## III.6: Ext Groups and Sheaves

### III.6.1.  #to-work

Let $\left(X, \mathcal{O}_X\right)$ be a ringed space, and let $\mathcal{F}^{\prime}, \mathcal{F}^{\prime \prime} \in \Mod(X)$. An **extension** of $\mathcal{F}^{\prime \prime}$ by $\mathcal{F}^{\prime}$ is a short exact sequence
\[
0 \rightarrow \mathcal{F}^{\prime} \rightarrow \mathcal{F} \rightarrow \mathcal{F}^{\prime \prime} \rightarrow 0
\]
in $\Mod(X)$. 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
\[
\delta: \operatorname{Hom}\left(\mathcal{F}^{\prime \prime}, \mathcal{F}^{\prime \prime}\right) \rightarrow \operatorname{Ext}^1\left(\mathcal{F}^{\prime \prime}, \mathcal{F}^{\prime}\right),
\]
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.

a. Show that there does not exist a projective object $\mathcal{P} \in \Mod(X)$, together with a surjective map $\mathcal{P} \rightarrow \mathcal{O}_X \rightarrow 0$.[^hint.3.6.2]

[^hint.3.6.2]: 
Hint: Consider surjections of the form $\mathcal{O}_V \rightarrow$ $k(x) \rightarrow 0$, where $x \in X$ is a closed point, $V$ is an open neighborhood of $x$, and $\mathcal{O}_V=j_{!}\left(\left.\mathcal{O}_X\right|_V\right)$, where $j: V \rightarrow X$ is the inclusion.


b. Show that there does not exist a projective object $\mathcal{P}$ in either $\QCoh(X)$ or $\Coh(X)$ together with a surjection $\mathcal{P} \rightarrow \mathcal{O}_X \rightarrow 0$.[^hint.3.6.2.b]

[^hint.3.6.2.b]: 
Hint: Consider surjections of the form $\mathcal{L} \rightarrow \mathcal{L} \otimes k(x) \rightarrow 0$, where $x \in X$ is a closed point, and $\mathcal{L}$ is an invertible sheaf on $X$.


### III.6.3.  #to-work

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

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

b. 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 $\Coh(X)$ has enough locally frees. Then for any $\mathcal{G} \in \Mod(X)$, show that the $\delta$-functor $\left(\mathcal{E} x t^i(\cdot, \mathcal{G})\right)$, from $\Coh(X)$ to $\Mod(X)$ is a contravariant universal $\delta$-functor.[^hint.3.6.4]

[^hint.3.6.4]: 
Hint: Show $\mathcal{E} x t^i(\cdot, \mathcal{G})$ is coeffaceable for $i>0$.


### III.6.5.  #to-work

Let $X$ be a noetherian scheme, and assume that $\Coh(X)$ 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:

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

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

c. $\operatorname{hd}(\mathcal{F})=\sup _x \operatorname{pd}_{\OO_x} \mathcal{F}_x$.

### 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.

a. $M$ is projective if and only if $\operatorname{Ext}^i(M, A)=0$ for all $i>0$.[^hint.3.6.6.a]



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

[^hint.3.6.6.a]: 
Hint: Use (6.11A) and descending induction on $i$ to show that $\operatorname{Ext}^i(M, N)=0$ for all $i>0$ and all finitely generated $A$-modules $N$. Then show $M$ is a direct summand of a free $A$-module (Matsumura $[2, p. 129]$).

### 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
\[
\operatorname{Ext}_X^i(\tilde{M}, \tilde{N}) \cong \operatorname{Ext}_A^i(M, N)
\]
and
\[
\mathcal{E} x t_X^i(\tilde{M}, \tilde{N}) \cong \operatorname{Ext}_A^i(M \cdot N)^{\sim} .
\]

### 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).

a. 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 $X$.[^hint.3.6.8.a]


b. 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 $n_i$.


[^hint.3.6.8.a]: 
Hint: Given a closed point $x \in X$ and an open neighborhood $U$ of $x$, to show there is an $\mathcal{L}, s$ such that $x \in X_s \subseteq U$, first reduce to the case that $Z=X-U$ is irreducible. Then let $\zeta$ be the generic point of $Z$. Let $f \in K(X)$ be a rational function with $f \in \mathcal{O}_x, f \notin \mathcal{O}_\zeta$. Let $D=(f)_{\infty}$, and let $\mathcal{L}=\mathcal{L}(D), s \in \Gamma(X, \mathcal{L}(D))$ correspond to $D$ (II, §6).

### 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.

a. 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
\[
0 \rightarrow \mathcal{E}_n \rightarrow \ldots \rightarrow \mathcal{E}_1 \rightarrow \mathcal{E}_0 \rightarrow \mathcal{F} \rightarrow 0 .
\]

b. 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

a. Let $f: X \rightarrow Y$ be a finite morphism of noetherian schemes. For any quasicoherent $\mathcal{O}_Y$-module $\mathcal{G}, \sheafhom_Y(f_* \OO_X, \mcg)$ 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).

b. Show that for any coherent $\mathcal{F}$ on $X$ and any quasi-coherent $\mathcal{G}$ on $Y$, there is a natural isomorphism
\[
f_* \sheafhom_X\left(\mathcal{F}, f^{\prime} \mathcal{G}\right) \iso \sheafhom_Y\left(f_* \mathcal{F}, \mathcal{G}\right) .
\]

c. For each $i \geqslant 0$, there is a natural map[^hint.3.6.10.c]
\[
\varphi_i: \operatorname{Ext}_X^i\left(\mathcal{F}, f^{!} \mathcal{G}\right) \rightarrow \operatorname{Ext}_Y^i\left(f_* \mathcal{F}, \mathcal{G}\right) .
\] 

d. Now assume that $X$ and $Y$ are separated, $\Coh(X)$ 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 $Y$.[^hint.3.6.10.d]

[^hint.3.6.10.c]: 
Hint: First construct a map
\[
\operatorname{Ext}_X^i\left(\mathcal{F}, f^{!} \mathcal{G}\right) \rightarrow \operatorname{Ext}_Y^i\left(f_* \mathcal{F}, f_* f^{!} \mathcal{G}\right)
.\]
Then compose with a suitable map from $f_* f^{!} \mathcal{G}$ to $\mathcal{G}$.

[^hint.3.6.10.d]: 
Hints: First do $i=0$. Then do $\mathcal{F}=\mathcal{O}_X$, using (Ex. 4.1). Then do $\mathcal{F}$ locally free. Do the general case by induction on $i$, writing $\mathcal{F}$ as a quotient of a locally free sheaf.