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Matthew Morrow

Publications and source records attributed to Matthew Morrow.

31 records · Page 2Linked to original sources
arXiv2015

Pro unitality and pro excision in algebraic K-theory and cyclic homology

We study pro excision in algebraic K-theory, following Suslin--Wodzicki, Cuntz--Quillen, Cortiñas, and Geisser--Hesselholt, as well as Artin--Rees and continuity properties of André--Quillen, Hochschild, and cyclic homology. Our key tool is to first establish the equivalence of various pro Tor vanishing conditions which appear in the literature. Using this we prove that all ideals of commutative, Noetherian rings are pro unital in a certain sense, and show that such ideals satisfy pro excision in $K$-theory as well as in cyclic and topological cyclic homology. In addition, our techniques yield a strong form of the pro Hochschild--Kostant--Rosenberg theorem, an extension to general base rings of the Cuntz--Quillen excision theorem in periodic cyclic homology, and a generalisation of the Fe\uıgin--Tsygan theorem.

math.KT↗
arXiv2015

A Variational Tate Conjecture in crystalline cohomology

Given a smooth, proper family of varieties in characteristic $p>0$, and a cycle $z$ on a fibre of the family, we formulate a Variational Tate Conjecture characterising, in terms of the crystalline cycle class of $z$, whether $z$ extends cohomologically to the entire family. This is a characteristic $p$ analogue of Grothendieck's Variational Hodge Conjecture. We prove the conjecture for divisors, and an infinitesimal variant of the conjecture for cycles of higher codimension. This can be used to reduce the $\ell$-adic Tate conjecture for divisors over finite fields to the case of surfaces.

math.AG↗
arXiv2014

Pro cdh-descent for cyclic homology and $K$-theory

In this paper we prove that cyclic homology, topological cyclic homology, and algebraic $K$-theory satisfy a pro Mayer--Vietoris property with respect to abstract blow-up squares of varieties, in both zero and finite characteristic. This may be interpreted as the well-definedness of $K$-theory with compact support.

math.KT↗
arXiv2014

Finite generation and continuity of topological Hochschild and cyclic homology

The goal of this paper is to establish fundamental properties of the Hochschild, topological Hochschild, and topological cyclic homologies of commutative, Noetherian rings, which are assumed only to be F-finite in the majority of our results. This mild hypothesis is satisfied in all cases of interest in finite and mixed characteristic algebraic geometry. We prove firstly that the topological Hochschild homology groups, and the homotopy groups of the fixed point spectra $TR^r$, are finitely generated modules. We use this to establish the continuity of these homology theories for any given ideal. A consequence of such continuity results is the pro Hochschild-Kostant-Rosenberg theorem for topological Hochschild and cyclic homology. Finally, we show more generally that the aforementioned finite generation and continuity properties remain true for any proper scheme over such a ring.

math.KT↗
arXiv2013

K-theory of one-dimensional rings via pro-excision

This paper studies "pro-excision" for the K-theory of one-dimensional (usually semi-local) rings and its various applications. In particular, we prove Geller's conjecture for equal characteristic rings over a perfect field of finite characteristic, give the first results towards Geller's conjecture in mixed characteristic, and we establish various finiteness results for the K-groups of singularities (covering both orders in number fields and singular curves over finite fields).

math.KT↗
arXiv2012

A singular analogue of Gersten's conjecture and applications to K-theoretic adeles

The first part of this paper introduces an analogue, for one-dimensional, singular, complete local rings, of Gersten's injectivity conjecture for discrete valuation rings. Our main theorem is the verification of this conjecture when the ring is reduced and contains Q, using methods from cyclic/Hochschild homology and Artin-Rees type results due to A. Krishna. The second part of the paper describes the relationship between adele type resolutions of K-theory on a one-dimensional scheme and more classical questions in K-theory such as localisation and descent. In particular, we construct a new resolution of sheafified K-theory, conditionally upon the conjecture.

math.KT↗
arXiv2012

An introduction to higher dimensional local fields and adeles

These notes are an introduction to higher dimensional local fields and higher dimensional adeles. As well as the foundational theory, we summarise the theory of topologies on higher dimensional local fields and higher dimensional local class field theory.

math.AG↗
arXiv2012

K2 of localisations of local rings

We show that K2 of "sufficiently regular" localisations of local rings (e.g. inverting a sequence of regular parameters) can be described by the Steinberg presentation. The proof is inductive on the number of irreducible elements being inverted, successively using a generalisation of a co-Cartesian square first exploited by Dennis and Stein.

math.KT↗
arXiv2011

Grothendieck's trace map for arithmetic surfaces via residues and higher adeles

We establish the reciprocity law along a vertical curve for residues of differential forms on arithmetic surfaces, and describe Grothendieck's trace map of the surface as a sum of residues. Points at infinity are then incorporated into the theory and the reciprocity law is extended to all curves on the surface. Applications to adelic duality for the arithmetic surface are discussed.

math.AG↗
arXiv2011

An explicit approach to residues on and dualizing sheaves of arithmetic surfaces

We develop a theory of residues for arithmetic surfaces, establish the reciprocity law around a point, and use the residue maps to explicitly construct the dualizing sheaf of the surface. These are generalisations of known results for surfaces over a perfect field. In an appendix, explicit local ramification theory is used to recover the fact that in the case of a local complete intersection the dualizing and canonical sheaves coincide.

math.NT↗
arXiv2009

Euler characteristics, Fubini's theorem, and the Riemann-Hurwitz formula

We relate Fubini's theorem for Euler characteristics to Riemann-Hurwtiz formulae, and reprove a classical result of Iversen. The techniques used include algebraic geometry, complex geometry, and model theory. Possible applications to the study of wild ramification in finite characteristic are discussed.

math.AG↗