# stacks/stacks-project

Determinant of perfect complexes

On schemes only and only for tor-amplitude in [-1, 0]
 @@ -8224,6 +8224,108 @@ \section{K-groups} \section{Determinants of complexes} \label{section-det-two-terms} \noindent This section is the continuation of More on Algebra, Section \ref{more-algebra-section-determinants-complexes}. 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 \cite{determinant} for the original exposition. We will add this material later (insert future reference). \medskip\noindent 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 $[-1, 0]$. \begin{lemma} \label{lemma-determinant-two-term-complexes} 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 \ref{more-algebra-section-determinants-complexes}. \end{lemma} \begin{proof} Let $L$ be an object of the left hand side. If $\Spec(A) = U \subset X$ 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 \ref{lemma-affine-compare-bounded}, \ref{lemma-tor-dimension-affine}, and \ref{lemma-perfect-affine}. Then we can consider the invertible $A$-module $\det(L^\bullet)$ constructed in More on Algebra, Lemma \ref{more-algebra-lemma-determinant-two-term-complexes}. If $\Spec(B) = V \subset U$ 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 \ref{lemma-quasi-coherence-pullback} 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. \end{proof} \input{chapters} \bibliography{my}