Samuel multiplicity and Fredholm theory

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 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 of the associated complex of analytic sheaves at z = 0.