# universal property of the determinant bundle

Let $X$ be a nontrivial ringed space (i.e. all stalks are nonzero). To every locally free module $M$ on $X$ of constant rank $n$ we can associated it's determinant $\det(M)$, which is a line bundle and is defined as the $n$th exterior power of $M$. We can also define $\det(M)$ if the rank is not assumed to be constant. It's surely locally constant, so $X$ is partitioned into the open subsets $(X_n)_{n \geq 0}$, where $M$ has constant rank $n$, and we can glue the $\det(M|_{X_n})$ to get the determinant $\det(M)$. There is no doubt that this is well-defined, but it seems to me a bit uncanonical.

For example look at the functoriality: If $f : M \to N$ is a homomorphism, then $\det(f) : \det(M) \to \det(N)$ is first defined on the intersections on the open subsets on which $\det(M)$ and $\det(N)$ are defined, and then glued. Then a small argument is needed to prove that this is, indeed, a functor. Isn't this ugly? Therefore:

Question Is there a characterization of the functor $\det(-)$ from locally finite free modules to line bundles which does not depend on partitions of $X$?

For example, for every $n \geq 0$, the functor $\det(-)$ on locally free modules of rank $n$ has a universal property, namely $\text{Hom}(\det(M),L)$ corresponds to alternating maps $M^n \to L$. You can write down a similar universal property when $n$ is a fixed locally constant function on $X$. But my question is if we can do it without fixing $n$.

• I don't think there's anyting uncanonical about this... Unless I'm confused, you're just partitioning $X$ into its connected components and "gluing" along their empty intersection. So the issue in the second paragraph never arises. Feb 7 '11 at 17:42
• Universal properties of the determinant are discussed in this MO question: <a href="mathoverflow.net/questions/7124/…>. The point is rather than glue you simply resolve your $M$ by a finite complex of locally free sheaves, and thus avoid any jumping in rank. Feb 7 '11 at 18:04
• Dave and David, thanks for the comments! Feb 7 '11 at 19:32
• Does Dmitri's suggestion work more generally? e.g. suppose I have a finite complex of vector bundles on a singular variety. (I realize this should be a comment, but I don't have enough rep!) Actually, that direct sum seems unfortunate if the goal is to always get a line bundle.... Dec 5 '11 at 19:48

• This paper characterizes the functor $\det$ with a list of properties. Really interesting! Feb 8 '11 at 10:18