可控扩张的相对整体维数
Relative Global Dimension of Controllable Extensions
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中文总结 AI 辅助
本文研究 Artin 代数扩张的相对整体维数，引入可控扩张与同调可控扩张概念，证明后者在张量积下封闭并给出维数加性公式，同时构造了可控与不可控扩张的实例。
中文摘要 AI 辅助
我们研究了完美域上 Artin 代数扩张 $B \subseteq A$ 的相对整体维数。引入了可控扩张的概念，即其相对整体维数由商代数 $A/AJ(B)A$ 的普通整体维数决定。建立了可控性的一般不等式和充分条件，包括 $J(B)$ 是 $A$ 的双边理想的情形。随后研究了相对整体维数与正则双模的相对投射维数之间的相互作用。引入了同调可控扩张类，并证明该类在张量积下封闭。因此，对于同调可控扩张 $B \subseteq A$ 和 $D \subseteq C$，我们得到 \\[\operatorname{gldim}(A \otimes C, B \otimes D)=\operatorname{gldim}(A,B) + \operatorname{gldim}(C,D).\\] 证明依赖于复形张量积的相对 Künneth 型结果，该结果可能具有独立意义。最后，构造了具有指定相对整体维数的可控扩张，以及一族不可控扩张，其相对整体维数与 $A/AJ(B)A$ 的整体维数可任意不同。
英文摘要
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 $A/AJ(B)A$.
发表机构
- Univ. de São Paulo（圣保罗大学）
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