Tensor triangular geometry of filtered objects and sheaves

DOI10.1007/S00209-023-03210-ZzbMATH Open1516.18007arXiv2001.00319MaRDI QIDQ6332155

Ko Aoki

Publication date: 1 January 2020

Abstract: We compute the the Balmer spectra of compact objects of tensor triangulated categories whose objects are filtered or graded objects of (or sheaves valued in) another tensor triangulated category. Notable examples include the filtered derived category of a scheme as well as the homotopy category of filtered spectra. We use an $infty$-categorical method to properly formulate and deal with the problem. Our computations are based on a point-free approach, so that distributive lattices and semilattices are used as key tools. In the appendix, we prove that the $infty$-topos of hypercomplete sheaves on an $infty$-site is recovered from a basis, which may be of independent interest.

Mathematics Subject Classification ID

Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) (18F20) Derived categories, triangulated categories (18G80) Frames and locales, pointfree topology, Stone duality (18F70) ((infty,1))-categories (quasi-categories, Segal spaces, etc.); (infty)-topoi, stable (infty)-categories (18N60)

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