Lemma 57.12.6. Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is regular. Let $F : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ X)$ be a $k$-linear exact functor. Assume for every coherent $\mathcal{O}_ X$-module $\mathcal{F}$ with $\dim (\text{Supp}(\mathcal{F})) = 0$ there is an isomorphism $\mathcal{F} \cong F(\mathcal{F})$. Then $F$ is fully faithful.

Email from Noah Olander of Jun 9, 2020

**Proof.**
By Lemma 57.12.5 it suffices to show that the maps

are isomorphisms for all $i \in \mathbf{Z}$ and all closed points $x, x' \in X$. By assumption, the source and the target are isomorphic. If $x \not= x'$, then both sides are zero and the result is true. If $x = x'$, then it suffices to prove that the map is either injective or surjective. For $i < 0$ both sides are zero and the result is true. For $i = 0$ any nonzero map $\alpha : \mathcal{O}_ x \to \mathcal{O}_ x$ of $\mathcal{O}_ X$-modules is an isomorphism. Hence $F(\alpha )$ is an isomorphism too and so $F(\alpha )$ is nonzero. Thus the result for $i = 0$. For $i = 1$ a nonzero element $\xi $ in $\mathop{\mathrm{Ext}}\nolimits ^1(\mathcal{O}_ x, \mathcal{O}_ x)$ corresponds to a nonsplit short exact sequence

Since $F(\mathcal{F}) \cong \mathcal{F}$ we see that $F(\mathcal{F})$ is a nonsplit extension of $\mathcal{O}_ x$ by $\mathcal{O}_ x$ as well. Since $\mathcal{O}_ x \cong F(\mathcal{O}_ x)$ is a simple $\mathcal{O}_ X$-module and $\mathcal{F} \cong F(\mathcal{F})$ has length $2$, we see that in the distinguished triangle

the first two arrows must form a short exact sequence which must be isomorphic to the above short exact sequence and hence is nonsplit. It follows that $F(\xi )$ is nonzero and we conclude for $i = 1$. For $i > 1$ composition of ext classes defines a surjection

See Duality for Schemes, Lemma 48.15.4. Hence surjectivity in degree $1$ implies surjectivity for $i > 0$. This finishes the proof. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)