zbMATH — the first resource for mathematics

Strong $$F$$-regularity and generating morphisms of local cohomology modules. (English) Zbl 1415.13006
Let $$k$$ be a field of positive chracteristic $$p>0$$ and $$R$$ a formal power series ring over $$k.$$ Let $$A=R/I$$ be a reduced Cohen-Macaulay ring of dimension $$d\geq 1.$$ Let also $$\Omega/I$$ be a canonical ideal of $$A.$$ Assume that $$d\geq 2, \Omega\neq R$$ and $$J$$ is a radical ideal defining the singular locus of $$R/\Omega.$$ The main result of the paper shows that if $$J(\Omega ^{[p]}:\Omega)\nsubseteq \mathfrak{m}^{[p]},$$ then $$A$$ is a strongly $$F$$-regular ring. Another result gives an explicit description of a generating morphism for the local cohomology module $$H^2_{I_{n-1}(X)}(k[[X]])$$, where $$X$$ is an $$n\times (n-1)$$ matrix of indeterminates over $$k.$$ As an application of the above results it is shown that, if $$k$$ is perfect and $$p\geq 5,$$ the generic determinantal ring $$k[[X]]/I_{n-1}(X)$$ is strongly $$F$$-regular.
MSC:
 13A35 Characteristic $$p$$ methods (Frobenius endomorphism) and reduction to characteristic $$p$$; tight closure 13D45 Local cohomology and commutative rings 13C40 Linkage, complete intersections and determinantal ideals 14B15 Local cohomology and algebraic geometry 14M12 Determinantal varieties
Full Text:
References:
 [1] Blickle, M., The Intersection Homology D-Module in Finite Characteristic, (2001), University of Michigan, PhD thesis · Zbl 1098.14502 [2] Brodmann, M. P.; Sharp, R. Y., Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge Stud. Adv. Math., vol. 60, (1998), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0903.13006 [3] Bruns, W.; Conca, A., F-rationality of determinantal rings and their Rees rings, Michigan Math. J., 45, 291-299, (1998) · Zbl 0967.13006 [4] Bruns, W.; Herzog, J., Cohen-Macaulay Rings, (1998), Cambridge University Press · Zbl 0909.13005 [5] Bruns, W.; Vetter, U., Determinantal Rings, Lecture Notes in Math., vol. 1327, (1988), Springer-Verlag: Springer-Verlag Berlin Heidelberg · Zbl 1079.14533 [6] Conca, A.; Herzog, J., Ladder determinantal rings have rational singularities, Adv. Math., 132, 120-147, (1997) · Zbl 0908.13005 [7] Conca, A.; Mostafazadehfard, M.; Singh, A. K.; Varbaro, M., Hankel determinantal rings have rational singularities, Adv. Math., 335, 111-129, (2018) · Zbl 1403.14082 [8] De Stefani, A.; Núñez-Betancourt, L., A sufficient condition for strong F-regularity, Proc. Amer. Math. Soc., 144, 21-29, (2016) · Zbl 1334.13002 [9] Fedder, R., F-purity and rational singularity, Trans. Amer. Math. Soc., 278, 461-480, (1983) · Zbl 0519.13017 [10] Fedder, R.; Watanabe, K., A characterization of F-regularity in terms of F-purity, (MSRI Publications, vol. 15, (1989), Springer: Springer New York), 227-245 · Zbl 0738.13004 [11] Glassbrenner, D., Strong F-regularity in images of regular rings, Proc. Amer. Math. Soc., 124, 345-353, (1996) · Zbl 0855.13002 [12] Glassbrenner, D.; Smith, K., Singularities of certain ladder determinantal varieties, J. Pure Appl. Algebra, 101, 59-75, (1995) · Zbl 0842.13008 [13] Goto, S., On the Gorensteinness of determinantal loci, J. Math. Kyoto Univ., 19, 371-374, (1979) · Zbl 0418.13008 [14] Hochster, M.; Huneke, C., Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc., 3, 31-116, (1990) · Zbl 0701.13002 [15] Hochster, M.; Huneke, C., F-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc., 346, 1-62, (1994) · Zbl 0844.13002 [16] Hochster, M.; Huneke, C., Tight closure of parameter ideals and splitting in module-finite extensions, J. Algebraic Geom., 3, 599-670, (1994) · Zbl 0832.13007 [17] Huneke, C., Tight Closure and Its Applications, CBMS Regional Conference Series in Mathematics, vol. 88, (1996), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0930.13004 [18] Iyengar, S. B.; Leuschke, G. J.; Leykin, A.; Miller, C.; Miller, E.; Singh, A. K.; Walther, U., Twenty-Four Hours of Local Cohomology, Graduate Studies in Mathematics, vol. 87, (2007), American Mathematical Society · Zbl 1129.13001 [19] Katzman, M., Parameter-test-ideals of Cohen-Macaulay rings, Compos. Math., 144, 933-948, (2008) · Zbl 1152.13005 [20] Katzman, M., Frobenius maps on injective hulls and their applications to tight closure, J. Lond. Math. Soc. (2), 81, 589-607, (2010) · Zbl 1197.13006 [21] Katzman, M.; Ma, L.; Smirnov, I.; Zhang, W., D-module and F-module length of local cohomology modules, Trans. Amer. Math. Soc., 370, 8551-8580, (2018) · Zbl 1440.13077 [22] Kunz, E., Characterizations of regular local rings of characteristic p, Amer. J. Math., 91, 772-784, (1969) · Zbl 0188.33702 [23] Lam, T. Y., A First Course in Noncommutative Rings, Graduate Texts in Mathematics, vol. 131, (2001), Springer-Verlag: Springer-Verlag New York · Zbl 0980.16001 [24] Lyubeznik, G., F-modules: applications to local cohomology and D-modules in characteristic $$p > 0$$, J. Reine Angew. Math., 491, 65-130, (1997) · Zbl 0904.13003 [25] Lyubeznik, G.; Singh, A. K.; Walther, U., Local cohomology modules supported at determinantal ideals, J. Eur. Math. Soc., 18, 2545-2578, (2016) · Zbl 1408.13047 [26] Lyubeznik, G.; Smith, K., On the commutation of the test ideal with localization and completion, Trans. Amer. Math. Soc., 353, 3149-3180, (2001) · Zbl 0977.13002 [27] Ma, L., A sufficient condition for F-purity, J. Pure Appl. Algebra, 218, 1179-1183, (2014) · Zbl 1283.13006 [28] Peskine, C.; Szpiro, L., Dimension projective finie at cohomologie locale, Publ. Math. Inst. Hautes Études Sci., 42, 323-395, (1973) [29] Raicu, C.; Weyman, J., Local cohomology with support in generic determinantal ideals, Algebra Number Theory, 8, 1231-1257, (2014) · Zbl 1303.13018 [30] Raicu, C.; Weyman, J.; Witt, E. E., Local cohomology with support in ideals of maximal minors and sub-maximal Pfaffians, Adv. Math., 250, 596-610, (2014) · Zbl 1295.13025 [31] Sharp, R., Graded annihilators of modules over the Frobenius skew polynomial ring, and tight closure, Trans. Amer. Math. Soc., 359, 4237-4258, (2007) · Zbl 1130.13002 [32] Singh, A. K., F-regularity does not deform, Amer. J. Math., 121, 919-929, (1999) · Zbl 0946.13002 [33] Witt, E. E., Local cohomology with support in ideals of maximal minors, Adv. Math., 231, 1998-2012, (2012) · Zbl 1253.13019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.