# Reference-Request: Symmetric Product Schemes

Is there a good reference for the theory of symmetric product schemes? (I only need a few basic things, the construction, etc.)

Googling it turned up a lot of papers which use it as if it's common knowledge, so I suspect there should be a reference somewhere, but I can't find any.

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Do you know how to construct products of schemes? Do you know how to quotient schemes by the action of a finite group? –  Qiaochu Yuan Aug 3 '12 at 17:09
I didn't know that you could quotient schemes by the action of a finite group. How would you do that? –  only Aug 3 '12 at 17:21
If $R$ is a commutative ring, then an action of $G$ on $\text{Spec } R$ by scheme automorphisms is equivalent to an action of $G$ on $R$ by ring automorphisms, and $\text{Spec } R^G$ is a sensible model of the quotient scheme (where $R^G$ denotes the invariant subring of $R$). The inclusion $R^G \to R$ dualizes to the quotient map $\text{Spec } R \to \text{Spec } R^G$. To extend this to schemes it suffices to find a covering by $G$-invariant affine opens; unfortunately I am not sure if this is always possible... –  Qiaochu Yuan Aug 3 '12 at 17:25
Hmm, the answers to mathoverflow.net/questions/1558/… seem to suggest that it's not always possible to quotient by a finite group... –  only Aug 3 '12 at 18:17
Yes, okay, the issue is that orbits may not be contained in affine opens. For any scheme $S$ such that a finite subset of $S$ is contained in an affine open this is not a problem for the action of the symmetric group on $S \times S \times ... \times S$. –  Qiaochu Yuan Aug 3 '12 at 18:25
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