Change of Base for Commutative Algebras

Abstract

In this paper we examine on a pair of adjoint functors $({\varphi }^{\ast },{\varphi }_{\ast })$ for a subcategory of the category of crossed modules over commutative algebras where ${\varphi }^{\ast }:\mathbf{XMod}$\textbf{/} $XMod/P$, pullback, which enables us to move from crossed $Q$-modules to crossed $P$-modules by an algebra morphism $\varphi :P\to Q$ and ${\varphi }_{\ast }:\mathbf{XMod}$\textbf{/}$P\to$ $XMod/Q$, induced. We note that this adjoint functor pair $({\varphi }^{\ast },{\varphi }_{\ast })$ makes $p:\mathbf{XMod}\to$ $k$\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.

22 pages, 26 figures