Upper bound of multiplicity in Cohen-Macaulay rings of prime characteristic
Main Article Content
Abstract
Let (R,m) be a local ring of prime characteristic p and of dimension d with the embedding dimension v, type s and the Frobenius test exponent for parameter ideals Fte(R). We will give an upper bound for the multiplicity of Cohen-Macaulay rings in prime characteristic in terms of Fte(R), d, v and s. Our result extends the main results for Gorenstein rings due to Huneke and Watanabe [8].
Article Details
Keywords
Multiplicity, the Frobenius test exponent, Cohen-Macaulay
References
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[10] D. T. Huong and P. H. Quy, Upper bound of multiplicity in primecharacteristic, Forum Math., 32(2) (2020), 393–397.
[11] D. T. Huong and P. H. Quy, Frobenius test exponent for ideals generated by filter regular sequences, Acta Math. Vietnam, 47(1) (2022), 151–159.
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Comm.Algebra, 47:6 (2019), 2450–2456.
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[17] P.H. Quy and K. Shimomoto, F-injectivity and Frobenius closure of ideals in Noetherian rings of characteristic p > 0, Adv. Math. 313 (2017), 127–166.
[18] P.H. Quy, On the uniform bound of Frobenius test exponents, J. Algebra 518 (2019), 119–128.
[19] R.Y. Sharp, On the Hartshorne-Speiser-Lyubeznik theorem about Ar- tinian modules with a Frobenius action, Proc. Amer. Math. Soc. 135 (2007), 665–670.
[20] I. Swanson and C. Huneke, Integral Closure of Ideals, Rings and Mod- ules, London Math. Soc. Lecture Notes 336, Cambridge University Press, 2006.
[2] W. Brun, J. Herzog, Cohen-Macaulay rings, Cambridge University Press, Cambridge, 1993.
[3] R. Hartshorne and R. Speiser, Local cohomological dimension in char- acteristic p, Ann. of Math. 105 (1977), 45–79.
[4] J. Herzog, T. Hibi, Monomial ideals, GTM 260, Springer, 2011.
[5] M. Hochster and C. Huneke, Tight Closure, Invariant Theory, and the Brian ̧con-Skoda Theorem, J. Amer. Math. Soc. 3 (1990), 31–116.
[6] C. Huneke, Tight closure and its applications, CBMS Lecture Notes in Mathematics, Vol.88, Amer. Math. Soc., Providence, (1996).
[7] C. Huneke, M. Katzman, R.Y. Sharp and Y. Yao, Frobenius test exponents for parameter ideals in generalized Cohen-Macaulay local rings, J. Algebra, 305 (2006), 516–539.
[8] C. Huneke and K.-i. Watanabe, Upper bound of multiplicity of F -pure rings, Proc. Amer. Math. Soc. 143 (2015), 5021–5026.
[9] D. T. Huong and P. H. Quy, Notes on the Frobenius test exponents, Comm. Algebra, 47:7 (2019), 2702–2710.
[10] D. T. Huong and P. H. Quy, Upper bound of multiplicity in primecharacteristic, Forum Math., 32(2) (2020), 393–397.
[11] D. T. Huong and P. H. Quy, Frobenius test exponent for ideals generated by filter regular sequences, Acta Math. Vietnam, 47(1) (2022), 151–159.
[12] M. Katzman and R.Y. Sharp, Uniform behaviour of the Frobenius closures of ideals generated by regular sequences, J. Algebra 295 (2006) 231–246.
[13] M. Katzman and W. Zhang, Multiplicity bounds in prime characteristic,
Comm.Algebra, 47:6 (2019), 2450–2456.
[14] G. Lyubeznik, F-modules: applications to local cohomology and D-modules in characteristic p > 0, J. reine angew. Math. 491 (1997), 65–130.
[15] K. Maddox, A sufficient condition for the finiteness of Frobenius test exponents, Proc. Amer. Math. Soc., 147 (2019), 5083–5092.
[16] T. Polstra and P. H. Quy, Nilpotence of Frobenius actions on local cohomology and Frobenius closure of ideals, J. Algebra, 529 (2019), 196–225.
[17] P.H. Quy and K. Shimomoto, F-injectivity and Frobenius closure of ideals in Noetherian rings of characteristic p > 0, Adv. Math. 313 (2017), 127–166.
[18] P.H. Quy, On the uniform bound of Frobenius test exponents, J. Algebra 518 (2019), 119–128.
[19] R.Y. Sharp, On the Hartshorne-Speiser-Lyubeznik theorem about Ar- tinian modules with a Frobenius action, Proc. Amer. Math. Soc. 135 (2007), 665–670.
[20] I. Swanson and C. Huneke, Integral Closure of Ideals, Rings and Mod- ules, London Math. Soc. Lecture Notes 336, Cambridge University Press, 2006.