在這篇文章內，給定 R 是一個有 1 的交換諾特局部環且特徵 p>0 的情況下，我們用希爾伯特-昆茲重數和科恩結構定理來研究 F-標誌 s(R)。假設 R 是一個整環且剩餘體是完全體，R^{1/p^e}代表 所有 R 的 p^e 次方根，那 F-標誌可以用 R^{1/p} 中最大的自由模的秩來計算。 另外 F-標誌可以用來判斷是不是正則局部環，s(R) = 1 時 R是正則，s(R) > 0 時 R 是強 F-正則。

In this article, we study the F-signature s(R) of a commutative Noetherian local ring R with characteristic p>0, by using the Hilbert-Kunz multiplicity and the Cohen structure theorem. If R is a domain with a perfect residue field and the ring R^{1/p^e} denote the p^e-th roots of R, then the Hilbert-Kunz multiplicity is related by the numbers of minimal generators of R^{1/p} as R-module for all e>0, the F-signature is calculated from the maximal rank of free R-submodule of R^{1/p^e} for all e>0. One use of the F-signature is to determine whether the ring is a regular local ring or not : s(R) = 1 if and only if R is regular. In fact, more is true : s(R)>0 if and only if R is strong F-regular, which defines a class of singularities for R.

[AL03] Ian M. Aberbach and Graham J. Leuschke, The F-signature and strong Fregularity, Math. Res. Lett. 10 (2003), no. 1, 51–56. MR 1960123
[AM69] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra,
Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.
MR 0242802
[Coh46] I. S. Cohen, On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. 59 (1946), 54–106. MR 16094
[Con96] Aldo Conca, Hilbert-Kunz function of monomial ideals and binomial hypersurfaces, Manuscripta Math. 90 (1996), no. 3, 287–300. MR 1397658
[Fed83] Richard Fedder, F-purity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), no. 2, 461–480. MR 701505
[HH89] Melvin Hochster and Craig Huneke, Tight closure and strong F-regularity, no. 38,1989, Colloque en l'honneur de Pierre Samuel (Orsay, 1987), pp. 119–133. MR1044348
[Hun13] Craig Huneke, Hilbert-Kunz multiplicity and the F-signature, Commutative algebra, Springer, New York, 2013, pp. 485–525. MR 3051383
[HW02] Nobuo Hara and Kei-Ichi Watanabe, F-regular and F-pure rings vs. log terminal and log canonical singularities, J. Algebraic Geom. 11 (2002), no. 2, 363–392. MR 1874118
[Kat] D Katz, A guide to cohen's structure theorem for complete local rings.
[Kun69] Ernst Kunz, Characterizations of regular local rings of characteristic p, Amer .J. Math. 91 (1969), 772–784. MR 252389
[Kun76] Ernst Kunz, On Noetherian rings of characteristic p, Amer. J. Math. 98 (1976), no. 4, 999–1013. MR 432625
[Mat80] Hideyuki Matsumura, Commutative algebra, second ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass. 1980. MR 575344
[Mat89] Hideyuki Matsumura, Commutative ring theory, second ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989, Translated from the Japanese by M. Reid. MR 1011461
[Mon83] P. Monsky, The Hilbert-Kunz function, Math. Ann. 263 (1983), no. 1, 43–49.
MR 697329
[Nag62] Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.),
New York-London, 1962. MR 0155856
[PT18] Thomas Polstra and Kevin Tucker, F-signature and Hilbert-Kunz multiplicity: a combined approach and comparison, Algebra Number Theory 12 (2018), no. 1,
61–97. MR 3781433
[Sch17] Karl Schwede, Notes on characteristic p, commutative algebra Math (Spring 2017).
[Tuc12] Kevin Tucker, F-signature exists, Invent. Math. 190 (2012), no. 3, 743–765. MR 2995185
[Urs] Andoni Zozaya Ursuegui, The cohen structure theorem.
[WY00] Kei-ichi Watanabe and Ken-ichi Yoshida, Hilbert–kunz multiplicity and an inequality between multiplicity and colength, Journal of Algebra 230 (2000), no. 1,
295–317.