
TL;DR
This paper develops a formal matrix calculus over rings, generalizing linear algebra and providing new presentations and homological interpretations of the Grothendieck group of finitely presented modules, with applications to fields.
Contribution
Introduces a formal matrix calculus over rings, unifies linear algebra over rings with module theory, and offers new homological and algebraic insights into the Grothendieck group.
Findings
Formal matrix calculus subsumes homogeneous systems over rings.
Provides two new presentations for the Grothendieck group K_0(R-mod).
Computes the first homology group for fields, relating it to the abelianization of the multiplicative group.
Abstract
Given an associative, not necessarily commutative, ring R with identity, a formal matrix calculus is introduced and developed for pairs of matrices over R. This calculus subsumes the theory of homogeneous systems of linear equations with coefficients in R. In the case when the ring R is a field, every pair is equivalent to a homogeneous system. Using the formal matrix calculus, two alternate presentations are given for the Grothendieck group of the category R-mod of finitely presented modules. One of these presentations suggests a homological interpretation, and so a complex is introduced whose 0-dimensional homology is naturally isomorphic to A computation shows that if R = k is a field, then the 1-dimensional homology group is given by the abelianization of the multiplicative group of k, modulo the subgroup {1, -1}. The formal matrix…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Advanced Topics in Algebra
