Vol. 4, No. 4, 2019

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Witt groups of abelian categories and perverse sheaves

Jörg Schürmann and Jon Woolf

Vol. 4 (2019), No. 4, 621–670
Abstract

We study the Witt groups W±(PervX) of perverse sheaves on a finite-dimensional topologically stratified space X with even-dimensional strata. We show that W±(PervX) has a canonical decomposition as a direct sum of the Witt groups of shifted local systems on strata. We compare this with another “splitting decomposition” for Witt classes of perverse sheaves obtained inductively from our main new tool, a “splitting relation” which is a generalisation of isotropic reduction.

The Witt groups W±(PervX) are identified with the (nontrivial) Balmer–Witt groups of the constructible derived category Dcb(X) of sheaves on X, and also with the corresponding cobordism groups defined by Youssin.

Our methods are primarily algebraic and apply more widely. The general context in which we work is that of a triangulated category with duality, equipped with a self-dual t-structure with noetherian heart, glued from self-dual t-structures on a thick subcategory and its quotient.

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Keywords
Witt group, perverse sheaf, triangulated category with duality
Mathematical Subject Classification 2010
Primary: 32S60
Secondary: 18E30, 19G99
Milestones
Received: 22 March 2018
Revised: 20 March 2019
Accepted: 10 April 2019
Published: 10 January 2020
Authors
Jörg Schürmann
Mathematisches Institut
Universität Münster
Münster
Germany
Jon Woolf
Department of Mathematical Sciences
University of Liverpool
Liverpool
United Kingdom