Abstract
In this paper, we prove that the Watanabe-Yoshida conjecture holds up to dimension 7. Our primary new tool is a function, \(\varphi _J(R;z^t),\) that interpolates between the Hilbert-Kunz multiplicities of a base ring, R, and various radical extensions, \(R_n\). We prove that this function is concave and show that its rate of growth is related to the size of \(e_{\textrm{HK}}(R)\). We combine techniques from Celikbas et al. (Nagoya Math. J. 205, 149–165, 2012) and Aberbach and Enescu (Nagoya Math. J. 212, 59–85, 2013) to get effective lower bounds for \(\varphi ,\) which translate to improved bounds on the size of Hilbert-Kunz multiplicities of singular rings. The improved inequalities are powerful enough to show that the conjecture of Watanabe and Yoshida holds in dimension 7.
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Notes
This limit was shown to be well defined by Paul Monsky in [17].
The formally unmixed condition is necessary here; it is not difficult to construct examples of singular rings with embedded prime ideals that satisfy \(e_{\textrm{HK}}({R}) = 1.\) Recall that \((R, \mathfrak {m}_R)\) is formally unmixed if the \(\mathfrak {m}_R\)-adic completion, \(\widehat{R},\) is unmixed — that is, \(\dim \widehat{R}/\mathfrak {p} = \dim \widehat{R}\) for all associated primes of \(\widehat{R}.\)
See Sect. 2.2 for details about radical extensions.
This function was partly inspired by the convex function introduced by Blickle, Schwede, and Tucker in [3]. Our function is related to theirs when R is regular.
As far as the authors know, this is the first time this formula has appeared in print.
See [20, Section V.B.2] for details about the completed tensor product — and recall that \(\overline{\kappa }\) is the directed colimit of finite extensions of \(\kappa \).
This reduction, to the case where R is complete with algebraically closed residue field, was first observed by Kunz in Section 3 of his famous paper on the Frobenius endomorphism [14].
The inequality \(e_{\textrm{HK}}(R_{p,d})<2\) is easily verified from the identity \(e_{\textrm{HK}}(R_{p,d}) + s(R_{p,d}) = 2\) by observing that \(R_{p,d}\) is strongly F-Regular, and therefore \(s(R_{p, d})>0\) — where \(s(R_{p, d})\) denotes the F-signature of \(R_{p,d}.\)
See Lemma 2.4.
Note that we are assuming R is normal and \(z \not \in \mathfrak {m}_R^2\), and this ensures that \(X^n - z\) is the minimal polynomial for \(z^{1/n}\) over R — see [1, Remark 4.3] for details.
Note that S is complete with perfect residue field, so S is F-finite.
Observe that \(\nu _s\) is a non-negative increasing function with \(\nu _s = 1\) for \(s \ge d\) and \(\nu _s = 0\) for all \(s \le 0.\) Several interesting formulas are known for \(\nu _s\) — e.g., see [5, 16] — including the following, which hold for \(s \ge 0\):
$$\begin{aligned} \nu _s = \sum _{j=0}^{\lfloor {s}\rfloor } \left( -1 \right) ^j \dfrac{(s-j)^d}{j! ( d-j)!} = \frac{2}{\pi } \int _{0}^s\int _{0}^{\infty } \left( \frac{\sin u}{u}\right) ^d \cos \left( (d-2w)u\right) du \,\, dw. \end{aligned}$$Note that the conditions of Lemma 2.4 are satisfied and, therefore, R and all of the \(R_n\)’s are complete domains, and, in particular, unmixed. The statement of [2, Theorem 3.2] assumes that \(\left( \left( (\underline{x})^*\right) R_n\right) ^* = \left( (\underline{x})R_n\right) ^* \subset (J, z^{k/n} )R_n,\) but the argument works if we only assume the weaker inclusion \(((\underline{x})^* ) R_n \subset (J, z^{k/n} )R_n.\) Also, note that \(z^{k/n}, z_2, \dots , z_r\) are clearly generators for \((J, z^{k/n})R_n / \left( (\underline{x})^{*}\right) R_n\) and that the proof in [2] goes through even if they are not minimal generators. With these (possibly) weaker assumptions, the strength of [2, Theorem 3.2] is slightly weakened as well.
This holds because \(R \hookrightarrow R_n\) is split and therefore \(z_i^k \in (\underline{x})^\ell R_n \Longleftrightarrow z_i^k \in (\underline{x})^\ell \) in R.
Recall that \(\nu _{s}\) is an increasing function of s.
This is well known to be a generic condition — see [12, Theorem 8.6.6].
When treating the dimension 7 case in the next section, we use this result with \(k=1\) — this turns out to give a more streamlined argument than taking larger k.
Also, recall that \(z=z_1 \in \mathfrak {m}_S \setminus \mathfrak {m}_S^2\) is part of a minimal reduction of \(\mathfrak {m}_S\) in S, by construction.
This can be seen immediately from the equality
$$\frac{d}{dp} e_{\textrm{HK}}({R_{p, 7}}) = - \frac{488 p^5+896 p^3+128 p}{4725 p^8+8190 p^6+8589 p^4+4368 p^2+1344}.$$They show that either R is Cohen-Macaulay with minimal multiplicity, in which case the conjecture is known (see [2, Theorem 2.4 (iv)]), or the inequality \(\mu (R) \le e(R) - 2,\) holds.
The bounds listed for \(\mu (R) =1, 2\) and 3 are achieved at \(s = 4\), \(s = 3.56745\), and \(s = 3.32317\), respectively.
This is always possible as long as \(\mu (R) \ge 2\) — see [12, Theorem 8.6.6].
The bounds in this table are not the best that our methods achieve for each specific value of \(e \ge 13\). The particular values of \(e_1\) and \(e_2\), and the corresponding bounds on \(e_{HK}\), were chosen, in an “informed trial and error” process, to make the overall presentation of the data as palatable as possible.
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Acknowledgements
The authors are grateful to the University of Tulsa and the University of Missouri, Columbia. They would also like to thank ™ZOOM for making this collaboration possible without extensive commuting.
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Nicholas O. Cox-Steib and Ian M. Aberbach contributed equally to this work.
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This paper is dedicated to Ngo Viet Trung on the occasion of his 70th birthday.
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Cox-Steib, N.O., Aberbach, I.M. Bounds for the Hilbert-Kunz Multiplicity of Singular Rings. Acta Math Vietnam (2024). https://doi.org/10.1007/s40306-024-00525-9
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DOI: https://doi.org/10.1007/s40306-024-00525-9