We hope to cover the fundamentals of derived algebraic geometry, with an emphasis on the theory of ind-coherent sheaves.

Fridays 3 - 4:30pm. Science Center 232. Spring 2016.

Main reference

[GR]* A study in derived algebraic geometry * (Gaitsgory-Rozenblyum). See the bottom of this page.

Supplementary reading

(Suggestions welcome!)

[IndCoh]* Ind-coherent sheaves * (Gaitsgory). Link to pdf.

[HTT]* Higher topos theory * (Lurie). Link to pdf.

[HA]* Higher algebra* (Lurie). Link to pdf.

[AG]* Singular support of coherent sheaves, and the geometric Langlands conjecture * (Arinkin-Gaitsgory). Link to pdf.

[SAG]* Spectral algebraic geometry * (Lurie, under construction). Link to pdf.

Schedule

You are welcome to sign up for any week with a [speaker TBA]. This is a learning seminar, which means everyone is encouraged (and should be) giving talks on materials they don't already know.

Week 1. Overview [Justin Campbell]. January 29.

Part I: fundamentals of derived algebraic geometry

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This part is aimed to be an introduction to derived algebraic geometry. We will first go through some higher category theory and higher algebra. Then we will define (higher) derived Artin stacks, talk a bit about cotangent complexes and deformation theory, working out examples as we go along.
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Week 2. A crash course on ∞-categories [Yifei Zhao]. February 5. (The notes are still being completed!)

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Some examples of ∞-categories that are important for us, and how to construct them. Kan extensions along inclusions (§4.3.2 of [HTT]), straightening/unstraightening (§3.2 of [HTT]), adjoint functors (Chapter 5 of [HTT]).
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Week 3. Higher algebra [Alex Perry]. February 12.

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This talk should introduce (symmetric) monoidal structures, algebras, and modules. We should then move on to dualizability and its relation to adjoint functors (§I.1.4 of [GR]), stable cocomplete ∞-categories with the Lurie tensor product and spectra (§I.1.6 of [GR]), ind-completion and compact generation (§I.1.7 of [GR]) and DG categories (§I.1.10 of [GR]).
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Week 4. Stable ∞-categories [Justin Campbell] and objects of derived algebraic geometry [Changho Han]. February 19.

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Prestacks, DG schemes, k-Artin stacks (though you're welcome to substitute these with your preferred model of derived schemes/stacks, and not necessarily limited to characteristic zero), convergence, quasi-coherent sheaves, perfect complexes. Following §I.2 of [GR].
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Week 5. Deformation theory I [Dmitrii Kubrak]. February 26.

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Cotangent complex, (split) square-zero extensions, factorization of a derived scheme into square-zero extensions, smoothness. Following §III.1, sections 1-7 of [GR].
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Week 6. Examples and applications [Michael McBreen]. March 4.

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Here we give some applications of derived algebraic geometry. Applications to representation theory could include computing the cotangent complexes of classifying stacks, Bun(G), LocSys(G), etc (covered in §10 of [AG]), derived Springer fibers. Applications to algebraic geometry could include intersection theory, virtual fundamental classes, etc.
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Part II: Ind-coherent sheaves and inf-schemes

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In this part, we work towards the theory of ind-coherent sheaves on inf-schemes. The talks here will cover the bulk of chapters II-III of [GR].
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Week 7. Ind-coherent sheaves I [Justin Campbell]. March 11.

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Introduce ind-coherent sheaves on a scheme, following §II.1 of [GR]. Ignore all (∞,2)-categorical stuff.
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Week 8. Deformation theory II [Dmitrii Kubrak]. March 18.

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Cover the rest of §III.1 of [GR]. Introduce ind-schemes and inf-schemes, which is §III.2 of [GR].
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Week 9. Ind-coherent sheaves II and Serre duality [Akhil Mathew]. March 25.

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Introduce the category of correspondences as in Chapter V of [GR], then follow §II.2 of [GR]. Mention the relations between ind-coherent and quasi-coherent sheaves (see §II.3 of [GR]).
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Week 10. Ind-coherent sheaves III [Hunter Spink]. April 1.

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Extending the theory of ind-coherent sheaves to inf-schemes, following §III.3 of [GR].
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Week 11. Applications to crystals [David Yang]. April 8.

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Following §III.4 of [GR].
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Part III: Formal geometry and Lie theory

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This is arguably the coolest part of the book. We will see the full power of delooping put to use.
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Week 12. Formal moduli problems [Ravi Jagadeesan]. April 15.

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Explain the equivalence between formal moduli problems and formal groupoids, following §IV.1 of [GR]. Deduce that the loop functor defines an equivalence from pointed formal moduli problems to formal groups.
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Week 13. Operads and Koszul duality [Justin Campbell]. April 22.

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Define operads, co-operads, their algebras and coalgebras. Introduce the bar construction and Koszul duality; use this to define the Lie operad as the Koszul dual to the cocommutative co-operad. State the PBW theorem and the characterization of the universal enveloping algebra as the Chevalley functor composed with the loop functor. All of this is in §IV.2 of [GR].
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Week 14. Lie theory in formal geometry [Yifei Zhao]. April 29.

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Describe the equivalence between formal groups and Lie algebras; in particular explain how to obtain the Lie algebra structure on the tangent space at the identity. This is §IV.3 of [GR]. Then discuss Lie algebroids, following §IV.4 of loc.cit. Make sure to mention that a Lie algebroid shifted by -1 acquires a Lie algebra structure.
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Week 15. Infinitesimal differential geometry [Justin Campbell]. May 6.

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Introduce deformation to the normal bundle. Define infinitesimal neighborhoods of a morphism and use them to construct the Hodge filtration. This is in §IV.5 of [GR].
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