Talk:Algebraic stack

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