--- created: 2022-04-05T23:42 updated: 2023-12-16T00:00 --- --- date: 2022-02-23 18:45 modification date: Thursday 17th March 2022 23:06:08 title: derived algebraic geometry aliases: [derived algebraic geometry, DAG] --- --- - Tags - #higher-algebra/DAG #higher-algebra/derived #higher-algebra/stacks - Refs: - [Someone's thesis](https://hoyois.app.uni-regensburg.de/papers/chern.pdf#page=28&zoom=180,-95,775) - [Intro to spectral AG](https://etale.site/livetex/gepner-at-motives-and-DAG.pdf) #resources/notes #resources/summaries - [Thesis: A Localization Theorem for Derived Loop Spaces and Periodic Cyclic Homology](https://escholarship.org/content/qt05m960s2/qt05m960s2.pdf?t=phm3ss) #resources/theses - [A study in derived algebraic geometry Volume I: Correspondences and duality](http://people.math.harvard.edu/~gaitsgde/GL/Vol1.pdf) #resources/notes - [DAG I the cotangent complex and derived de Rham cohomology (Benjamin Antieau)](https://www.youtube.com/watch?v=zRPa-VAvl6Q&list=PLCe-H2N8-ny6KgQqZ_xLv0ZUg_8fnNjad&index=6) #resources/videos #resources/workshops - [attachments/What is a Derived Stack.pdf](attachments/What%20is%20a%20Derived%20Stack.pdf) #resources/notes #resources/summaries - [attachments/DerivedStacks 1.pdf](attachments/DerivedStacks%201.pdf) - LAGOON seminar: #resources/videos #projects/youtube-videos - [DAG XII](http://people.math.harvard.edu/~lurie/papers/DAG-XIII.pdf) #resources/books - #resources/theses - Intro course (in French!) #resources/course-notes - Links: - [stacks MOC](Unsorted/stacks%20MOC.md) - [E_infty rings](Unsorted/E_n%20ring%20spectrum.md) - [animated rings](animated%20rings) - [algebraic cobordism](algebraic%20cobordism) - [operad](Unsorted/operad.md) - [correspondences](correspondences) - [moduli of curves](moduli%20of%20curves) - [cotangent complex](cotangent%20complex.md) - [derived stack](derived%20stack.md) --- Some applications: > In derived algebraic geometry one constructs "derived schemes", "derived stacks", and so on. It was recently realized that such derived enhancements provide a natural explanation of structures in the [[Langlands correspondence]]. Two different types of examples are the derived [[Galois deformation]] spaces of Galatius-Venkatesh, which were used to explain the structure of cohomology of arithmetic groups, and the derived [[special cycles]] of Feng-Yun-Zhang and Madapusi, which were used to construct [virtual fundamental cycles](virtual%20fundamental%20class.md) comprising arithmetic [theta functions](theta%20function.md). Notation: - DAG: derived algebraic geometry - DDT: derived deformation theory - $\smooth\Var$: smooth (algebraic or projective) varieties - $\dg$: differential graded - $\der\Sch$: derived schems - $\der\St$: derived stacks. - $\IndCoh$: Ind-coherent sheaves - $\HoH(\wait)$: Hochschild homology - $\dgla$: differential graded Lie algebras - $\dg\Lie\Algebroid$: dg Lie algebroids - $\Corr(\cat C)$: correspondences over $\cat C$ - $\LL$: the cotangent complex Summary of abstracts: - **Introductory topics in DAG** - $\Aff\der\Sch, \der\Alg\St$, Artin-Lurie representability. - Reviews deformation theory of smooth proper schemes - Differential forms on $\der\St$ - Shifted symplectic forms on $\der\St$ - **Nonlinear traces** - DAG version of sheaf theories, e.g. $\IndCoh$ and $\Dmod$, without requiring smoothness or transversality - Formalize $\dim(A)$ for dualizable objects $A$ as $\HoH(A)$ - Correspondence categories, derived loop spaces, and fixed point loci - Nonlinear versions of: - Grothendieck-Riemann Roch theorem - Atiyah-Bott-Lefschetz trace formula - Frobenius-Weyl character formula - Applications to groups acting on categories - **Formal modular problems and formal derived stacks** - Lurie-Pridham's theorem formulating Drinfeld's DDT philosophy: formal moduli problems correspond to $\dg\Lie\Alg$ - Koszul duality in Lurie's deformation contexts - Applications to non-split formal moduli problems for a fixed derived scheme, replacing $\dg\Lie\Alg$ with $\dg\Lie\Algebroid$ - Globalize these results to formal thickenings of $\der\St$ - Alternative approach to Gaitsgory-Rozenblyum's result - **Gromov-Witten theory with DAG** - Regard $X\in \smooth\Var$ as an object in $\Corr(\der\St)$ - Construct an operad of stable curves that an act on $X\in \smooth\Var$, encoding the GW theory of $X$ - **Shifted Poisson geometry** - Poisson structures for $\der\St$. Relevant background topics: - Serre's Tor intersection formula, the first example of using $\der\Sch$ - Lie algebra cohomology: - $H^*(\lieg; M)$ for $M\in \mods{\lieg}$, - $\cocomplex{C}(\lieg)$ the Eilenberg-MacLane complex of $\lieg$. - Kontsevich's moduli space of stable maps - Virtual fundamental classes - The Riemann-Roch formula - The Hilbert and Quot schemes - Descent, as a form of gluing - The geometric Langlands correspondence - Geometric representation theory - $D\dash$modules - Quantization of symplectic and Poisson structures - \(p\dash \)adic geometry - Formal/infinitesimal geometry - The tangent complex, say of a moduli space - Integration of a Lie algebra - The cotangent complex - The Grothendieck-Riemann-Roch formula - Derived categories $\DD \cat C$ - Particularly for $\cat C = \QCoh(X), \IndCoh(X), \Dmod$. - The classical result: formal moduli problems over a field are classified by objects in $\dgla$ - A well-known correspondence between Lie algebroids and Lie groupoids - Formal completions of schemes - Operads - Spans and correspondences - \(\K\dash\)theory, periodic cyclic homology - Classical symplectic and Poisson structures - Classifying stacks $\BG$ for $G\in \Alg\Grp$ reductive - $(\infty, n)\dash$categories - Perfect complexes - Tor amplitudes - $\EE_n\dash$rings and algebras - Transfers (or norms) # Misc ![](attachments/Pasted%20image%2020220903173003.png) # Definitions ![](attachments/Pasted%20image%2020220320032610.png) ![](attachments/Pasted%20image%2020220420095704.png) ![](attachments/Pasted%20image%2020220420095716.png) # Motivations ![](attachments/Pasted%20image%2020220319000935.png) ![](attachments/Pasted%20image%2020220317230604.png) # Derived vector bundles ![](attachments/Pasted%20image%2020220320024245.png) Relation to [spec sym](spec%20sym) ![](attachments/Pasted%20image%2020220320024427.png) # Exterior algebra ![](attachments/Pasted%20image%2020220320024533.png) # QCoh over a stack ![](attachments/Pasted%20image%2020220320033114.png) # Cotangent complex How to define the [cotangent complex](Unsorted/cotangent%20complex.md) for a stack: ![](attachments/Pasted%20image%2020220320033140.png) # Tangent stack ![](attachments/Pasted%20image%2020220320033448.png)