Gr\"obner Bases: Connecting Linear Algebra with Homological and Homotopical Algebra

TL;DR

This paper explores how Gr"obner bases can be used to connect linear algebra, homological algebra, and homotopical algebra, particularly through constructing resolutions of graded algebras and analyzing their properties.

Contribution

It introduces methods to compute Anick's resolution for K2 algebras using Gr"obner bases, linking algebraic structures with homological and homotopical concepts.

Findings

Computed Anick's resolution for K2 algebras.

Identified combinatorial properties of resolutions.

Outlined construction of A_infinity structures on Ext-algebras.

Abstract

The main objective of this paper is to connect the theory of $Gr\"obner$ bases to concepts of homological algebra. $Gr\"obner$ bases, an important tool in algebraic system and in linear algebra help us to understand the structure of an algebra presented by its generators and relations by constructing a basis of its set of relations. In this paper we mainly deal with graded augmented algebras. Given a graded augmented algebra with its generators and relations, it is possible to construct a free resolution from its $Gr\"obner$ basis, known as Anick's resolution. Though rarely minimal, this resolution helps us to understand combinatorial properties of the algebra. The notion of a $K_2$ algebra was recently introduced by Cassidy and Shelton as a generalization of the notion of a Koszul algebra. We compute the Anick's resolution of $K_2$ algebra which shows several nice combinatorial properties. Later we compute some derived functors from Anick's resolution and outline how to construct $A_\infty$-algebra structure on $Ext$-algebra from a minimal graded projective resolution in which we can use Anick's resolution as a tool. Thus this paper provides an unique platform to connect concepts of linear algebra with homological algebra and homotopical algebra with the help of $Gr\"obner$ bases.