The basic geometry of Witt vectors, I: The affine case

We give a concrete description of the category of etale algebras over the ring of Witt vectors of a given finite length with entries in an arbitrary ring. We do this not only for the classical p-typical and big Witt vector functors but also for variants of these functors which are in a certain sense their analogues over arbitrary local and global fields. The basic theory of these generalized Witt vectors is developed from the point of view of commuting Frobenius lifts and their universal properties, which is a new approach even for the classical Witt vectors. The larger purpose of this paper is to provide the affine foundations for the algebraic geometry of generalized Witt schemes and arithmetic jet spaces. So the basics here are developed somewhat fully, with an eye toward future applications.Comment: Final versio

Contramodules

Contramodules are module-like algebraic structures endowed with infinite summation (or, occasionally, integration) operations satisfying natural axioms. Introduced originally by Eilenberg and Moore in 1965 in the case of coalgebras over commutative rings, contramodules experience a small renaissance now after being all but forgotten for three decades between 1970-2000. Here we present a review of various definitions and results related to contramodules (drawing mostly from our monographs and preprints arXiv:0708.3398, arXiv:0905.2621, arXiv:1202.2697, arXiv:1209.2995, arXiv:1512.08119, arXiv:1710.02230, arXiv:1705.04960, arXiv:1808.00937) - including contramodules over corings, topological associative rings, topological Lie algebras and topological groups, semicontramodules over semialgebras, and a "contra version" of the Bernstein-Gelfand-Gelfand category O. Several underived manifestations of the comodule-contramodule correspondence phenomenon are discussed.Comment: LaTeX 2e with pb-diagram and xy-pic; 93 pages, 3 commutative diagrams; v.4: updated to account for the development of the theory over the four years since Spring 2015: introduction updated, references added, Remark 2.2 inserted, Section 3.3 rewritten, Sections 3.7-3.8 adde

Obstructing extensions of the functor Spec to noncommutative rings

In this paper we study contravariant functors from the category of rings to the category of sets whose restriction to the full subcategory of commutative rings is isomorphic to the prime spectrum functor Spec. The main result reveals a common characteristic of these functors: every such functor assigns the empty set to M_n(C) for n >= 3. The proof relies, in part, on the Kochen-Specker Theorem of quantum mechanics. The analogous result for noncommutative extensions of the Gelfand spectrum functor for C*-algebras is also proved.Comment: 23 pages. To appear in Israel J. Math. Title was changed; introduction was rewritten; old Section 2 was removed to streamline the exposition; final section was rewritten to omit an error in the earlier proof of Theorem 1.

On some local cohomology spectral sequences

We introduce a formalism to produce several families of spectral sequences involving the derived functors of the limit and colimit functors over a finite partially ordered set. The first type of spectral sequences involves the left derived functors of the colimit of the direct system that we obtain applying a family of functors to a single module. For the second type we follow a completely different strategy as we start with the inverse system that we obtain by applying a covariant functor to an inverse system. The spectral sequences involve the right derived functors of the corresponding limit. We also have a version for contravariant functors. In all the introduced spectral sequences we provide sufficient conditions to ensure their degeneration at their second page. As a consequence we obtain some decomposition theorems that greatly generalize the well-known decomposition formula for local cohomology modules given by Hochster.Comment: 63 pages, comments are welcome. To appear in International Mathematics Research Notice