On the de Rham cohomology of differential and algebraic stacks

By K. Behrend. 2004

58 citations recorded by the source corpus.

Abstract We introduce the notion of cofoliation on a stack. A cofoliation is a change of the differentiable structure which amounts to giving a full representable smooth epimorphism. Cofoliations are uniquely determined by their associated Lie algebroids. Cofoliations on stacks arise from flat connections on groupoids. Connections on groupoids generalize connections on gerbes and bundles in a natural way. A flat connection on a groupoid is an integrable distribution of the morphism space compatible with the groupoid structure and complementary to both source and target fibres. A cofoliation of a stack determines the flat groupoid up to etale equivalence. We show how a cofoliation on a stack gives rise to a refinement of the Hodge to De Rham spectral sequence, where the E 1 -term consists entirely of vector bundle valued cohomology groups. Our theory works for differentiable, holomorphic and algebraic stacks.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1984c50deb5933af5ffd2bc09cb4d26b00002c26

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Citations in the corpus

Representations up to homotopy and Bott's spectral sequence for Lie groupoids (2009, background): “The notion of a connection on a Lie groupoid already appeared in Behrend’s paper [3] and in Tang’s [18].”
Riemannian metrics on Lie groupoids (2014, background): “Denote by [2] the pair groupoid on a set with three objects {0, 1, 2}.”
Multiplicative forms and Spencer operators (2012, methodology): “The flat Cartan connections are the same “flat connections on groupoids” used by Behrend in the context of equivariant cohomology [4].”