Noncommutative Fitting invariants and improved annihilation results. (Q2843985)

The theory of noncommutative Fitting invariants is important in arithmetic, for instance in the context of the Strong Brumer Conjecture and many related conjectures. In prior papers, Nickel developed an algebraic framework for noncommutative Fitting invariants and also put this to good use in arithmetic. The paper under review, joint with Henri Johnston, is an interesting contribution to the algebraic side of things.NEWLINENEWLINE Let \(\Lambda\) be an order in a finite-dimensional separable \(F\)-algebra, where \(F\) is the quotient field of a ``good enough'' local domain \(\mathfrak o\) (e.g.~a discrete valuation ring). To every finitely generated \(\Lambda\)-module \(M\) one attaches a certain ideal in the center of \(\Lambda\), denoted \(\mathrm{Fitt}_\Lambda^{\max}(M)\). One would dearly like this ideal to be contained in the \(\Lambda\)-annihilator ideal of \(M\), but unfortunately this containment can only be proved upon multiplying \(\mathrm{Fitt}_\Lambda^{\max}(M)\) with a certain ideal \(\mathcal H(\Lambda)\). This only depends on \(\Lambda\), not on the module \(M\). The introduction of the paper lists some very natural questions concerning \(\mathrm{Fitt}_\Lambda^{\max}(M)\) and \(\mathcal H(\Lambda)\), all inspired by the commutative situation. The body of the paper provides interesting (partial) answers. A key role is played by so-called nice Fitting orders. The requirements for \(\Lambda\) to be a Fitting order are technical and rather harmless; the usual group rings in arithmetic are Fitting orders. Such an order is called ``nice'' if it is a product of subrings which are either maximal orders or full matrix rings over commutative rings. For rings of the latter type, one employs Morita equivalence. Very loosely speaking, nice Fitting orders seem to be almost as good in the context of Fitting invariants as commutative rings. For example, the recalcitrant ideal \(\mathcal H(\Lambda)\) turns out to be 1, that is, the Fitting invariant is indeed a subset of the annihilators, if \(\Lambda\) is a nice Fitting order.NEWLINENEWLINE In \S4 we find (among many other things) one result that deserves mention because of its elegance: If \(\mathfrak o\) is an appropriate local domain with residual characteristic \(p>0\) (take \(\mathbb Z_p\) for example), then the Fitting order \(\mathfrak o[G]\) is nice iff \(p\) does not divide the order of the commutator group \(G'\). Observe the appealing link with the commutative case (\(G'=1\)). The last section \S6 of the paper discusses the central conductor ideal \(\mathcal F(\Lambda)\), its relation to \(\mathcal H(\Lambda)\), and proves many detailed results on annihilation of modules in terms of Fitting invariants.