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It is well known that Serre [FAC] gave us a nice categorical description for quasi coherent sheaves on projective scheme, it is a proj-category.(graded modules category localized by Serre subcategory)

Now, I wonder whether anybody has developed the proj-category formulation for ind-projective schemes.

It seems that Beilinson-Drinfeld have create some framework to deal with such kind of things, but I am not sure whether they talked about the categorical formulation for ind-Projective scheme in their book...

Of course, I can develop this by myself, however, If there exists some work, I will be happy to look at.

Thanks in advance

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Could you give some basic information (especially the definition) of ind-projective schemes? –  Martin Brandenburg Mar 21 '11 at 1:21

1 Answer 1

Often (perhaps even typically?) in the ind-projective case, people restrict attention to sheaves whose support is finite-dimensional, i.e. if the ind-projective scheme is the limit of the projective schemes $X_n$ under closed immersions, then they consider sheaves which live on one of the $X_n$ (which can then be thought of living on $X_{n'}$ for all higher $n'$ via pushforward). For such sheaves, one may be able to reduce things to Serre's desription, applied to $X_n$. (The complication being that $n$ is not uniquely determined, i.e. can be replaced by $n'$ for any $n' \geq n$.)

Sorry if this doesn't tell you anything that you didn't already know!

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