ttgeometry of Tate motives over algebraically closed fields
 Submitted (37 pages)
 arXiv:1708.00834
(pdf, source)
 Abstract. We study Tate motives with integral coefficients through the lens of tensor triangular geometry. For some base fields, including the field of algebraic numbers and the algebraic closure of a finite field, we arrive at a complete description of the tensor triangular spectrum and a classification of thick tensor ideals.
ttgeometry of filtered modules
 Submitted (27 pages)
 arXiv:1708.00833
(pdf, source)
 Abstract. We compute the tensor triangular spectrum of perfect complexes of filtered modules over a commutative ring, and deduce a classification of the thick tensor ideals. We give two proofs: one by reducing to perfect complexes of graded modules which have already been studied in the literature, and one more direct for which we develop some useful tools.
Homotopy theory of dg sheaves
joint work with
Utsav
Choudhury
 To appear in Comm. Algebra (29 pages)
 arXiv:1511.02828
(pdf, source)
 Abstract. In this note we study the local projective model structure on presheaves of complexes on a site, i.e. we describe its classes of cofibrations, fibrations and weak equivalences. In particular, we prove that the fibrant objects are those satisfying descent with respect to all hypercovers. We also describe cofibrant and fibrant replacement functors with pleasant properties.
 Comments. After completing this note we learned that this description of the fibrant objects also appeared in doi:10.1016/j.aim.2004.07.007 and in the Ph.D. thesis of Brad Drew.
An isomorphism of motivic Galois groups
joint work with
Utsav
Choudhury
 Adv. Math. 313 (2017), pp. 470536. 10.1016/j.aim.2017.04.006
 arXiv:1410.6104
(pdf, source). Last
updated: 18 May 2017
 Abstract. In characteristic 0 there are essentially two approaches to the conjectural theory of mixed motives, one due to Nori and the other one due to, independently, Hanamura, Levine, and Voevodsky. Although these approaches are apriori quite different it is expected that ultimately they can be reduced to one another. In this article we provide some evidence for this belief by proving that their associated motivic Galois groups are canonically isomorphic.
Traces in monoidal derivators, and homotopy colimits
 Adv. Math. 261 (2014), pp. 2684. 10.1016/j.aim.2014.03.029
 arXiv:1303.0153
(pdf, source). Last
updated: 16 July 2014
 Abstract. A variant of the trace in a monoidal
category is given in the setting of closed monoidal derivators,
which is applicable to endomorphisms of fiberwise dualizable
objects. Functoriality of this trace is established. As an
application, an explicit formula is deduced for the trace of the
homotopy colimit of endomorphisms over finite categories in which
all endomorphisms are invertible. This result can be seen as a
generalization of the additivity of traces in monoidal categories
with a compatible triangulation.
 Comments. In the meantime the same result has
been obtained independently
here.
Traces, homotopy theory, and motivic Galois groups
The LefschetzVerdier trace formula and a generalization of a
theorem of Fujiwara
after Y. Varshavsky
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