A Study in Derived Algebraic Geometry

2017 ◽  
Author(s):  
Dennis Gaitsgory ◽  
Nick Rozenblyum
Keyword(s):  
Algebraic Geometry ◽  
Derived Algebraic Geometry

scholarly journals CATEGORICAL PROOF OF HOLOMORPHIC ATIYAH–BOTT FORMULA

2018 ◽  
Vol 19 (5) ◽  
pp. 1739-1763 ◽  
Author(s):  
Grigory Kondyrev ◽  
Artem Prikhodko
Keyword(s):  
Fixed Point ◽  
Algebraic Geometry ◽  
Commutative Diagram ◽  
Natural Morphism ◽  
Lefschetz Fixed Point Formula ◽  
Derived Algebraic Geometry

Given a $2$-commutative diagramin a symmetric monoidal $(\infty ,2)$-category $\mathscr{E}$ where $X,Y\in \mathscr{E}$ are dualizable objects and $\unicode[STIX]{x1D711}$ admits a right adjoint we construct a natural morphism $\mathsf{Tr}_{\mathscr{E}}(F_{X})\xrightarrow[{}]{~~~~~}\mathsf{Tr}_{\mathscr{E}}(F_{Y})$ between the traces of $F_{X}$ and $F_{Y}$, respectively. We then apply this formalism to the case when $\mathscr{E}$ is the $(\infty ,2)$-category of $k$-linear presentable categories which in combination of various calculations in the setting of derived algebraic geometry gives a categorical proof of the classical Atiyah–Bott formula (also known as the Holomorphic Lefschetz fixed point formula).

scholarly journals Tannakization in derived algebraic geometry

2014 ◽  
Vol 14 (3) ◽  
pp. 642-700 ◽  
Author(s):  
Isamu Iwanari
Keyword(s):  
Algebraic Geometry ◽  
Galois Group ◽  
Monoidal Category ◽  
Group Scheme ◽  
Affine Group ◽  
Stable Category ◽  
Group Schemes ◽  
Symmetric Monoidal Category ◽  
Derived Algebraic Geometry ◽  
Mixed Motives

AbstractIn this paper we begin studying tannakian constructions in ∞-categories and combine them with the theory of motivic categories developed by Hanamura, Levine, and Voevodsky. This paper is the first in a series of papers. For the purposes above, we first construct a derived affine group scheme and its representation category from a symmetric monoidal ∞-category, which we shall call the tannakization of a symmetric monoidal ∞-category. It can be viewed as an ∞-categorical generalization of work of Joyal-Street and Nori. Next we apply it to the stable ∞-category of mixed motives equipped with the realization functor of a mixed Weil cohomology. We construct a derived motivic Galois group which represents the automorphism group of the realization functor, and whose representation category satisfies an appropriate universal property. As a consequence, we construct an underived motivic Galois group of mixed motives, which is a pro-algebraic group and has nice properties. Also, we present basic properties of derived affine group schemes in the Appendix.

scholarly journals Localizing virtual fundamental cycles for semi-perfect obstruction theories

2018 ◽  
Vol 29 (04) ◽  
pp. 1850032 ◽  
Author(s):  
Young-Hoon Kiem
Keyword(s):  
Algebraic Geometry ◽  
Local Existence ◽  
Obstruction Theory ◽  
Compatibility Conditions ◽  
Euler Characteristics ◽  
Smoothness Assumption ◽  
Fundamental Class ◽  
Derived Algebraic Geometry ◽  
Wall Crossing ◽  
Invent Math

Recently, Chang and Li generalized the theory of virtual fundamental class to the setting of semi-perfect obstruction theory. A semi-perfect obstruction theory requires only the local existence of a perfect obstruction theory with compatibility conditions. In this paper, we generalize the torus localization of Graber–Pandharipande [T. Graber and R. Pandharipande, Localization of virtual cycles, Invent. Math. 135(2) (1999) 487–518], the cosection localization [Y.-H. Kiem and J. Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26(4) (2013) 1025–1050] and their combination [H.-L. Chang, Y.-H. Kiem and J. Li, Torus localization and wall crossing for cosection localized virtual cycles, Adv. Math. 308 (2017) 964–986], to the setting of semi-perfect obstruction theory. As an application, we show that the Jiang-Thomas theory [Y. Jiang and R. Thomas, Virtual signed Euler characteristics, preprint (2014), arXiv:1408.2541] of virtual signed Euler characteristic works without the technical quasi-smoothness assumption from derived algebraic geometry.

scholarly journals Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes

2015 ◽  
Vol 2015 (702) ◽  
pp. 1-40 ◽  
Author(s):  
Timo Schürg ◽  
Bertrand Toën ◽  
Gabriele Vezzosi
Keyword(s):  
Algebraic Geometry ◽  
Projective Variety ◽  
Obstruction Theory ◽  
Important Ingredient ◽  
Smooth Complex ◽  
Perfect Complexes ◽  
Derived Algebraic Geometry ◽  
Complex Projective Variety ◽  
Complex Projective

AbstractA quasi-smooth derived enhancement of a Deligne–Mumford stack 𝒳 naturally endows 𝒳 with a functorial perfect obstruction theory in the sense of Behrend–Fantechi. We apply this result to moduli of maps and perfect complexes on a smooth complex projective variety.ForWe give two further applications toAn important ingredient of our construction is a

scholarly journals Tangent Lie algebra of derived Artin stacks

2018 ◽  
Vol 2018 (741) ◽  
pp. 1-45 ◽  
Author(s):  
Benjamin Hennion
Keyword(s):  
Algebraic Geometry ◽  
Lie Algebra ◽  
Algebraic Variety ◽  
Category Theory ◽  
Smoothness Assumption ◽  
Coherent Sheaves ◽  
Derived Algebraic Geometry ◽  
Atiyah Class ◽  
Artin Stacks

Abstract Since the work of Mikhail Kapranov in [Compos. Math. 115 (1999), no. 1, 71–113], it is known that the shifted tangent complex \mathbb{T}_{X} [-1] of a smooth algebraic variety X is endowed with a weak Lie structure. Moreover, any complex of quasi-coherent sheaves E on X is endowed with a weak Lie action of this tangent Lie algebra. This Lie action is given by the Atiyah class of E. We will generalise this result to (finite enough) derived Artin stacks, without any smoothness assumption. This in particular applies to singular schemes. This work uses tools of both derived algebraic geometry and {\infty} -category theory.

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