Change of Base for Commutative Algebras

Details Change of Base for Commutative Algebras Change of Base for Commutative Algebras

2009-10-12

arxiv.org/abs/0910.2152v2

Mathematics

Category Theory

22 pages, 26 figures

Scientific paper

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.

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