Samuel multiplicity and Fredholm theory

Abstract

Abstract. In this note we prove that, for a given Fredholm tuple T = (T1, . . . , Tn) of commuting bounded operators on a complex Banach space X, the limits cp(T ) = lim k→∞ 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, O X C n ) of the associated complex of analytic sheaves at z = 0.

Estimates for Hilbertian Koszul homology

“…• Clearly, our result implies that e i = lim k→∞ h i (T k 1 ,...,T k n ) k n (see Corollary 2.3 in [12]) and that index index(T ) = n i=0 (−1) i e i by the multiplicity formula index(T k 1 1 , . .…”

Fredholm spectrum and growth of cohomology groups

“…The results obtained so far, together with a base change theorem proved in [4], allow us to prove the first main result of this paper.…”

“…as well as the induced cochain maps between the associated complexes of sheaves O On the other hand, it is well-known (see [6] or [4]) that the rank of the coherent sheaf H p at z = 0 is given by the Samuel multiplicity of its stalk…”

“…Using the fact that c p is the rank of the coherent sheaf H p on C, and hence can be computed as the Samuel multiplicity of its stalks, it was shown in [4] that…”

An Introduction to Hilbert Module Approach to Multivariable Operator Theory

“…Because of its strong interaction with commutative algebra and complex analytic geometry, Hilbert module approach to Hilbert-Samuel polynomial and Samuel multiplicity has had a spectacular development since its origin. The reader is referred to the recent work by Eschmeier [Es08a], [Es08b], [Es07b], [Es07a] and Fang [Fa06], [Fa08], [Fa09].…”