Tensor Triangular Geometry and Applications

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

``Tensor triangular geometry'' is a recent term coined by the PI for the ongoing unification by means of tensor triangulated categories of geometric aspects of very different areas of mathematics and theoretical physics, among which algebraic geometry, modular representation theory, stable homotopy theory, motivic theory, noncommutative geometry, string theory, and probably more in the future. The geometric foundation of this new theory is a construction proposed by the PI, called the ``spectrum of a tensor triangulated category'', which produces a locally ringed space. For example, the spectrum of the category of perfect complexes over a scheme recovers the scheme. Heuristically, this says that algebraic geometry embeds into tensor triangular geometry. In modular representation theory, the spectrum of the stable category recovers the ``projective support variety''. For the triangulated categories discovered more recently, the computation of the spectrum is one of the challenges proposed here by the PI, e.g. for mixed Tate motives or equivariant KK-theory of C*-algebras. This will establish unexpected bridges between these areas and algebraic geometry. The PI wants to extend well-understood concepts from the above numerous examples to tensor triangular geometry (e.g. invariants, like Chow groups, K-theory, Brauer groups, Dade groups,... etc). Then, he wants to prove general theorems at the astract level that can be applied to the various areas listed above to obtain new results. This ideal strategy has already been successfully started by the PI, like with filtrations by dimension of support yielding a new local-global spectral sequence for K-theory of singular varieties, or like in the joint work with Benson and Carlson on the Picard group, a standard invariant in algebraic geometry but a more subtle one in modular representation theory. The PI also applies such ideas to quadratic forms (via ``triangular Witt groups''), allowing many new results, like the proof of the 1980 Gersten-Witt Conjecture, and more computations to come.