The Hilbert-Kunz function of some quadratic quotients of the Rees algebra. (English) Zbl 1480.13015
Authors’ abstract: Given a commutative local ring \((R, m)\) and an ideal \(I\) of \(R\), a family of quotients of the Rees algebra \(R[It]\) has been recently studied as a unified approach to the Nagata’s idealization and the amalgamated duplication and as a way to construct interesting examples, especially integral domains. When \(R\) is noetherian of prime characteristic, we compute the Hilbert-Kunz function of the members of this family and, provided that either \(I\) is \(m\)-primary or \(R\) is regular and \(F\)-finite, we also find their Hilbert-Kunz multiplicity. Some consequences and examples are explored.
Reviewer: Amir Mafi (Sanandaj and Tehran)
MSC:
13D40 | Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series |
13H15 | Multiplicity theory and related topics |
13A30 | Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics |
Keywords:
Hilbert-Kunz functionReferences:
[1] | Anderson, D. D.; Winders, Michael, Idealization of a module, J. Commut. Algebra, 1, 1, 3-56 (2009) · Zbl 1194.13002 · doi:10.1216/JCA-2009-1-1-3 |
[2] | Barucci, V.; D’Anna, M.; Strazzanti, F., A family of quotients of the Rees algebra, Comm. Algebra, 43, 1, 130-142 (2015) · Zbl 1327.13087 · doi:10.1080/00927872.2014.897549 |
[3] | Barucci, V.; D’Anna, M.; Strazzanti, F., Families of Gorenstein and almost Gorenstein rings, Ark. Mat., 54, 2, 321-338 (2016) · Zbl 1372.13017 · doi:10.1007/s11512-016-0235-5 |
[4] | Borz\`\i , Alessio, A characterization of the Arf property for quadratic quotients of the Rees algebra, J. Algebra Appl., 19, 7, 2050127, 14 pp. (2020) · Zbl 1453.20078 · doi:10.1142/S0219498820501273 |
[5] | Br H. Brenner, Irrational Hilbert-Kunz multiplicities, 1305.5873, 2013. |
[6] | Chan, C.-Y. Jean; Kurano, Kazuhiko, Hilbert-Kunz functions over rings regular in codimension one, Comm. Algebra, 44, 1, 141-163 (2016) · Zbl 1346.13007 · doi:10.1080/00927872.2014.974247 |
[7] | D’Anna, Marco, A construction of Gorenstein rings, J. Algebra, 306, 2, 507-519 (2006) · Zbl 1120.13022 · doi:10.1016/j.jalgebra.2005.12.023 |
[8] | D’Anna, Marco; Finocchiaro, Carmelo A.; Fontana, Marco, New algebraic properties of an amalgamated algebra along an ideal, Comm. Algebra, 44, 5, 1836-1851 (2016) · Zbl 1345.13002 · doi:10.1080/00927872.2015.1033628 |
[9] | D’Anna, Marco; Fontana, Marco, An amalgamated duplication of a ring along an ideal: the basic properties, J. Algebra Appl., 6, 3, 443-459 (2007) · Zbl 1126.13002 · doi:10.1142/S0219498807002326 |
[10] | D’Anna, Marco; Jafari, Raheleh; Strazzanti, Francesco, Tangent cones of monomial curves obtained by numerical duplication, Collect. Math., 70, 3, 461-477 (2019) · Zbl 1495.13012 · doi:10.1007/s13348-019-00241-w |
[11] | D’Anna, Marco; Strazzanti, Francesco, New algebraic properties of quadratic quotients of the Rees algebra, J. Algebra Appl., 18, 3, 1950047, 14 pp. (2019) · Zbl 1451.13029 · doi:10.1142/S0219498819500476 |
[12] | Enescu, Florian, Applications of pseudocanonical covers to tight closure problems, J. Pure Appl. Algebra, 178, 2, 159-167 (2003) · Zbl 1018.13002 · doi:10.1016/S0022-4049(02)00172-X |
[13] | Enescu, Florian, A finiteness condition on local cohomology in positive characteristic, J. Pure Appl. Algebra, 216, 1, 115-118 (2012) · Zbl 1238.13013 · doi:10.1016/j.jpaa.2011.05.008 |
[14] | Finocchiaro, Carmelo Antonio, A construction of Pr\"{u}fer rings involving quotients of Rees algebras, J. Algebra Appl., 17, 6, 1850098, 16 pp. (2018) · Zbl 1454.13027 · doi:10.1142/S0219498818500986 |
[15] | HY M. Hochster and Y. Yao, Second coefficients of Hilbert-Kunz functions for domains, preliminary preprint, http://www.math.lsa.umich.edu/ hochster/hk.pdf. |
[16] | Huneke, Craig, Hilbert-Kunz multiplicity and the F-signature. Commutative algebra, 485-525 (2013), Springer, New York · Zbl 1275.13012 · doi:10.1007/978-1-4614-5292-8\_15 |
[17] | Huneke, Craig; McDermott, Moira A.; Monsky, Paul, Hilbert-Kunz functions for normal rings, Math. Res. Lett., 11, 4, 539-546 (2004) · Zbl 1099.13508 · doi:10.4310/MRL.2004.v11.n4.a11 |
[18] | Kunz, Ernst, Characterizations of regular local rings of characteristic \(p\), Amer. J. Math., 91, 772-784 (1969) · Zbl 0188.33702 · doi:10.2307/2373351 |
[19] | Kunz, Ernst, On Noetherian rings of characteristic \(p\), Amer. J. Math., 98, 4, 999-1013 (1976) · Zbl 0341.13009 · doi:10.2307/2374038 |
[20] | Monsky, P., The Hilbert-Kunz function, Math. Ann., 263, 1, 43-49 (1983) · Zbl 0509.13023 · doi:10.1007/BF01457082 |
[21] | Nagata, Masayoshi, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, xiii+234 pp. (1962), Interscience Publishers a division of John Wiley & Sons New York-London · Zbl 0123.03402 |
[22] | Oneto, Anna; Strazzanti, Francesco; Tamone, Grazia, One-dimensional Gorenstein local rings with decreasing Hilbert function, J. Algebra, 489, 91-114 (2017) · Zbl 1443.13021 · doi:10.1016/j.jalgebra.2017.05.038 |
[23] | Salimi, Maryam, Family of quotients of some special rings, J. Algebra Appl., 17, 12, 1850233, 8 pp. (2018) · Zbl 1445.13023 · doi:10.1142/S021949881850233X |
[24] | Tavasoli, Elham, Some properties of a family of quotients of the Rees algebra, J. Algebra Appl., 18, 6, 1950113, 11 pp. (2019) · Zbl 1412.13019 · doi:10.1142/S0219498819501135 |
[25] | Watanabe, Kei-ichi; Yoshida, Ken-ichi, Hilbert-Kunz multiplicity and an inequality between multiplicity and colength, J. Algebra, 230, 1, 295-317 (2000) · Zbl 0964.13008 · doi:10.1006/jabr.1999.7956 |
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