Higher-dimensional absolute versions of symmetric, Frobenius, and quasi-Frobenius algebras

TL;DR

This paper introduces higher-dimensional and absolute generalizations of symmetric, Frobenius, and quasi-Frobenius algebras, comparing them with existing relative notions and proving a generalized codimension two-argument for modules over certain sheaves of algebras.

Contribution

It defines new higher-dimensional algebraic structures and establishes a generalized codimension two-argument, expanding the theoretical framework of these algebras.

Findings

Defined higher-dimensional absolute versions of key algebraic structures

Compared new notions with existing relative concepts

Proved a generalized codimension two-argument for modules

Abstract

In this paper, we define and discuss higher-dimensional and absolute versions of symmetric, Frobenius, and quasi-Frobenius algebras. In particular, we compare these with the relative notions defined by Scheja and Storch. We also prove the validity of codimension two-argument for modules over a coherent sheaf of algebras with a $2$-canonical module, generalizing a result of the author.