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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 $\nu(a^{\otimes n})=\text{norm}_{A/k}(a)$.

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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.

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