Talk:Algebraic stack

"fppf scheme?"

I think the phrase "fppf scheme" in item 3 in the definition is misleading, since the fppf site used here is the big one, not the small one. (It suggests that we should restrict attention to schemes (over the base) satisfying the "fppf" property, but this is not the case.) — Preceding unsigned comment added by 5.13.18.163 (talk) 09:52, 6 August 2024 (UTC)

Untitled

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

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

This section is a sketch of ways the article can be improved

Defining morphisms

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 $[X_{ss}/G]\to X_{ss}//G$ http://www.numdam.org/item/AIF_2013__63_6_2349_0/

Theorems

• 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

• 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

• 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

• 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

• Smooth atlas (https://arxiv.org/abs/math/9911199v1)

References

• 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

• 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