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III.6: Ext Groups and Sheaves

III.6.1. #to_work

Let \(\left(X, \mathcal{O}_X\right)\) be a ringed space, and let \(\mathcal{F}^{\prime}, \mathcal{F}^{\prime \prime} \in {\mathsf{Mod}}(X)\). An extension of \(\mathcal{F}^{\prime \prime}\) by \(\mathcal{F}^{\prime}\) is a short exact sequence \begin{align*} 0 \rightarrow \mathcal{F}^{\prime} \rightarrow \mathcal{F} \rightarrow \mathcal{F}^{\prime \prime} \rightarrow 0 \end{align*} in \({\mathsf{Mod}}(X)\). Two extensions are isomorphic if there is an isomorphism of the short exact sequences, inducing the identity maps on \(\mathcal{F}^{\prime}\) and \(\mathcal{F}^{\prime \prime}\). Given an extension as above consider the long exact sequence arising from \(\operatorname{Hom}\left(\mathcal{F}^{\prime \prime}, \cdot\right)\), in particular the map \begin{align*} \delta: \operatorname{Hom}\left(\mathcal{F}^{\prime \prime}, \mathcal{F}^{\prime \prime}\right) \rightarrow \operatorname{Ext}^1\left(\mathcal{F}^{\prime \prime}, \mathcal{F}^{\prime}\right), \end{align*} and let \(\xi \in \operatorname{Ext}^1\left(\mathcal{F}^{\prime \prime}, \mathcal{F}^{\prime}\right)\) be \(\delta\left(1_{\mathcal{F}^{\prime \prime}}\right)\). Show that this process gives a one-to-one correspondence between isomorphism classes of extensions of \(\mathcal{F}^{\prime \prime}\) by \(\mathcal{F}^{\prime}\), and elements of the group \(\operatorname{Ext}^1\left(\mathcal{F}^{\prime \prime}, \mathcal{F}^{\prime}\right)\).

III.6.2. #to_work

Let \(X=\mathbf{P}_k^1\), with \(k\) an infinite field.

• Show that there does not exist a projective object \(\mathcal{P} \in {\mathsf{Mod}}(X)\), together with a surjective map \(\mathcal{P} \rightarrow \mathcal{O}_X \rightarrow 0\). ¹

• Show that there does not exist a projective object \(\mathcal{P}\) in either \({\mathsf{QCoh}}(X)\) or \({\mathsf{Coh}}(X)\) together with a surjection \(\mathcal{P} \rightarrow \mathcal{O}_X \rightarrow 0\). ²

III.6.3. #to_work

Let \(X\) be a noetherian scheme, and let \(\mathcal{F}, \mathcal{G} \in {\mathsf{Mod}}(X)\).

• If \(\mathcal{F}, \mathcal{G}\) are both coherent, then \(\mathcal{E x t}(\mathcal{F}, \mathcal{G})\) is coherent, for all \(i \geqslant 0\).

• If \(\mathcal{F}\) is coherent and \(\mathcal{G}\) is quasi-coherent, then \(\mathcal{E} x t^i(\mathcal{F}, \mathcal{G})\) is quasi-coherent, for all \(i \geqslant 0\).

III.6.4. #to_work

Let \(X\) be a noetherian scheme, and suppose that every coherent sheaf on \(X\) is a quotient of a locally free sheaf. In this case we say \({\mathsf{Coh}}(X)\) has enough locally frees. Then for any \(\mathcal{G} \in {\mathsf{Mod}}(X)\), show that the \(\delta\)-functor \(\left(\mathcal{E} x t^i(\cdot, \mathcal{G})\right)\), from \({\mathsf{Coh}}(X)\) to \({\mathsf{Mod}}(X)\) is a contravariant universal \(\delta\)-functor. ³

III.6.5. #to_work

Let \(X\) be a noetherian scheme, and assume that \({\mathsf{Coh}}(X)\) has enough locally frees (Ex. 6.4). Then for any coherent sheaf \(\mathcal{F}\) we define the homological dimension of \(\mathcal{F}\), denoted \(\mathrm{hd} (\mathcal{F})\), to be the least length of a locally free resolution of \(\mathcal{F}\) (or \(+\infty\) if there is no finite one). Show:

• \(\mathcal{F}\) is locally free \(\Leftrightarrow \mathcal{E} x t^1(\mathcal{F}, \mathcal{G})=0\) for all \(\mathcal{G} \in {\mathsf{Mod}}(X)\);

• \(\operatorname{hd}(\mathcal{F}) \leqslant n \Leftrightarrow \mathcal{E x t}(\mathcal{F}, \mathcal{G})=0\) for all \(i>n\) and all \(\mathcal{G} \in {\mathsf{Mod}}(X)\);

• \(\operatorname{hd}(\mathcal{F})=\sup _x \operatorname{pd}_{{\mathcal{O}}_x} \mathcal{F}_x\).

III.6.6. #to_work

Let \(A\) be a regular local ring, and let \(M\) be a finitely generated \(A\)-module. In this case, strengthen the result \((6.10 \mathrm{~A})\) as follows.

• \(M\) is projective if and only if \(\operatorname{Ext}^i(M, A)=0\) for all \(i>0\). ⁴

• Use (a) to show that for any \(n\), pd \(M \leqslant n\) if and only if \(\operatorname{Ext}^i(M, A)=0\) for all \(i>n\).

III.6.7. #to_work

Let \(X=\operatorname{Spec} A\) be an affine noetherian scheme. Let \(M, N\) be \(A\)-modules, with \(M\) finitely generated. Then \begin{align*} \operatorname{Ext}_X^i(\tilde{M}, \tilde{N}) \cong \operatorname{Ext}_A^i(M, N) \end{align*} and \begin{align*} \mathcal{E} x t_X^i(\tilde{M}, \tilde{N}) \cong \operatorname{Ext}_A^i(M \cdot N)^{\sim} . \end{align*}

III.6.8. #to_work

Prove the following theorem of Kleiman (see Borelli \([1]\)): if \(X\) is a noetherian, integral, separated, locally factorial scheme, then every coherent sheaf on \(X\) is a quotient of a locally free sheaf (of finite rank).

• First show that open sets of the form \(X_s\), for various \(s \in \Gamma(X, \mathcal{L})\), and various invertible sheaves \(\mathcal{L}\) on \(X\), form a base for the topology of \(X\). ⁵

• Now use (II, 5.14) to show that any coherent sheaf is a quotient of a direct sum \(\bigoplus \mathcal{L}_i^{n_i}\) for various invertible sheaves \(\mathcal{L}_i\) and various integers \(n_i\).

III.6.9. #to_work

Let \(X\) be a noetherian, integral, separated, regular scheme. (We say a scheme is regular if all of its local rings are regular local rings.) Recall the definition of the Grothendieck group \(K(X)\) from (II, Ex. 6.10).

We define similarly another group \(K_1(X)\) using locally free sheaves: it is the quotient of the free abelian group generated by all locally free (coherent) sheaves, by the subgroup generated by all expressions of the form \(\mathcal{E}-\mathcal{E}^{\prime}-\mathcal{E}^{\prime \prime}\), whenever \(0 \rightarrow \mathcal{E}^{\prime} \rightarrow \mathcal{E} \rightarrow \mathcal{E}^{\prime \prime} \rightarrow 0\) is a short exact sequence of locally free sheaves.

Clearly there is a natural group homomorphism \(\varepsilon: K_1(X) \rightarrow K(X)\). Show that \(\varepsilon\) is an isomorphism (Borel and Serre \([1, \S 4])\) as follows.

• Given a coherent sheaf \(\mathcal{F}\), use (Ex. 6.8) to show that it has a locally free resolution \(\mathcal{E} . \rightarrow \mathcal{F} \rightarrow 0\). Then use (6.11A) and (Ex. 6.5) to show that it has a finite locally free resolution \begin{align*} 0 \rightarrow \mathcal{E}_n \rightarrow \ldots \rightarrow \mathcal{E}_1 \rightarrow \mathcal{E}_0 \rightarrow \mathcal{F} \rightarrow 0 . \end{align*}

• For each \(\mathcal{F}\), choose a finite locally free resolution \(\mathcal{E}\). \(\rightarrow \mathcal{F} \rightarrow 0\), and let \(\delta(\mathcal{F})=\sum(-1)^i \gamma\left(\mathcal{E}_i\right)\) in \(K_1(X)\). Show that \(\delta(\mathcal{F})\) is independent of the resolution chosen, that it defines a homomorphism of \(K(X)\) to \(K_1(X)\), and finally, that it is an inverse to \(\varepsilon\).

III.6.10. Duality for a Finite Flat Morphism. #to_work

• Let \(f: X \rightarrow Y\) be a finite morphism of noetherian schemes. For any quasicoherent \(\mathcal{O}_Y\)-module \(\mathcal{G}, \mathop{\mathcal{H}\! \mathit{om}}_Y(f_* {\mathcal{O}}_X, {\mathcal{G}})\) is a quasi-coherent \(f_* \mathcal{O}_X\)-module, hence corresponds to a quasi-coherent \(\mathcal{O}_X\)-module, which we call \(f ! \mathcal{G}\) (II, Ex. 5.17e).

• Show that for any coherent \(\mathcal{F}\) on \(X\) and any quasi-coherent \(\mathcal{G}\) on \(Y\), there is a natural isomorphism \begin{align*} f_* \mathop{\mathcal{H}\! \mathit{om}}_X\left(\mathcal{F}, f^{\prime} \mathcal{G}\right) { \, \xrightarrow{\sim}\, }\mathop{\mathcal{H}\! \mathit{om}}_Y\left(f_* \mathcal{F}, \mathcal{G}\right) . \end{align*}

• For each \(i \geqslant 0\), there is a natural map ⁶ \begin{align*} \varphi_i: \operatorname{Ext}_X^i\left(\mathcal{F}, f^{!} \mathcal{G}\right) \rightarrow \operatorname{Ext}_Y^i\left(f_* \mathcal{F}, \mathcal{G}\right) . \end{align*}

• Now assume that \(X\) and \(Y\) are separated, \({\mathsf{Coh}}(X)\) has enough locally frees, and assume that \(f_* \mathcal{O}_X\) is locally free on \(Y\) (this is equivalent to saying \(f\) flat-see §9). Show that \(\varphi_i\) is an isomorphism for all \(i\), all \(\mathcal{F}\) coherent on \(X\), and all \(\mathcal{G}\) quasi-coherent on \(Y\). ⁷