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Determinant of perfect complexes

On schemes only and only for tor-amplitude in [-1, 0]
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aisejohan committed Sep 10, 2019
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@@ -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}

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