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Let $X=Proj(A)$ be a projective scheme, one can the moduli space of coherent sheaves on $X$ with fixed Hilbert polynomial and stability. Since coherent sheaves on $X$ are all obtained as the sheafification $\widetilde{M}$ of a graded $A$-module $M$, it is reasonable to ask whether we can construct the moduli space of sheaves on non-commutative space, i.e. the moduli space of graded right $B$-modules for some $good$ non-commutative graded ring $B$ with some fixed data. I am pretty sure that some works in this direction have been done (what condition on $B$ and what data of modules should be fixed, etc).

Could anyone tell me a reference or paper which discuss the construction? Thank you very much.

Edit I have for example a quantum plane in my mind.

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2 Answers 2

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I think the paper "Abstract Hilbert Schemes" by Artin and Zhang might be useful. This may be one of the other papers Peter had in mind in his answer.

The basic idea as far as I understand it seems to be the following; things work nicely if your ring is strongly noetherian! This condition means that whenever you tensor your ring with a commutative noetherian ring your ring is still noetherian. There are examples of rings which are noetherian but not strongly noetherian (the naive blow-ups of Rogalski), but the quantum plane is certainly strongly noetherian - it is an Ore extension and when you tensor with a commutative ring you can express the result as another Ore extension. Then you can use the version of Hilbert's Theorem for Ore extensions, Theorem 2.6 in Goodearl and Warfield's "Introduction to Noncommutative Noetherian rings".

Back to Artin and Zhang's paper, quoting from the abstract, they show that "For the category of the graded modules over a strongly Noetherian graded ring, the Hilbert functor of graded modules with a fixed Hilbert series is represented by a commutative projective scheme."

There is a recent paper of Nevins and Sierra which I think looks at what can be said for non-strongly noetherian rings, http://arxiv.org/abs/1009.2061. I think that they may work with just point modules though, these being the graded modules with Hilbert series that of a polynomial ring in one variable.

I hope this is the kind of thing you were looking for. Also, I'm sure that someone more expert than myself would be able to explain in more detail what Artin and Zhang do.

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Thank you for the detailed answer, Andrew. This is exactly what I was looking for. I have heard of Artin and Zhang's "Abstract Hilbert Schemes" before but I totally forgot about it. This seems a good staring point, or they might even solve my question. Your information is extremely useful, thank you! –  user2013 Sep 14 '12 at 17:29

This question is very open-ended. The paper by Berest and Wilson below address one possible interpretation of the question, but it may not be exactly what you're looking for. The paper gives an explicit description of the moduli space $M$ of isomorphism classes of left ideals in the Weyl algebra $\mathbb C \langle x,y\rangle / (xy-yx-1)$. The space $M$ is diffeomorphic (but not isomorphic as schemes) to the disjoint union $\sqcup_n H(n,\mathbb C^2)$ of the Hilbert schemes of $n$ points in the plane. There are a number of other papers in this direction.

http://arxiv.org/abs/math/0104248

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Thank you for the comment. This paper is interesting as they think all modules over the Weyl algebra. I expected that one would need some invariants fixed (Hilbert polynomial etc). –  user2013 Sep 14 '12 at 3:42
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In the commutative case, to get a finite dimensional moduli space you can fix $n$, i.e. look at the moduli space of modules over $\mathbb C[x,y]$ whose Hilbert polynomial is the constant $n$. In the noncommutative case there is also an invariant that picks out the piece of $M$ corresponding to $H(n,\mathbb C^2)$, but its definition is more subtle than in the commutative case. –  Peter Samuelson Sep 14 '12 at 21:29

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