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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeDec 28th 2010
• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeDec 28th 2010

never heard :)

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeDec 28th 2010

It’s kind of important in the context of the discusison of “2-affine” geometry that we are having on the $n$Cafe: a projective variety $X/\mathbb{G}_m$ is not affine, but if we think of it as a weak quotient $X // \mathbb{G}_m$ instead thus realizing it as a geometric stack, we see that it is “2-affine” (in the sense of Tannaka duality of geometric stacks.)

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeDec 28th 2010

never heard :)

by the way: if you ask Google, you’ll see that the term is used by a bunch of autors.

• CommentRowNumber5.
• CommentAuthorzskoda
• CommentTimeDec 29th 2010

It’s kind of important in the context of the discusison of “2-affine” geometry that we are having on the $n$Cafe: a projective variety $X/\mathbb{G}_m$ is not affine, but if we think of it as a weak quotient $X // \mathbb{G}_m$ instead thus realizing it as a geometric stack, we see that it is “2-affine”

Right, but one can not think of one object as something else. Most of algebraic geometers find many substantial advantages and applications of coarse moduli spaces compared with stack solutions like fine moduli spaces. Resolutions are nicer than the objects themselves, but the object of the study (say in algebraic geometry), most of the time is the object itself. I understand that it is easier to take the trivial replacement (stack) and this is what I do most of the time in my work in noncommutative geometry, and that is one of the reasons why most of algebraic geometers are not interested in my work.

by the way: if you ask Google, you’ll see that the term is used by a bunch of autors

Has nothing to do with me.

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeDec 29th 2010

One of the areas where the difficulty of coarse moduli spaces is very apparent is GIT (geometric invariant theory). It is a well defined area of mathematics, with important 19-th century problems making its core. While generalizing stacks to noncommutative setup is not difficult, I would be very happy if I would know anything about how to generalize GIT to noncommutative setup to use it in noncommutative invariant theory which is in its infancy.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeDec 29th 2010
• (edited Dec 29th 2010)

Maybe help me: isn’t the action of $\mathbb{G}_m$ on $\mathbb{A}^{n+1}-\{0\}$ free (and transitive)? Doesn’t that make the weak quotient equivalent to the strict quotient?

• CommentRowNumber8.
• CommentAuthorzskoda
• CommentTimeDec 29th 2010

Yes, but already the simplest and most widely used generalizations, like the weighted projective spaces do not share this feature.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeDec 29th 2010

Yes, but that’s not the case that I mentioned in the paragraoh that you quoted and replied to.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeDec 29th 2010

Oh, I see, it is. I said “projective variety”. I should have said “projective space”. Sorry.

• CommentRowNumber11.
• CommentAuthorzskoda
• CommentTimeDec 29th 2010

The matter is just the relation between the emphasis on two kinds of content; one very specific and another aimed at the simplest gadgets expressing quotients. Both ideas aree very powerful, I just object to either side if one claims that one of the two ideas replaces another. Grothendieck was master of both. Not only he devised stacks but also solved the question of representability of a number of interesting functors, the questions which were open for about a century, what lead to solutions as Hilbert schemes, Quot schemes etc.