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



The universal multiset for a finite scheme - reference request
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](https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/singular_homology_published.pdf)


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.


2. 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](http://www.numdam.org/article/BSMF_1998__126_1_1_0.pdf)


I suspect this construction also appears somewhere in [David Rydh's thesis](https://people.kth.se/%7Edary/thesis/thesis.pdf), which is related to both of the above papers, although I couldn't immediately locate it.


