so the functor is coeffeaceable for $i>0$, and so universal. By Definition 2 there exist maps generalizing the map $\Ext^0$ from Definition 4.
First iff $\Leftarrow$: Since universal $\delta$-functors are unique (up to isomorphism), we show $H^{n-i}(X,-)^\vee:\text{Coh}(X)\to \text{Mod}$ is also universal contravariant, which follows as it is coeffaceable for $i>0$. Using the same sequence and sheaf as in the equation above, we have that
\[H^{n-i}(X,\mathcal E) = \bigoplus_{j=1}^N H^{n-i}(X,\mathcal O_X(-q)) = 0\]
whenever $n-i<n$ by hypothesis, or equivalently, when $i>0$. The dual module is then also zero for $i>0$, so we are done.
First iff $\Rightarrow$: Assume the hypothesis with index $n-i$ and a locally free sheaf $\mathcal F(-q)$ for $q\gg 0$, for which
\begin{align*}
H^i(X,\mathcal F(-q))^\vee & \cong \Ext^{n-i}(\mathcal F(-q),\omega_X^\circ) & (\text{hypothesis}) \\
& \cong \Ext^{n-i}(\mathcal O_X,\mathcal F^\vee \otimes \mathcal O_X(q) \otimes\omega_X^\circ) & (\text{Remark 1.2}) \\
& \cong H^{n-i}(X,(\mathcal F^\vee \otimes\omega_X^\circ)\otimes \mathcal O_X(q)). & (\text{Remark 1.1})
\end{align*}
Tensoring with $\mathcal O_X(q)$ is twisting by $q$, and Lemma 2 says that $H^{n-i}(X,\mathcal G(q))=0$ for $\mathcal G$ coherent, for all $n-i>0$, for $q$ large enough. So for $i<n$ and $q$ large enough $H^i(X,\mathcal F(-q))^\vee=0$, and so its dual, the original cohomology group, is also trivial.
Second iff $\Leftarrow$: Embed $X\hookrightarrow \P^N$. As $X$ is Cohen--Macaulay and equidimensional of dimension $n$, for $\mathcal F$ locally free on $X$, a stalk $\mathcal F_x$ of a closed point $x\in X$ has depth $n$. Also, $\mathcal F_x\subset \mathcal O_{\P^N,x}$, and $\mathcal O_{\P^n,x}$ is regular as $\P^N$ is smooth over $k$. By Lemma 4.2, we have that
\[\text{pd}(\mathcal F_x) +n \leqslant \text{pd}(\mathcal O_{\P^N,x})+n=N, \]
so Lemma 4.1 and Remark 1.3 gives us that, for $i>N-n$,
\[\Ext^i(\mathcal F_x,-)=0
\ \ \implies\ \
\mathcal Ext^i(\mathcal F_x,-)=0
\ \ \implies\ \
\mathcal Ext^i(\mathcal F,-)=0.\]
Applying Theorem 1, Remark 1.4, and letting the functor $\mathcal Ext^i(\mathcal F,-)$ act on $\omega_{\P^N}(q)$, we have
\[H^i(X,\mathcal F(-q))^\vee \cong \Ext_{\P^n}^{N-i}(\mathcal F,\omega_{\P^N}(q)) \cong \Gamma(\P^N,\mathcal Ext_{\P^N}^{N-i}(\mathcal F, \omega_{\P^n}(q))) \cong \Gamma(\P^N, 0) = 0\]
for $q\gg0$ and $N-i>N-n$, or $i<n$. Since the dual is trivial, the cohomology group $H^i(X,\mathcal F(-q))$ is also trivial.
Second iff $\Rightarrow$: Omitted (techniques are similar to previous step, but use many others not used elsewhere). $\square$
Addendum
In certain cases, Serre duality holds for the canonical sheaf instead of the dualizing sheaf.
Proposition 1: For $X$ a smooth projective variety over $k=\overline k$, $\omega_X^\circ \cong \omega_X$.
References: Grothendieck (Tohoku paper), Hartshorne (Algebraic Geometry, Section III.7), Weibel (An introduction to homological algebra, Section 2.1), Matsumura (Commutative algebra, Chapter 6)