arXiv subjects

Lorenzo Ramero

Publications and source records attributed to Lorenzo Ramero.

6 recordsLinked to original sources

arXiv2018

Foundations for almost ring theory -- Release 7.5

This is release 7.5 of our project, aiming to provide a complete treatment of the foundations of almost ring theory, following and extending Faltings's method of "almost etale extensions". The central result is the "almost purity theorem", for whose proof we adapt Scholze's method, based on his perfectoid spaces. This release provides the foundations for our generalization of Scholze's perfectoid spaces, and reduces the proof of the almost purity theorem to a general assertion concerning the étale topology of adic spaces, whose proof uses previous work by the first author. As usual, this new release is a mix of corrections and various improvements, with a final chapter dedicated to applications; notably, we include a generalization of Y.André's "perfectoid Abhyankar's lemma" which we use to give a proof of a generalization of the "direct summand conjecture", extending André's recent work.

math.AG↗

arXiv2009

Hasse-Arf filtrations in $p$-adic analytic geometry -- Fourth Release

We the study of the monodromy of local systems with bounded ramification on a punctured disc defined over a non-archimedean valued field of characteristic zero. First, we construct the local Fourier transforms and we establish their main properties. Also, we show that the Fourier transform of a "perverse sheaf" with bounded ramification on the affine line, is again a "perverse sheaf" with bounded ramification (we put that in quotes, because actually the language of perversity is not used). These foundations are then used to exhibit a natural break decomposition for local systems with bounded ramification, analogous to the classical one for Galois representation. This is the paper that was previously posted with the title "Local monodromy in non-archimedean analytic geometry -- II". This revised release contains a new section, with an application to the problem of localization of the determinant of cohomology.

math.AG↗

arXiv2004

Local monodromy in non-archimedean analytic geometry -- fifth release

We study the topology of the punctured disc defined over a non-archimedean field of characteristic zero. Chapter two includes a new proof of the so-called p-adic Riemann existence theorem. This release completes the study of breaks and break decompositions of the monodromy representation of a sheaf around the origin of the punctured disc.

math.AG↗

arXiv2002

Springer correspondence via p-adic analytic methods

We show that the standard proof of the Springer correspondence in positive characteristic (via Deligne-Fourier transform) works verbatim in characteristic zero, up to replacing Deligne-Fourier transform by another etale Fourier transform introduced by the author in a previous work. The construction of this Fourier transform uses methods from p-adic analytic geometry.

math.RT↗

arXiv2002

Almost ring theory - sixth release

We develop almost ring theory, which is a domain of mathematics somewhere halfway between ring theory and category theory (whence the difficulty of finding appropriate MSC-class numbers). We apply this theory to valuation theory and to p-adic analytic geometry. You should really have a look at the introductions (each chapter has one).

math.AG↗

arXiv2000

Almost ring theory

The categories of almost modules and almost algebras are introduced as a convenient setting for the development of Faltings' method of almost etale extensions. After some preliminaries of general "almost homological algebra" we construct the almost version of the cotangent complex and we use it to generalise some results of Faltings on the lifting of almost etale morphisms and almost etale algebras over nilpotent extensions. We also study the "almost trace" of an almost flat and almost finitely presented morphism, in particular we show that the almost trace is (almost) perfect if and only if the morphism is almost etale. Finally we study some cases of non-flat descent for almost rings, and establish the invariance of almost etale morphisms under Frobenius.

math.AG↗