Commutative algebra

Summary

Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers \mathbb{Z}; and p-adic integers. Commutative algebra is the main technical tool in the local study of schemes. The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras. Overview Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry. In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings. Considerations related to modular arithmetic have l

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The theory of persistence, which arises from topological data analysis, has been intensively studied in the one-parameter case both theoretically and in its applications. However, its extension to the multi-parameter case raises numerous difficulties, where it has been proven that no barcode-like decomposition exists. To tackle this problem, algebraic invariants have been proposed to summarize multi-parameter persistence modules, adapting classical ideas from commutative algebra and algebraic geometry to this context. Nevertheless, the crucial question of their stability has raised little attention so far, and many of the proposed invariants do not satisfy a naive form of stability. In this paper, we equip the homotopy and the derived category of multi-parameter persistence modules with an appropriate interleaving distance. We prove that resolution functors are always isometric with respect to this distance. As an application, this explains why the graded-Betti numbers of a persistence module do not satisfy a naive form of stability. This opens the door to performing homological algebra operations while keeping track of stability. We believe this approach can lead to the definition of new stable invariants for multi-parameter persistence, and to new computable lower bounds for the interleaving distance (which has been recently shown to be NP-hard to compute in [2]).

SPRINGER INTERNATIONAL PUBLISHING AG2021

Embedding functor for classical groups and Brauer–Manin obstruction

Eva Bayer Fluckiger

Let K be a global field of characteristic not 2. The embedding problem for maximal tori in a classical group G can be described in terms of algebras with involution. The aim of this paper is to give an explicit description of the obstruction group to the Hasse principle in terms of ramification properties of certain commutative étale algebras, and to show that this group is isomorphic to one previously defined by the second author. This builds on our previous work as well as on results of Borovoi. In particular, we show that this explicit obstruction group can be identified with the group of Borovoi (J. Reine Angew. Math. 473 (1996), 181–194), where X is the homogeneous space associated to the embedding functor defined by the second author (Comment. Math. Helv. 89 (2014), 671–717).

2015

Hopf algebras and Hopf-Galois extensions in infinity-categories

Aras Ergus

In this thesis, we study interactions between algebraic and coalgebraic structures in infinity-categories (more precisely, in the quasicategorical model of (infinity, 1)-categories). We define a notion of a Hopf algebra H in an E-2-monoidal infinity-category and lift some results about ordinary Hopf algebras, such as the fundamental theorem of Hopf modules, to this setting. We also study Hopf-Galois extensions in this context. Given a candidate Hopf-Galois extension, i.e., a map f : A -> B of H-comodule algebras where H coacts on A trivially, we construct a structured version of the comparison map B (x)_A B -> H (x) B that allows us to compare the category of descent data for f with a category of "B-modules equipped with a semilinear coaction of H". We provide further insights into the case of commutative (i.e., E-infinity) comodule algebras over a commutative Hopf algebra, for instance a description of the aforementioned category of modules equipped with a semilinear coaction as the limit of a "categorified cobar construction". Moreover, we provide a simple description of comodules over a space in slice categories of the infinity-category of spaces, which enables us to realize multiplicative Thom objects as comodule algebras and thus incorporate them into the aforementioned framework.

EPFL2022