The Stacks project

36.37 Determinants of complexes

This section is the continuation of More on Algebra, Section 15.111. For any ringed space $(X, \mathcal{O}_ X)$ there is a functor

\[ \det : \left\{ \begin{matrix} \text{category of perfect complexes} \\ \text{morphisms are isomorphisms} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{category of invertible modules} \\ \text{morphisms are isomorphisms} \end{matrix} \right\} \]

Moreover, given an object $(L, F)$ of the filtered derived category $DF(\mathcal{O}_ X)$ whose filtration is finite and whose graded parts are perfect complexes, there is a canonical isomorphism $\det (\text{gr}L) \to \det (L)$. See [determinant] for the original exposition. We will add this material later (insert future reference).

For the moment we will present an ad hoc construction in the case where $X$ is a scheme and where we consider perfect objects $L$ in $D(\mathcal{O}_ X)$ of tor-amplitude in $[-1, 0]$.

Lemma 36.37.1. Let $X$ be a scheme. There is a functor

\[ \det : \left\{ \begin{matrix} \text{category of perfect complexes} \\ \text{with tor amplitude in }[-1, 0] \\ \text{morphisms are isomorphisms} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{category of invertible modules} \\ \text{morphisms are isomorphisms} \end{matrix} \right\} \]

In addition, given a rank $0$ perfect object $L$ of $D(\mathcal{O}_ X)$ with tor-amplitude in $[-1, 0]$ there is a canonical element $\delta (L) \in \Gamma (X, \det (L))$ such that for any isomorphism $a : L \to K$ in $D(\mathcal{O}_ X)$ we have $\det (a)(\delta (L)) = \delta (K)$. Moreover, the construction is affine locally given by the construction of More on Algebra, Section 15.111.

Proof. Let $L$ be an object of the left hand side. If $\mathop{\mathrm{Spec}}(A) = U \subset X$ is an affine open, then $L|_ U$ corresponds to a perfect complex $L^\bullet $ of $A$-modules with tor-amplitude in $[-1, 0]$, see Lemmas 36.3.5, 36.9.4, and 36.9.7. Then we can consider the invertible $A$-module $\det (L^\bullet )$ constructed in More on Algebra, Lemma 15.111.4. If $\mathop{\mathrm{Spec}}(B) = V \subset U$ is another affine open contained in $U$, then $\det (L^\bullet ) \otimes _ A B = \det (L^\bullet \otimes _ A B)$ and hence this construction is compatible with restriction mappings (see Lemma 36.3.8 and note $A \to B$ is flat). Thus we can glue these invertible modules to obtain an invertible module $\det (L)$ on $X$. The functoriality and canonical sections are constructed in exactly the same manner. Details omitted. $\square$

Remark 36.37.2. The construction of Lemma 36.37.1 is compatible with pullbacks. More precisely, given a morphism $f : X \to Y$ of schemes and a perfect object $K$ of $D(\mathcal{O}_ Y)$ of tor-amplitude in $[-1, 0]$ then $Lf^*K$ is a perfect object $K$ of $D(\mathcal{O}_ X)$ of tor-amplitude in $[-1, 0]$ and we have a canonical identification

\[ f^*\det (K) \longrightarrow \det (Lf^*K) \]

Moreover, if $K$ has rank $0$, then $\delta (K)$ pulls back to $\delta (Lf^*K)$ via this map. This is clear from the affine local construction of the determinant.


Comments (0)


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 0FJW. Beware of the difference between the letter 'O' and the digit '0'.