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Observations on the \(F\)-signature of local rings of characteristic \(p\). (English) Zbl 1102.13027

Let \((R,\mathfrak{m}, k)\) be a \(d\)-dimensional noetherian reduced local ring of prime characteristic \(p\). If \(R^{1/p}\) is finite over \(R\) (i.e., \(R\) is \(F\)-finite), C. Huneke and G. Leuschke [Math. Ann. 324, No. 2, 391–404 (2002; Zbl 1007.13005)] have introduced a numerical invariant \(s(R)\) as the limit of the quotient of certain numerical data associated to the ring \(R\). The present author defines \(s^+(R)\) and \(s^-(R)\) as limsup and respectively liminf of the same quotient, which, unlike \(s(R)\), always exist. A first contribution of the paper under review is an alternative definition for these invariants that does not require \(R\) to be \(F\)-finite. Moreover, the definitions apply to finitely generated \(R\)-modules. The values of \(s^+(R)\), \(s^-(R)\), \(s(R)\) are between 0 and 1. The author shows that if \(s^+(R)\) is big enough, then \(R\) is regular.
The invariant \(s(M)\) being defined as a limit, its existence is not always guaranteed. Several sufficient conditions for the existence of the limit are given. In particular, it is shown that \(s(M)\) exists when \(M\) is finitely generated over a regular ring or is a maximal Cohen-Macaulay module over an excellent local ring which is Gorenstein on the punctured spectrum. The behavior of \(F\)-signature under localizations and faithfully flat ring extensions is also studied.

MSC:

13H05 Regular local rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure

Citations:

Zbl 1007.13005
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References:

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