# On the notion of Krull super-dimension

## TL;DR

This paper introduces the concept of Krull super-dimension for super-commutative super-rings, extending classical notions to superschemes and providing tools to analyze their regularity and nonsingularity.

## Contribution

It defines Krull super-dimension, applies it to regular super-rings and superschemes, and characterizes nonsingular superschemes using sheaves of Kähler superdifferentials.

## Key findings

- Defined Krull super-dimension for super-rings
- Calculated super-dimensions of completions of super-rings
- Characterized nonsingular superschemes via sheaves of Kähler superdifferentials

## Abstract

We introduce the notion of Krull super-dimension of a super-commutative super-ring. This notion is used to describe regular super-rings and calculate Krull super-dimensions of completions of super-rings. Moreover, we use this notion to introduce the notion of super-dimension of any irreducible superscheme of finite type. Finally, we describe nonsingular superschemes in terms of sheaves of K\"{a}hler superdifferentials.