# determinant of a perfect complex

Say $K_\bullet$ is a bounded complex of vector bundles. I seem to want the determinant of $K_\bullet$ to be the alternating tensor product of the terms of the complex:

$\det(K) = \bigotimes_n \det(K_n)^{(-1)^n}$.

Is there a reason why this is the right definition (or the wrong definition)? Is there a better definition?

It is a theorem of Deligne that this is essentialy the only possible formula if you ask for the determinant functor to satisfy some natural properties (mainly det has to be compatible with exact sequences). See these Slides by Fernando Muro for example.

• Thanks for your very pertinent link to Fernando Murro's slides: they are splendid and he has others on his homepege. – Georges Elencwajg Nov 29 '09 at 15:52
• Is it really a theorem of Deligne that this is "canonical"? I thought it was Knudsen-Mumford. – user5172 Apr 6 '10 at 14:13
• And I thought it was essentially due to Tate (modulo the equivalence of traces and determinants) – Reimundo Heluani May 20 '16 at 13:11

One place this is used beautifully (and where I learned it) is Beilinson's work on epsilon factors.

• Beilinson's article contains a puzzling remark: "Arguably, the common language of category theory may be inadequate for describing the homotopy world."(p.4). What do you think about that? – Thomas Riepe Dec 1 '09 at 0:58
• I think it's his way of saying we need oo-categories - he's pointing out the inadequacies of both homotopy categories and model categories for doing algebra and geometry the way he thinks they ought to be done (Lurie's work provides a satisfying solution IMHO). – David Ben-Zvi Dec 1 '09 at 3:48

You should take a look at the appendix A of "Discriminants, Resultants, and Multidimensional Determinants" by Gelfand, Kapranov and Zelevinsky

As I understand the construction of the determinant of a perfect complex, this definition is quite straightforward, following from the fact that in a short exact sequence, say $$0\rightarrow S\rightarrow E\rightarrow Q\rightarrow 0$$ defining the determinant of the sequence to be the alternating tensor is the canonical way to make it isomorphic to $\mathbb{1}$.

Also, I think good references to this may be the original paper by Knudsen-Mumford, a book by Kato, and also a paper by Kings which are listed below:

Finn Faye Knudsen and David Mumford, The projectivity of the moduli space of stable curves. I. Preliminaries on ''det'' and ''Div'' (pdf). The part about determinants appears in Chapter I, but note that there is a typo defining the determinant, namely in the map of the transposition of tensor product, there should be $\alpha\cdot\beta$ instead the sum of these two as a power of $-1$;

Guido Kings, An introduction to the equivariant Tamagawa number conjecture: the relation to the Birch-Swinnerton-Dyer conjecture (pdf)

There is a part about determinants in lecture 1 section 5, where there are not a lot of details but it provides a good view towards the construction of determinant.

Kazuya Kato, Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via $B_{dR}$, part I Springer LNM 1553 pp 50-163 (doi:10.1007/BFb0084729), which mentions determinant in 2.1.