On dimension theory of supermodules, super-rings and superschemes

TL;DR

This paper develops a new concept of Krull super-dimension for supermodules over super-commutative Noetherian super-rings, exploring its properties and applications to superschemes and their morphisms.

Contribution

It introduces the Krull super-dimension, analyzes its relation to odd regular sequences, and studies its behavior under various super-structure transformations.

Findings

Krull super-dimension defined for supermodules.

Relation established between super-dimension and odd regular sequences.

Applications to super-scheme dimension theory and morphisms.

Abstract

We introduce the notion of Krull super-dimension of supermodules over certain super-commutative Noetherian super-rings. We investigate how this notion relates to the notion of odd regular sequence introduced by T.Schmitt and how it behaves with respect to the transition to the graded and bigraded supermodules and super-rings associated with the original ones. We also apply these results to the super-dimension theory of superschemes of finite type and their morphisms.