# Quasicoherent sheaves on superschemes

I am interested in learning about super algebraic geometry (some objects I am studying seem to be naturally superstacks, at least in some sense). What would be the best reference for the subject? I am particularly interested in (quasi)coherent sheaves over superschemes and on criteria for when a global section is $\otimes$-nilpotent (that is $s^{\otimes n}=0$ for some n).

EDIT: I am intersted mainly in places where the theory is developed integrally, so you can have elements in odd degree whose square is 2-torsion but not 0.

• By the way, do you have any reference about superstacks? May 6 '15 at 16:25
• Only the first hit on google: arxiv.org/abs/1204.4369. I do not vouch for it :). May 6 '15 at 16:32

Some basic references are:

1. the book by Manin Gauge Field Theory and Complex Geometry ;
2. the first paper by Delinge in Quantum Fields and Strings: A Course for Mathematicians ;
3. the recent paper on the Arxiv by Donagi and Witten Supermoduli Space Is Not Projected .

About your question, my guess would be the following. Let $X$ be a super-scheme and $X^{red}$ the reduced space. There is a natural the embedding $$\iota\colon X^{red} \to X$$ then $s$ is nilpotent if and only if $\iota^* s=0$, or it is nilpotent.

(I might have been naif on your definition of $\otimes$-nilpotent)

An easy example. Consider the affine line $\mathbb{A}^{0|1}$. Take the odd function $\eta$, then $\eta\otimes\eta=\eta^2(1\otimes 1)=0$ but $\eta$ is not. In this case, $X^{red}$ is a point and $\iota^*\eta=0$. (The sheaf I am considering is just the structure sheaf.)

In general, if $\iota^*s=0$, then, locally, all terms of $s$ will have non-trivial odd factors; so $s^{\otimes n}=0$ for $n$ big enough.

• Thanks for the references! I meant a global section of any quasicoherent module. I was hoping something along the lines "if it is zero on every point", but I do not know what a point is in supergeometry :). May 6 '15 at 16:33
• By the way I am not accepting your answer for a while in the hope that someone one will write something but if I forget about it feel free to remind me :) May 6 '15 at 16:33
• It's ok. With respect to your question, it is a bit like you are working on the spectrum of $\mathbb{C}[x]/(x^2)$, then $x$ is zero at every point. May 6 '15 at 16:37
• I've skimmed the references you gave and it seems that they always work over $\mathbb{C}$. Is there anything about what happens integrally? Note that here for example you may have odd elements whose square is 2-torsion but not 0. May 6 '15 at 22:30
• As a far as I know, nothing has been written over $\mathbb{Z}$. Of course this can happen, you can check it by working with $\mathcal{F}^{red}$ on $X^{red}$. May 7 '15 at 13:32