A length characterization of \(*\)-spread. (English) Zbl 1145.13002

Epstein, Neil; Vraciu, Adela

Osaka J. Math. 45, No. 2, 445-456 (2008).

Several closure operations in a noetherian local ring \((R, \mathfrak{m},k)\) have been defined and studied by numerous authors. For a such operation, say \(\phi\), one can try to define the \(\phi\)-spread of an ideal \(I\subseteq R\) as the minimal number of generators of a minimal reduction of \(I\) with respect to \(\phi\), i.e. an ideal \(J \subseteq I\), minimal with respect to inclusion, such that \(\phi(J)=\phi(I)\). Clearly some problems could arrive: in fact maybe a minimal reduction does not exist, or the spread depends from the choice of the minimal reduction. Anyway for many closure operations make sense the concept of spread. As an example one can think about the integral closure; for this is defined the bar-spread of an ideal \(I\), and it is a classical result, provided the residue field \(k\) is infinite, that the bar-spread of \(I\) is equal to its analytic spread.
In this paper the authors study the \(*\)-spread (the spread corresponding to tight closure) for a local ring of characteristic \(p>0\). The fact that the minimal number of generators is independent from the choice of the minimal reduction (with respect to tight closure) was proved by the first author in N. Epstein [Math. Proc. Camb. Philos. Soc. 139, 371–383 (2005; Zbl 1091.13008)]. The main result is an asymptotic characterization of the \(*\)-spread of an ideal \(I\), \(l^*(I)\), in terms of the length (the formula generalizes a result of the second author obtained in A. Vraciu [J. Algebra 249, 544–565 (2002; Zbl 1057.13004)]). More precisely they showed, under some restrictive hypothesis on the ring \(R\), that \[ l^*(I)=\frac{1}{\operatorname{e_{HK}}(\mathfrak{a})}\lim_{e \rightarrow \infty}\frac{\lambda(I^{[p^{e+e_0}]}/\mathfrak{a}^{[p^{e}]}I^{[p^{e+e_0}]})}{p^{de}} \] where \(\mathfrak{a}\) is an \(\mathfrak{m}\)-primary ideal, \(\operatorname{e_{HK}}\) denotes the Hilbert-Kunz multiplicity, \(e_0\) is a positive integer big enough and \(d\) is the Krull dimension of \(R\).
In the theory of tight-closure, the Hilbert-Kunz multiplicity is an important numerical invariant of an \(\mathfrak{m}\)-primary ideal; this is a real number, and it is a problem to understand whether it is rational. As an application of their main result, the authors connect the rationality of the Hilbert-Kunz multiplicity of \(\mathfrak{m}\)-primary ideals \(I\) and \(J\) with that of \(IJ^{[p^e]}\) for \(e\gg0\).
The authors prove also a change of base formula under flat local homomorphism.

Reviewer: Matteo Varbaro (Genova)

MSC:

13A35	Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
13A30	Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics

Keywords:

\(*\)-spread; Tight closure; Hilbert-Kunz multiplicity

Citations:

Zbl 1091.13008; Zbl 1057.13004

References:

[1]	I.M. Aberbach: Extension of weakly and strongly F-regular rings by flat maps , J. Algebra 241 (2001), 799–807. · Zbl 1007.13002 · doi:10.1006/jabr.2001.8785
[2]	N.M. Epstein: A tight closure analogue of analytic spread , Math. Proc. Cambridge Philos. Soc. 139 (2005), 371–383. · Zbl 1091.13008 · doi:10.1017/S0305004105008546
[3]	M. Hochster and C. Huneke: Tight closure, invariant theory, and the Briançon-Skoda theorem , J. Amer. Math. Soc. 3 (1990), 31–116. JSTOR: · Zbl 0701.13002 · doi:10.2307/1990984
[4]	C. Lech: On the associativity formula for multiplicities , Ark. Mat. 3 (1957), 301–314. · Zbl 0089.26002 · doi:10.1007/BF02589424
[5]	P. Monsky: The Hilbert-Kunz function , Math. Ann. 263 (1983), 43–49. · Zbl 0509.13023 · doi:10.1007/BF01457082
[6]	D.G. Northcott and D. Rees: Reductions of ideals in local rings , Proc. Cambridge Philos. Soc. 50 (1954), 145–158. · Zbl 0057.02601 · doi:10.1017/S0305004100029194
[7]	A. Vraciu: \(\ast\)-independence and special tight closure , J. Algebra 249 (2002), 544–565. · Zbl 1057.13004 · doi:10.1006/jabr.2001.9074