# stacks/stacks-project

Clarify what can be done with det_\kappa

 are given a finite filtration (see Homology, Definition \ref{homology-definition-filtered}) $$M = F^n \supset F^{n + 1} \supset \ldots \supset F^{m - 1} \supset F^m = 0. 0 = F^m \subset F^{m - 1} \subset \ldots \subset F^{n + 1} \subset F^n = M$$ Then there is a canonical isomorphism then there is a well defined and canonical isomorphism $$\gamma_{(M, F)} : \bigotimes\nolimits_i \det\nolimits_\kappa(F^i/F^{i + 1}) \det\nolimits_\kappa(F^{m - 1}/F^m) \otimes_\kappa \ldots \otimes_k \det\nolimits_\kappa(F^n/F^{n + 1}) \longrightarrow \det\nolimits_\kappa(M)$$ well defined up to sign(!). One can make the sign explicit either by giving a well defined order of the terms in the tensor product (starting with higher indices unfortunately), and by thinking of the target category for the functor $\det_\kappa$ as the category of $1$-dimensional super vector spaces. See \cite[Section 1]{determinant}. To construct it we use isomorphisms of Lemma \ref{lemma-det-exact-sequences} coming from the short exact sequences $0 \to F^{i - 1}/F^i \to M/F^i \to M/F^{i - 1} \to 0$. Part (2) of Lemma \ref{lemma-det-exact-sequences} with $G = 0$ shows we obtain the same isomorphism if we use the short exact sequences $0 \to F^i \to F^{i - 1} \to F^{i - 1}/F^i \to 0$. \medskip\noindent Here is another typical result for determinant functors.