Serre duality for coherent sheaves on projective space

If $n=0$, then $X=\mathrm{Spec}k$ and every twist is a one-dimensional trivial bundle by [F6]. A coherent sheaf is a finite-dimensional vector space $V$, ${\omega }_X\cong k$ and the stated trace is the identity. The only asserted degree is $q=0$, and the pairing is ${\mathrm{Hom}}_k(V,k)\times V\to k$, $(\phi ,v)\mapsto \phi (v)$. It is natural and perfect by a basis and its dual basis, including $V=0$. This proves the complete statement for $n=0$. In steps 1.2–8.1 assume $n\ge 1$.

The pairing is well defined and bilinear. By [F1] the maps ${\chi }_{\mathcal{G}}^q$ are canonical isomorphisms, so ${\chi }_F^q{}^{-1}(\eta )$ is a well-defined class in ${\mathrm{Ext}}_{{\mathcal{O}}_X}^q({\mathcal{O}}_X,F)$; by [F4] the Yoneda product with $\alpha$ is the composition of derived morphisms and is additive in each variable, so $\alpha \cdot {\chi }_F^q{}^{-1}(\eta )$ is a well-defined class in ${\mathrm{Ext}}_{{\mathcal{O}}_X}^n({\mathcal{O}}_X,{\omega }_X)$; applying ${\chi }_{{\omega }_X}^n$ and the $k$-linear trace $t_X$ gives an element of $k$. These maps are additive, and multiplication by any $\lambda \in k$ on $F$ or ${\omega }_X$ commutes with the Yoneda composition by naturality [F4]; because $t_X$ is $k$-linear, the pairing is $k$-bilinear. Naturality in $F$: a morphism $u:F\to F^{\mathrm{\prime }}$ of coherent modules induces $H^q(X,u):H^q(X,F)\to H^q(X,F^{\mathrm{\prime }})$ and the pullback $u^{\ast }:{\mathrm{Ext}}^{n-q}(F^{\mathrm{\prime }},{\omega }_X)\to {\mathrm{Ext}}^{n-q}(F,{\omega }_X)$ given by precomposition, and the square expressing $⟨u^{\ast }\alpha ,\eta {⟩}_F=⟨\alpha ,H^q(u)\eta {⟩}_{F^{\mathrm{\prime }}}$ commutes because ${\chi }^q$ is natural in the sheaf variable by [F1] and composition of derived morphisms is associative by [F4].

Relative local computation for a line bundle. Let $L$ be an invertible ${\mathcal{O}}_X$-module with dual $L^{\vee }$ and let ${\omega }_X\to I^{\bullet }$ be the functorial ${\mathcal{O}}_X$-injective resolution. By [F9] there is a canonical isomorphism ${\mathrm{Hom}}_{{\mathcal{O}}_X}(L,M)\cong \mathrm{\Gamma }(X,L^{\vee }\otimes M)$ for every ${\mathcal{O}}_X$-module $M$, so degreewise ${\mathrm{Hom}}_{{\mathcal{O}}_X}(L,I^p)\cong \mathrm{\Gamma }(X,L^{\vee }\otimes I^p)\mathrm{.}$ Each $L^{\vee }\otimes I^p$ is injective in $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)$ because tensor product with the invertible sheaf $L^{\vee }$ is exact with exact inverse and carries injectives to injectives [F9], and the complex $L^{\vee }\otimes I^{\bullet }$ is the coaugmented exact complex ${\omega }_X\otimes L^{\vee }\to L^{\vee }\otimes I^{\bullet }$; from [F10] and [F11] applied to the underlying abelian sheaves (injective ${\mathcal{O}}_X$-modules are flasque, and flasque sheaves are $\mathrm{\Gamma }$-acyclic) it computes the cohomology of ${\omega }_X\otimes L^{\vee }$. Hence ${\mathrm{Ext}}_{{\mathcal{O}}_X}^j(L,{\omega }_X)\cong H^j(X,{\omega }_X\otimes L^{\vee })(j\ge 0),$ canonically. For $L=\mathcal{O}(e)$ and $e=-d$ this reads ${\mathrm{Ext}}_{{\mathcal{O}}_X}^j(\mathcal{O}(-d),{\omega }_X)\cong H^j(X,\mathcal{O}(d-n-1))$, which vanishes for every $j\ge 1$ and every $d\ge 1$ by [F6].

Description by composition in the derived category. Under the identifications of [F4] the pairing reads as follows: $\alpha$ corresponds to a morphism $\tilde{\alpha }:F\to {\omega }_X[n-q]$ in $D(\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X))$ and ${\chi }_F^q{}^{-1}(\eta )$ corresponds to a morphism $\tilde{\eta }:{\mathcal{O}}_X\to F[q]$; their Yoneda product corresponds to the composite $\tilde{\alpha }[q]\circ \tilde{\eta }:{\mathcal{O}}_X\to {\omega }_X[n]$; applying the identification ${\mathrm{Ext}}_{{\mathcal{O}}_X}^n({\mathcal{O}}_X,{\omega }_X)\cong H^n(X,{\omega }_X)$ of [F1] and the trace gives $⟨\alpha ,\eta {⟩}_F=t_X\circ {\chi }_{{\omega }_X}^n(\tilde{\alpha }[q]\circ \tilde{\eta })\mathrm{.}$ In particular the pairing depends only on the two classes and not on the choices of resolutions, and it is natural in $F$ in the sense of step 1.2.

Adjoint maps. For $j\in \{0,\dots ,n\}$ and a coherent $F$ let ${\mathrm{\Phi }}_F^j:{\mathrm{Ext}}_{{\mathcal{O}}_X}^j(F,{\omega }_X)⟶H^{n-j}(X,F{)}^{\vee },{\mathrm{\Phi }}_F^j(\alpha )(\eta ):=⟨\alpha ,\eta {⟩}_F,$ be the first adjoint map of the pairing in degree $q=n-j$. By step 1.2 these are $k$-linear, and for $j=0$ the identifications ${\mathrm{Ext}}^0=\mathrm{Hom}$ and ${\chi }_{{\omega }_X}^n$ show that ${\mathrm{\Phi }}_F^0(\phi )(\eta )=t_X(H^n(X,\phi )(\eta )),$ the trace map of the classical statement: a morphism $\phi :F\to {\omega }_X$ induces $H^n(X,\phi )$ on cohomology by [F3] and therefore the stated functional.

The case of a single twisting sheaf. Let $e\in \mathbb{Z}$ and put $d=-e-n-1$, so that ${\omega }_X(-e)=\mathcal{O}(-n-1-e)=\mathcal{O}(d)$ and $\mathcal{O}(-d-n-1)=\mathcal{O}(e)$. Under the canonical isomorphism ${\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{O}(e),{\omega }_X)\cong \mathrm{\Gamma }(X,{\omega }_X(-e))=H^0(X,\mathcal{O}(d))$ of [F9] a morphism $\phi :\mathcal{O}(e)\to {\omega }_X$ corresponds to the global section $s$ giving multiplication by $s$, and the induced map $H^n(X,\phi ):H^n(X,\mathcal{O}(e))\to H^n(X,{\omega }_X)$ is the cup product $H^0(X,\mathcal{O}(d))\times H^n(X,\mathcal{O}(e))\to H^n(X,{\omega }_X)$ of [F12] with the global section $s$; this is the composite $H^n(X,\mathcal{O}(e))\to H^n(X,\mathcal{O}(d)\otimes \mathcal{O}(e))=H^n(X,{\omega }_X)$ under the canonical isomorphism $\mathcal{O}(d)\otimes \mathcal{O}(e)\cong {\omega }_X$. Therefore ${\mathrm{\Phi }}_{\mathcal{O}(e)}^0$ is, under these identifications, the pairing of [F5] in degree $q=0$ and with twist $e$, namely $H^0(X,\mathcal{O}(d))\times H^n(X,\mathcal{O}(-d-n-1))\to k$, which is perfect: for $d\ge 0$ it is the residue pairing of [F5], and for $d<0$ both spaces are zero by [F6]. Hence ${\mathrm{\Phi }}_{\mathcal{O}(e)}^0$ is an isomorphism for every $e$.

Compatibility with connecting maps. Let $0\to K\to P\to F\to 0$ be a short exact sequence of coherent ${\mathcal{O}}_X$-modules, with connecting maps ${\mathrm{\partial }}^j:{\mathrm{Ext}}^j(K,{\omega }_X)\to {\mathrm{Ext}}^{j+1}(F,{\omega }_X)$ from [F2] and ${\delta }^m:H^m(X,F)\to H^{m+1}(X,K)$ from [F3]. For $j\ge 0$ and $m=n-j-1\ge 0$, the identity is $⟨{\mathrm{\partial }}^j\alpha ,\eta {⟩}_F=⟨\alpha ,{\delta }^m\eta {⟩}_K(\alpha \in {\mathrm{Ext}}^j(K,{\omega }_X),\eta \in H^m(X,F));$ when $m<0$ both sides are vacuous. To prove it, work throughout in $D(\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X))$, where the short exact sequence gives a triangle $K\to P\to F\stackrel{\gamma }{\to }K[1]$. Under [F4] the Ext boundary sends $\tilde{\alpha }:K\to {\omega }_X[j]$ to $\tilde{\alpha }[1]\circ \gamma :F\to {\omega }_X[j+1]$. Put $\tilde{\eta }={\chi }_F^m{}^{-1}(\eta )$, viewed by [F4] as a morphism ${\mathcal{O}}_X\to F[m]$. The comparison [F15] commutes with connecting maps, so ${\chi }_K^{m+1}{}^{-1}({\delta }^m\eta )$ is represented by $\gamma [m]\circ \tilde{\eta }:{\mathcal{O}}_X\to K[m+1]$. The left Yoneda product in the pairing is therefore $(\tilde{\alpha }[1]\circ \gamma )[m]\circ \tilde{\eta }=\tilde{\alpha }[m+1]\circ \gamma [m]\circ \tilde{\eta }$, which is exactly the right Yoneda product. Applying ${\chi }_{{\omega }_X}^n$ and $t_X$ proves the identity without mixing derived categories of modules and abelian sheaves.

Finite sums of twists. For a finite direct sum $P={⨁}_i\mathcal{O}(e_i)$ the Hom module is the direct sum ${⨁}_i\mathrm{Hom}(\mathcal{O}(e_i),{\omega }_X)$, cohomology is the direct sum ${⨁}_i{H}^n(X,\mathcal{O}(e_i))$, and the pairing is the orthogonal sum of the pairings of step 3.1, so ${\mathrm{\Phi }}_P^0$ is an isomorphism as a direct sum of isomorphisms; the same isomorphisms of steps 1.3 and [F6] give ${\mathrm{Ext}}_{{\mathcal{O}}_X}^j(P,{\omega }_X)=0(j\ge 1)\text{and}{H}^{n-j}(X,P)=0(1\le j\le n)$ whenever every $\mathcal{O}(e_i)$ satisfies $e_i\le -1$, in particular for $P={⨁}_i\mathcal{O}(-d_i)$ with all $d_i\ge 1$.

The compatibility square. With the notation of step 3.2 and $0\le j<n$, the connecting map relevant to the pairing is ${\delta }^{n-j-1}:H^{n-j-1}(X,F)\to H^{n-j}(X,K)$. Its linear dual has the direction $({\delta }^{n-j-1}{)}^{\vee }:H^{n-j}(X,K{)}^{\vee }\to H^{n-j-1}(X,F{)}^{\vee }$. The identity of step 3.2 says that the square $\begin{array}{ccc}{\mathrm{Ext}}^j(K,{\omega }_X)& \stackrel{{\mathrm{\partial }}^j}{\to }& {\mathrm{Ext}}^{j+1}(F,{\omega }_X)\\ \downarrow {\mathrm{\Phi }}_K^j& & \downarrow {\mathrm{\Phi }}_F^{j+1}\\ H^{n-j}(X,K{)}^{\vee }& \stackrel{({\delta }^{n-j-1}{)}^{\vee }}{\to }& H^{n-j-1}(X,F{)}^{\vee }\end{array}$ commutes. This step asserts commutativity only; additional vanishing hypotheses are needed to make the horizontal maps isomorphisms.

Degree zero for all coherent modules. By [F7] choose a right-exact presentation $E_1\stackrel{\psi }{\to }{E}_0\to F\to 0$ with $E_0,E_1$ finite sums of twists. Put $K=\ker (E_0\to F)$ and $L=\ker (E_1\to K)$; both are coherent by [F13], and the presentation splits into the two genuine short exact sequences $0\to K\to E_0\to F\to 0$ and $0\to L\to E_1\to K\to 0$. Contravariant left exactness gives $0\to \mathrm{Hom}(F,{\omega }_X)\to \mathrm{Hom}(E_0,{\omega }_X)\stackrel{{\psi }^{\ast }}{\to }\mathrm{Hom}(E_1,{\omega }_X),$ exact because a map $E_0\to {\omega }_X$ killed by precomposition with $\psi$ kills $K=\mathrm{im}\psi$ and therefore factors uniquely through $F$. By [F14], $H^{n+1}(X,L)=H^{n+1}(X,K)=0$. The long exact sequence of $0\to L\to E_1\to K\to 0$ thus makes $H^n(E_1)\to H^n(K)$ surjective; that of $0\to K\to E_0\to F\to 0$ makes $H^n(E_0)\to H^n(F)$ surjective. Composing the two exact segments proves $H^n(X,E_1)\stackrel{H^n(\psi )}{\to }{H}^n(X,E_0)\to H^n(X,F)\to 0$ is exact. Dualizing over $k$ yields $0\to H^n(X,F{)}^{\vee }\to H^n(X,E_0{)}^{\vee }\stackrel{H^n(\psi {)}^{\vee }}{\to }{H}^n(X,E_1{)}^{\vee }\mathrm{.}$ Naturality of ${\mathrm{\Phi }}^0$ from step 1.2 gives a commutative diagram between these two left-exact rows. The vertical maps for $E_0,E_1$ are isomorphisms by step 4.1, so the induced map between their kernels, ${\mathrm{\Phi }}_F^0$, is an isomorphism. This argument uses the two short exact sequences above and never treats $E_1\to E_0\to F\to 0$ as short exact.

Effacing presentations. Let $F$ be coherent. By [F8] there is $m\ge 1$ with $F(m)$ globally generated, so that the evaluation map $\mathrm{\Gamma }(X,F(m)){\otimes }_{\mathbb{Z}}{\mathcal{O}}_X\to F(m)$ is surjective. By [F14] the $k$-vector space $\mathrm{\Gamma }(X,F(m))$ is finite-dimensional; choose a $k$-basis $s_1,\dots ,s_N$. Since every $f\in \mathrm{\Gamma }(X,F(m))$ is a $k$-linear combination $f={\sum }_i{c}_i{s}_i$ with $c_i\in k$, the evaluation map factors through the morphism $\sigma :{\mathcal{O}}_X^{\oplus N}\to F(m)$, $(g_1,\dots ,g_N)\mapsto {\sum }_i{g}_i{s}_i$, and $f\otimes g\mapsto \sigma ((c_{1}g,\dots ,c_{N}g))$ on $U$; as the evaluation map is surjective, so is $\sigma$. Twisting by the invertible sheaf ${\mathcal{O}}_X(-m)$ is exact by [F9], so $\sigma \otimes {\mathcal{O}}_X(-m)$ is a surjection $P:={\mathcal{O}}_X(-m{)}^{\oplus N}\to F$ with $m\ge 1$. Its kernel $K$ is coherent by [F13], so $K$ is again of the kind considered. By step 1.3 with $d=m\ge 1$ and [F6], ${\mathrm{Ext}}_{{\mathcal{O}}_X}^j(\mathcal{O}(-m),{\omega }_X)\cong H^j(X,\mathcal{O}(m-n-1))=0$ for every $j\ge 1$ since $m-n-1\ge -n$, and $H^0(X,\mathcal{O}(-m))=0$ since $-m<0$; hence by step 4.1, $P$ has ${\mathrm{Ext}}_{{\mathcal{O}}_X}^j(P,{\omega }_X)=0(j\ge 1)\text{and}{H}^{n-j}(X,P)=0(1\le j\le n)\mathrm{.}$

Degree-one case from an effacing presentation, with $n\ge 1$. Let $0\to K\to P\stackrel{\pi }{\to }F\to 0$ be as in step 5.2. From the long exact sequences of [F2] and [F3] and the vanishing of step 5.2 we obtain exact sequences $\mathrm{Hom}(P,{\omega }_X)\stackrel{{\pi }^{\ast }}{\to }\mathrm{Hom}(K,{\omega }_X)\stackrel{{\mathrm{\partial }}^0}{\to }{\mathrm{Ext}}^1(F,{\omega }_X)\to 0,$ $H^{n-1}(X,K)\to H^{n-1}(X,P)=0\to H^{n-1}(X,F)\stackrel{{\delta }^{n-1}}{\to }{H}^n(X,K)\to H^n(X,P),$ where the vanishing ${\mathrm{Ext}}^1(P,{\omega }_X)=0$ and $H^{n-1}(X,P)=0$ are those of step 5.2. Dualizing the second gives the exact sequence $H^n(X,P{)}^{\vee }\to H^n(X,K{)}^{\vee }\stackrel{({\delta }^{n-1}{)}^{\vee }}{\to }{H}^{n-1}(X,F{)}^{\vee }\to 0.$ By step 4.2 the square relating ${\mathrm{\partial }}^0$ and ${\delta }^{n-1}$ commutes, and both ${\mathrm{\Phi }}_P^0$ and ${\mathrm{\Phi }}_K^0$ are isomorphisms by step 5.1; passing to cokernels, ${\mathrm{\Phi }}_F^1$ is the induced isomorphism ${\mathrm{Ext}}^1(F,{\omega }_X)\cong \mathrm{coker}({\pi }^{\ast })\stackrel{\sim }{\to }\mathrm{coker}(H^n(X,P{)}^{\vee }\to H^n(X,K{)}^{\vee })\cong H^{n-1}(X,F{)}^{\vee },$ so ${\mathrm{\Phi }}_F^1$ is an isomorphism.

Higher degrees by the dimension shift. Let $2\le j\le n$ and let $0\to K\to P\to F\to 0$ be as in step 5.2. Since ${\mathrm{Ext}}^{j-1}(P,{\omega }_X)={\mathrm{Ext}}^j(P,{\omega }_X)=0$ by step 5.2 and $H^{n-j}(X,P)=H^{n-j+1}(X,P)=0$ because $n-j+1\le n-1$, the exact sequences of [F2] and [F3] give isomorphisms ${\mathrm{\partial }}^{j-1}:{\mathrm{Ext}}^{j-1}(K,{\omega }_X)\stackrel{\sim }{\to }{\mathrm{Ext}}^j(F,{\omega }_X),{\delta }^{n-j}:H^{n-j}(X,F)\stackrel{\sim }{\to }{H}^{n-j+1}(X,K)\mathrm{.}$ By the compatibility square of step 4.2 the diagram $\begin{array}{ccc}{\mathrm{Ext}}^{j-1}(K,{\omega }_X)& \stackrel{\sim }{\to }& {\mathrm{Ext}}^j(F,{\omega }_X)\\ \downarrow {\mathrm{\Phi }}_K^{j-1}& & \downarrow {\mathrm{\Phi }}_F^j\\ H^{n-j+1}(X,K{)}^{\vee }& \stackrel{\sim }{\to }& H^{n-j}(X,F{)}^{\vee }\end{array}$ commutes with isomorphisms in the horizontal directions; hence ${\mathrm{\Phi }}_F^j$ is an isomorphism if and only if ${\mathrm{\Phi }}_K^{j-1}$ is.

Induction. We prove by induction on $j=0,\dots ,n$ that ${\mathrm{\Phi }}_F^j$ is an isomorphism for every coherent ${\mathcal{O}}_X$-module $F$. The case $j=0$ is step 5.1. Assume the statement known for $j-1$ and let $F$ be coherent; if $F=0$ both sides vanish, and otherwise step 5.2 supplies an effacing presentation $0\to K\to P\to F\to 0$ with $K$ coherent. For $j=1$ the claim is step 6.1, and for $j\ge 2$ it is step 6.2 combined with the induction hypothesis applied to the coherent module $K$.

Perfectness. Let $F$ be coherent and $q\in \{0,\dots ,n\}$, and put $j=n-q$, so that ${\mathrm{\Phi }}_F^{n-q}:{\mathrm{Ext}}^{n-q}(F,{\omega }_X)\to H^q(X,F{)}^{\vee }$ is an isomorphism by step 7.1. By [F14] the $k$-vector space $H^q(X,F)$ is finite-dimensional, so the transpose ${\mathrm{\Phi }}_F^{n-q\vee }$ is an isomorphism and the second adjoint map $H^q(X,F)\to {\mathrm{Ext}}^{n-q}(F,{\omega }_X{)}^{\vee }$ is the composite of ${\mathrm{\Phi }}_F^{n-q\vee }$ with the double-duality isomorphism of finite-dimensional vector spaces, which is an isomorphism; hence both adjoint maps of the pairing are isomorphisms and the pairing is perfect. For $q=n$ this is the classical $j=0$ case of step 5.1; for $q=0$ it is the dual statement that ${\mathrm{Ext}}^n(F,{\omega }_X)\cong H^0(X,F{)}^{\vee }$, and for $F={\mathcal{O}}_X$ it specializes by step 1.3 to ${\mathrm{Ext}}^{n-q}({\mathcal{O}}_X,{\omega }_X)\cong H^{n-q}(X,{\omega }_X)\cong H^q(X,{\mathcal{O}}_X{)}^{\vee }$.

Boundary cases and the Axiom of Choice. If $F=0$ both sides of the pairing are zero for every $q$, so the pairing is perfect vacuously, in agreement with step 7.1. If $n=0$ then $X=\mathrm{Spec}k$, ${\omega }_X={\mathcal{O}}_X$ and $t_X$ is the identity $k\to k$ under the convention of Dualizing line bundle and trace datum of a smooth projective variety; a coherent ${\mathcal{O}}_X$-module is a finite-dimensional $k$-vector space $V$, ${\mathrm{Ext}}^0(V,{\mathcal{O}}_X)=V^{\vee }$ and the pairing is the evaluation $V^{\vee }\times V\to k$, which is perfect, and [F6] gives $H^q=0$ for $q>0$. If $q=0$ and $n\ge 1$ clause 2 reads ${\mathrm{Ext}}^n(F,{\omega }_X)\cong H^0(X,F{)}^{\vee }$, and if $q=n$ it reads $\mathrm{Hom}(F,{\omega }_X)\cong H^n(X,F{)}^{\vee }$; both are the two ends of the same comparison by step 7.1. If $F$ is a finite direct sum of twisting sheaves the statement is step 3.1 with step 4.1, and the general coherent case is obtained from it by the effacing presentations of step 5.2, so no hypothesis on $F$ beyond coherence is used. Finally, the Axiom of Choice [A1] is assumed in the statement and is used exactly through the functorial injective resolution data of [F10] for modules and abelian sheaves, through the Dependent Choice instances of [F11] used in step 1.3, through the DC hypotheses of the derived-category identifications [F4] used in steps 2.1, 2.2 and 3.2, and through [F12]; the only selection made beyond the functorial data is the finite $k$-basis of $\mathrm{\Gamma }(X,F(m))$ chosen in step 5.2, which is finite choice and hence available in ZF, all other constructions being canonical from the fixed data. The comparison of the connecting maps is proved within the module derived category in step 3.2 using the delta-functor compatibility established in [F15].