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36.39 Determinants of complexes

This section is the continuation of More on Algebra, Section 15.122. 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 36.39.1. 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 15.122.

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, 36.10.4, and 36.10.7. Then we can consider the invertible $A$-module $\det (L^\bullet )$ constructed in More on Algebra, Lemma 15.122.4. 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 36.39.2. The construction of Lemma 36.39.1 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.

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