Noncommutative Fitting invariants and improved annihilation results

Abstract

To each finitely presented module M over a commutative ring R one can associate an R-ideal Fit_R(M) which is called the (zeroth) Fitting ideal of M over R and which is always contained in the R-annihilator of M. In an earlier article, the second author generalised this notion by replacing R with a (not necessarily commutative) o-order Lambda in a finite dimensional separable algebra, where o is an integrally closed complete commutative noetherian local domain. To obtain annihilators, one has to multiply the Fitting invariant of a (left) Lambda-module M by a certain ideal H(Lambda) of the centre of Lambda. In contrast to the commutative case, this ideal can be properly contained in the centre of Lambda. In the present article, we determine explicit lower bounds for H(Lambda) in many cases. Furthermore, we define a class of `nice' orders Lambda over which Fitting invariants have several useful properties such as good behaviour with respect to direct sums of modules, computability in a certain sense, and H(Lambda) being the best possible.

24 pages; appendix deleted, many corrections and improvements following referee's report. To appear in J. Lond. Math. Soc

References in corpus (2)

• Equivariant Iwasawa theory and non-abelian Stark-type conjectures
• A conductor formula for completed group algebras

Cited by in corpus (7)

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• On the non-abelian Brumer-Stark conjecture and the equivariant Iwasawa main conjecture
• Notes on noncommutative Fitting invariants
• An equivariant Iwasawa main conjecture for local fields
• On non-commutative Euler systems, I: preliminaries on `det' and `Fit'
• Graduated orders over completed group rings and conductor formulæ
• On the equivariant Tamagawa number conjecture for Tate motives and unconditional annihilation results