Relative Global Dimension of Controllable Extensions

Abstract

We study relative global dimensions of extensions $B\subseteq A$ of Artin algebras over a perfect field. A controllable extension is introduced as one for which the relative global dimension is determined by the ordinary global dimension of the quotient $A/AJ(B)A$. General inequalities and sufficient conditions for controllability are established, including the case in which $J(B)$ is a two-sided ideal of $A$. The interaction between relative global dimensions and relative projective dimensions of the regular bimodule is then studied. The class of homologically controllable extensions is introduced, and it is shown to be closed under tensor products. Consequently, for homologically controllable extensions $B\subseteq A$ and $D\subseteq C$, we obtain \[\operatorname{gldim}(A \otimes C, B \otimes D)=\operatorname{gldim}(A,B) + \operatorname{gldim}(C,D).\] The proof relies on a relative Künneth-type result for tensor products of complexes, which may be of independent interest. Finally, controllable extensions with prescribed relative global dimension are constructed, along with families of non-controllable extensions for which the relative global dimension differs arbitrarily from the global dimension of $A/AJ(B)A$.

arXiv comment32 pages, 1 figure