Abstract
In this note we combine methods from commutative algebra and complex analytic geometry to calculate the generic values of the cohomology dimensions of a commuting multioperator on its Fredholm domain. More precisely, we prove that, for a given Fredholm tuple T = (T 1, ..., T n ) of commuting bounded operators on a complex Banach space X, the limits \(c_p(T) = {\rm lim}_{k \rightarrow \infty} {\rm dim} H^p(T^k,X)/k^n\) exist and calculate the generic dimension of the cohomology groups H p(z − T, X) of the Koszul complex of T near z = 0. To deduce this result we show that the above limits coincide with the Samuel multiplicities of the stalks of the cohomology sheaves \(H^p(z-T,{\mathcal{O}}^{X}_{{\mathbb{C}}^n})\) of the associated complex of analytic sheaves at z = 0.
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