Cosupport in tensor triangular geometry

Abstract

We develop a theory of cosupport and costratification in tensor triangular geometry. We study the geometric relationship between support and cosupport, provide a conceptual foundation for cosupport as categorically dual to support, and discover surprising relations between the theory of costratification and the theory of stratification. We prove that many categories in algebra, topology and geometry are costratified by developing and applying descent techniques. An overarching theme is that cosupport is relevant for diverse questions in tensor triangular geometry and that a full understanding of a category requires knowledge of both its support and its cosupport.

Stratification and the comparison between homological and tensor triangular support

“…It is an open question whether the converse to Theorem 1.3 holds, that is, whether the surjectivity of ϕ in (1.4) implies that the family {f * i } i∈I is jointly nil-conservative. It is known that the family need not be jointly conservative (see [BCHS23,Example 14.26]). In light of Theorem 1.8, the converse of Theorem 1.3 would follow from Balmer's "Nerves of Steel" Conjecture that the homological and tensor triangular spectra always coincide; see [BHS21a].…”

“…Remark. Using the Balmer-Favi support [BF11] and the techniques of [BCHS23], it is possible to prove Theorem 1.3 for arbitrary indexing sets I under the additional hypothesis that Spc(T c ) is weakly noetherian. However, since the construction of a surjective map as in (1.4) is often the first step in understanding Spc(T c ), making any assumption on its topology is not desirable.…”