Change of Base for Commutative Algebras

TL;DR

This paper investigates the change of base functors between categories of crossed modules over commutative algebras, establishing their adjoint relationships and exploring their categorical properties.

Contribution

It introduces and analyzes adjoint functors for changing the base algebra in crossed modules, demonstrating their role in the bifibred category structure over commutative algebras.

Findings

The functors form an adjoint pair enabling base change.

The category of crossed modules is bifibred over commutative algebras.

Examples of induced crossed modules are provided.

Abstract

In this paper we examine on a pair of adjoint functors $(\phi ^{\ast},\phi_{\ast})$ for a subcategory of the category of crossed modules over commutative algebras where $\phi ^{\ast}:\mathbf{XMod}$\textbf{/}$% Q\rightarrow $ $\mathbf{XMod/}P$, pullback, which enables us to move from crossed $Q$-modules to crossed $P$-modules by an algebra morphism $\phi :P\rightarrow Q$ and $\phi_{\ast}:\mathbf{XMod}$\textbf{/}$P\rightarrow $ $\mathbf{XMod/}Q$, induced. We note that this adjoint functor pair $(\phi ^{\ast},\phi_{\ast})$ makes $p:\mathbf{XMod}\rightarrow $ $k$\textbf{-Alg}into a bifibred category over $k$\textbf{-Alg}, the category of commutative algebras, where $p$ is given by $p(C,R,\partial)=R.$ Also, some examples and results on induced crossed modules are given.