# The universal multiset for a finite scheme - reference request

If $$X$$ is a finite set of size $$n$$, then by listing the elements of $$X$$ we get a canonical element of the symmetric power $$X^n/\Sigma_n$$, which we can call the universal multiset for $$X$$.

Now let $$X$$ instead be an affine scheme $$\text{spec}(A)$$ over a base scheme $$S=\text{spec}(k)$$, and suppose that $$A$$ is a free (or just projective) module of rank $$n$$ over $$k$$. There is then a canonical section $$S\to X^n/\Sigma_n$$, which is strongly analogous to the universal multiset as described above. The corresponding ring map $$\nu\colon(A^{\otimes n})^{\Sigma_n}\to k$$ is almost characterised by the fact that $$\nu(a^{\otimes n})=\text{norm}_{A/k}(a)$$ for all $$a\in A$$.

When I thought of this the other day, I was amazed by the fact that it had never occurred to me before and that I did not remember seeing it in the literature, although it is adjacent to things that I have thought about many times. I think that the closest thing that I have seen is the notion of a "full set of sections" as described in Section 1.8 of Katz and Mazur's "Arithmetic moduli of elliptic curves", but even that is not all that close.

My question is: does this map $$\nu$$ appear in the literature, and if so, under what name?

For completeness, here is the actual definition. Let $$P$$ be the graded symmetric algebra over $$k$$ generated by $$P_1=A^\vee=\text{Hom}_k(A,k)$$, and let $$u\in P_1\otimes A$$ correspond to the identity under the usual isomorphism $$P_1\otimes A=A^\vee\otimes A=\text{Hom}_k(A,A)$$. Put $$\tilde{u}=\text{norm}_{(P\otimes A)/P}(u)\in P_n$$. There is a standard perfect pairing between $$P_n=(A^\vee)^{\otimes n}_{\Sigma_n}$$ and $$(A^{\otimes n})^{\Sigma_n}$$, and we define $$\nu(a)=\langle\tilde{u},a\rangle$$. With a little work one can check that this is a ring homomorphism, and that it satisfies $$\nu(a^{\otimes n})=\text{norm}_{A/k}(a)$$.

I have seen the morphishm $$\nu\colon (A^{\otimes n})^{\Sigma_n}\to k$$ in a couple of places:

1. On page 81 of

A. Suslin, V. Voevodsky, Singular homology of abstract algebraic varieties

It is defined when $$A$$ is finite locally free over $$k$$, but also when $$k$$ is a normal, $$A$$ is an integral domain, and $$\operatorname{Spec}(A)\to\operatorname{Spec}(k)$$ is finite and surjective. The latter construction is used there to show that symmetric powers of schemes represent finite correspondences in the sense of Voevodsky (the "motivic Dold-Thom theorem"), which is a standard result in motivic homotopy theory.

1. It is used to construct the norm functor $$N_{A/k}\colon \mathrm{Mod}_A \to \mathrm{Mod}_k$$ in Section 3.1 of

D. Ferrand, Un foncteur norme

I suspect this construction also appears somewhere in David Rydh's thesis, which is related to both of the above papers, although I couldn't immediately locate it.