Talk:Algebraic stack
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Untitled[edit]
The final sentence in the motivation section is broken. —Preceding unsigned comment added by 129.240.21.51 (talk) 08:35, 2 February 2010 (UTC)
What even are these??? —Keenan Pepper 02:18, 16 November 2005 (UTC)
Proposing the article to be split[edit]
The article's title suggests that it covers stacks (i.e., fibred categories over a site (or a topos, or a topological space) for which "descent works"). However, the present article is essentially a stub for algebraic stacks, i.e., very particular types of stacks in groupoids that are suitably "covered" by algebraic spaces. Whereas stacks in general are a formalisation of a key concept in general descent theory, algberaic stacks (of the D-M and Artin types) are a generalisation of schemes. The two topics are in fact quite dissimilar in terms of questions considered and techniques used.
Thus, the article should be split into a new article on general stacks, and another one to cover algebraic stacks. For the general article a title like Stack (descent theory) would be better than Stack (category theory) as the framework for stacks includes also a Grothendieck topology and not only categories. Stca74 17:30, 3 September 2007 (UTC)
- With no objections having emerged, I'm moving ahead with the split. I've requested speedy deletion of the redirect Algebraic stack to move this page there. I shall next establish the stub Stack (descent theory). Stca74 17:49, 9 September 2007 (UTC)
- Split completed. We now have two stubs to work on — this one and Stack (descent theory). Stca74 05:56, 10 September 2007 (UTC)
Article improvements[edit]
This section is a sketch of ways the article can be improved
Defining morphisms[edit]
Defining morphisms of algebraic stacks and giving some examples. This could include
- maps between classifying stacks induced from group morphisms
- coarse maps from good moduli spaces http://www.numdam.org/item/AIF_2013__63_6_2349_0/
Theorems[edit]
- Keel, Mori - construction of coarse moduli spaces - https://arxiv.org/abs/alg-geom/9508012 - reference Keel Mori theorem as main article
- A Luna étale slice theorem for algebraic stacks - migrate content from stacks page - change url to - https://annals.math.princeton.edu/2020/191-3/p01
- Stratification by Gerbes - https://stacks.math.columbia.edu/tag/06RB
- Relation with Hopf algebroids - https://www.mathi.uni-heidelberg.de/~tsitte/thesis.pdf
- Perfect complexes and Derived noncommutative algebraic geometry - https://www.math.arizona.edu/~jackhall/papers/perfect_stacks.pdf
Examples[edit]
- Classifying spaces
- Quotient stacks
- Weighted projective stacks - Cite weighted projective space and include material about stacky enhancement
- Tame stacks, http://www.numdam.org/item/AIF_2008__58_4_1057_0/
- Inertia stack - https://stacks.math.columbia.edu/tag/0373 - Heinloth - https://www.uni-due.de/~hm0002/Artikel/StacksCourse_v2.pdf
- Quotient Stacks and String Orbifolds - Eric Sharpe - https://cds.cern.ch/record/489543/files/0103008.pdf
- Gerbes and Brauer groups over stacks - https://arxiv.org/abs/1705.01382v4
Moduli stacks[edit]
- Moduli of root stacks
- Picard stack
- Hilbert stacks of weighted projective stacks
- page 23 of "Degenerations of Abelian Varieties" sketches two constructions of the moduli stack of abelian varieties
- moduli stacks of vector bundles on curves -
Define various sheaves[edit]
- Use http://individual.utoronto.ca/groechenig/stacks.pdf for defining quasi-coherent stacks
- Note: LMB sheaves on Lisse-Etale site has issues, Martin Olsson fixes this problem here: https://math.berkeley.edu/~molsson/qcohrevised.pdf
- Sheaves of abelian groups
- (Quasi-)coherent sheaves
- l-adic sheaves
Moduli of vector bundles[edit]
- Smooth atlas (https://arxiv.org/abs/math/9911199v1)
References[edit]
- https://web.archive.org/web/20201006181315/http://individual.utoronto.ca/groechenig/stacks.pdf - Excellent resource
- https://math.berkeley.edu/~molsson/qcohrevised.pdf - Sheaves on Artin stacks
- https://math.berkeley.edu/~molsson/pervers3.pdf - Perverse sheaves on Artin stacks
Other topics to touch upon[edit]
- Dimension theory of stacks - https://stacks.math.columbia.edu/tag/0AFL
- Properties of morphisms of stacks
- Fibered products of stacks and delooping - http://homepages.math.uic.edu/~henri/notes/Line_Bundles_on_stacks.pdf
- Properties of algebraic stacks - https://stacks.math.columbia.edu/tag/04X8
- Quasi-Coherent sheaves - https://stacks.math.columbia.edu/tag/06WU
- Presentation - https://stacks.math.columbia.edu/tag/04T5
- Gerbes and quotient stacks - https://stacks.math.columbia.edu/tag/06PD
- https://impa.br/wp-content/uploads/2017/04/PM_36.pdf - gives definition of sheaves/vector bundles + cohomology
- https://link.springer.com/book/10.1007/978-3-540-24899-6 - another reference