Homological dimensions of modules of holomorphic functions on submanifolds of Stein manifolds

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• 作者：A.Yu. Pirkovskii ^{aupirkovskii@hse.ru" class="auth_mail pirkosha@gmail.com" class="auth_mail}
• 关键词：46M18 ; 46H25 ; 46E25 ; 16E10 ; 16E30 ; 16E40
• 刊名：Journal of Functional Analysis
• 出版年：15 June 2014
• 年：2014
• 卷：266
• 期：12
• 页码：6663-6683
• 全文大小：388 K

文摘

Let X   be a Stein manifold, and let Y⊂X$Y\subset X$ be a closed complex submanifold. Denote by O(X)$\mathcal{O}(X)$ the algebra of holomorphic functions on X  . We show that the weak (i.e., flat) homological dimension of O(Y)$\mathcal{O}(Y)$ as a Fréchet O(X)$\mathcal{O}(X)$-module equals the codimension of Y   in X  . In the case where X   and Y   are of Liouville type, the same formula is proved for the projective homological dimension of O(Y)$\mathcal{O}(Y)$ over O(X)$\mathcal{O}(X)$. On the other hand, we show that if X   is of Liouville type and Y   is hyperconvex, then the projective homological dimension of O(Y)$\mathcal{O}(Y)$ over O(X)$\mathcal{O}(X)$ equals the dimension of X.