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Tates central extension originates from 1968 and has since found many applications to curves. In the 80s Beilinson found an n-dimensional generalization: cubically decomposed algebras, based on ideals of bounded and discrete operators in ind-pro limits of vector spaces. Kato and Beilinson independently defined (n-)Tate categories whose objects are formal iterated ind-pro limits in general exact categories. We show that the endomorphism algebras of such objects often carry a cubically decomposed structure, and thus a (higher) Tate central extension. Even better, under very strong assumptions on the base category, the n-Tate category turns out to be just a category of projective modules over this type of algebra.
Refining a theorem of Zarhin, we prove that given a $g$-dimensional abelian variety $X$ and an endomorphism $u$ of $X$, there exists a matrix $A in operatorname{M}_{2g}(mathbb{Z})$ such that each Tate module $T_ell X$ has a $mathbb{Z}_ell$-basis on which the action of $u$ is given by $A$.
We develop an analogue of Eisenbud-Floystad-Schreyers Tate resolutions for toric varieties. Our construction, which is given by a noncommutative analogue of a Fourier-Mukai transform, works quite generally and provides a new perspective on the relati
We study a generalization of Serre--Tate theory of ordinary abelian varieties and their deformation spaces. This generalization deals with abelian varieties equipped with additional structures. The additional structures can be not only an action of a
Let k be a number field, and let S be a finite set of k-rational points of P^1. We relate the Deligne-Goncharov contruction of the motivic fundamental group of X:=P^1_k- S to the Tannaka group scheme of the category of mixed Tate motives over X.
We prove that the minimal left ideals of the superextension $lambda(Z)$ of the discrete group $Z$ of integers are metrizable topological semigroups, topologically isomorphic to minimal left ideals of the superextension $lambda(Z_2)$ of the compact group $Z_2$ of integer 2-adic numbers.