Descent theory for open varieties

We present a Langlands dual realization of the putative category of affine character sheaves. Namely, we calculate the categorical center and trace (also known as the Drinfeld center and trace, or categorical Hochschild cohomology and homology) of the affine Hecke category starting from its spectral presentation. The resulting categories comprise coherent sheaves on the commuting stack of local systems on the two-torus satisfying prescribed support conditions, in particular singular support conditions, which appear in recent advances in the geometric Langlands program. The key technical tools in our arguments are a new descent theory for coherent sheaves or ${\mathcal{D}}$-modules with prescribed singular support and the theory of integral transforms for coherent sheaves developed in the companion paper by Ben-Zvi et al. [Integral transforms for coherent sheaves, J. Eur. Math. Soc. (JEMS), to appear].

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The Picard group of topological modular forms via descent theory

Geometry & Topology ◽

10.2140/gt.2016.20.3133 ◽

2016 ◽

Vol 20 (6) ◽

pp. 3133-3217 ◽

Cited By ~ 15

Author(s):

Akhil Mathew ◽

Vesna Stojanoska

Keyword(s):

Modular Forms ◽

Picard Group ◽

Descent Theory ◽

Topological Modular Forms

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Adelic descent theory

Compositio Mathematica ◽

10.1112/s0010437x17007217 ◽

2017 ◽

Vol 153 (8) ◽

pp. 1706-1746

Author(s):

Michael Groechenig

Keyword(s):

Vector Bundles ◽

Continuous Functions ◽

Topological Spaces ◽

Reductive Algebraic Group ◽

Perfect Complexes ◽

Descent Theory ◽

Closed Fields ◽

Ring Of Continuous Functions ◽

Reconstruction Theorem ◽

Algebraically Closed Fields

A result of André Weil allows one to describe rank $n$ vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set $\text{GL}_{n}(\mathbb{A})$ of regular matrices over the ring of adèles (over algebraically closed fields, this result is also known to extend to $G$-torsors for a reductive algebraic group $G$). In the present paper we develop analogous adelic descriptions for vector and principal bundles on arbitrary Noetherian schemes, by proving an adelic descent theorem for perfect complexes. We show that for Beilinson’s co-simplicial ring of adèles $\mathbb{A}_{X}^{\bullet }$, we have an equivalence $\mathsf{Perf}(X)\simeq |\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })|$ between perfect complexes on $X$ and cartesian perfect complexes for $\mathbb{A}_{X}^{\bullet }$. Using the Tannakian formalism for symmetric monoidal $\infty$-categories, we conclude that a Noetherian scheme can be reconstructed from the co-simplicial ring of adèles. We view this statement as a scheme-theoretic analogue of Gelfand–Naimark’s reconstruction theorem for locally compact topological spaces from their ring of continuous functions. Several results for categories of perfect complexes over (a strong form of) flasque sheaves of algebras are established, which might be of independent interest.

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