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

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

2012-01-13

arxiv.org/abs/1201.2828v1

Mathematics

Functional Analysis

17 pages

Scientific paper

Let X be a Stein manifold, and let Y be a closed complex submanifold of X. Denote by O(X) the algebra of holomorphic functions on X. We show that the weak (i.e., flat) homological dimension of O(Y) as a Fr'echet 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) over 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) over O(X) equals the dimension of X.

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