The Stacks project

Lemma 15.74.15. Let $A$ be a ring. Let $K \in D(A)$ be perfect. Then $K^\vee = R\mathop{\mathrm{Hom}}\nolimits _ A(K, A)$ is a perfect complex and $K \cong (K^\vee )^\vee $. There are functorial isomorphisms

\[ L \otimes _ A^\mathbf {L} K^\vee = R\mathop{\mathrm{Hom}}\nolimits _ A(K, L) \quad \text{and}\quad H^0(L \otimes _ A^\mathbf {L} K^\vee ) = \mathop{\mathrm{Ext}}\nolimits _ A^0(K, L) \]

for $L \in D(A)$.

Proof. We can represent $K$ by a complex $K^\bullet $ of finite projective $A$-modules. By Lemma 15.73.2 the object $K^\vee $ is represented by the complex $E^\bullet = \mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , A)$. Note that $E^ n = \mathop{\mathrm{Hom}}\nolimits _ A(K^{-n}, A)$ and the differentials of $E^\bullet $ are the transpose of the differentials of $K^\bullet $ up to sign. Observe that $E^\bullet $ is the left dual of $K^\bullet $ in the symmetric monoidal category of complexes of $R$-modules, see Lemma 15.72.2. There is a canonical map

\[ K^\bullet = \text{Tot}(\mathop{\mathrm{Hom}}\nolimits ^\bullet (A, A) \otimes _ A K^\bullet ) \longrightarrow \mathop{\mathrm{Hom}}\nolimits ^\bullet (\mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , A), A) \]

which up to sign uses the evaluation map in each degree, see Lemma 15.71.6. (For sign rules see Section 15.72.) Thus this map defines a canonical isomorphism $(K^\vee )^\vee \cong K$ as the double dual of a finite projective module is itself.

The second equality follows from the first by Lemma 15.73.1 and Derived Categories, Lemma 13.19.8 as well as the definition of Ext groups, see Derived Categories, Section 13.27. Let $L^\bullet $ be a complex of $A$-modules representing $L$. By Section 15.72 item (11) there is a canonical isomorphism

\[ \text{Tot}(L^\bullet \otimes _ A E^\bullet ) \longrightarrow \mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , L^\bullet ) \]

of complexes of $A$-modules. This proves the first displayed equality and the proof is complete. $\square$


Comments (0)

There are also:

  • 7 comment(s) on Section 15.74: Perfect complexes

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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07VI. Beware of the difference between the letter 'O' and the digit '0'.