Applications of perverse sheaves in commutative algebra

References

[1] D. Arinkin and R. Bezrukavnikov, Perverse coherent sheaves, Mosc. Math. J. 10 (2010), no. 1, 3–29. 10.17323/1609-4514-2010-10-1-3-29Search in Google Scholar

[2] J. Baudin, Duality between Cartier crystals and perverse F p -sheaves, and application to generic vanishing, preprint (2023), https://arxiv.org/abs/2306.05378. Search in Google Scholar

[3] J. Baudin, F. Bernasconi and T. Kawakami, The Frobenius–stable version of the Grauert–Riemenschneider vanishing theorem fails, preprint (2023), https://arxiv.org/abs/2312.13456. Search in Google Scholar

[4] J. Bernstein and V. Lunts, Equivariant sheaves and functors, Lecture Notes in Math. 1578, Springer, Berlin 1994. 10.1007/BFb0073549Search in Google Scholar

[5] P. Berthelot, S. Bloch and H. Esnault, On Witt vector cohomology for singular varieties, Compos. Math. 143 (2007), no. 2, 363–392. 10.1112/S0010437X06002533Search in Google Scholar

[6] A. A. Beĭlinson, J. Bernstein and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, Astérisque 100, Société Mathématique de France, Paris (1982), 5–171. Search in Google Scholar

[7] B. Bhatt, Derived splinters in positive characteristic, Compos. Math. 148 (2012), no. 6, 1757–1786. 10.1112/S0010437X12000309Search in Google Scholar

[8] B. Bhatt, Perverse sheaves, (2015), https://www.math.purdue.edu/~murayama/MATH731.pdf. Search in Google Scholar

[9] B. Bhatt, Cohen–Macaulayness of absolute integral closures, preprint (2020), https://arxiv.org/abs/2008.08070. Search in Google Scholar

[10] B. Bhatt and J. Lurie, A Riemann–Hilbert correspondence in positive characteristic, Camb. J. Math. 7 (2019), no. 1–2, 71–217. 10.4310/CJM.2019.v7.n1.a3Search in Google Scholar

[11] B. Bhatt and J. Lurie, A 𝑝-adic Riemann–Hilbert functor: Z / p n -coefficients, in preparation. Search in Google Scholar

[12] B. Bhatt and P. Scholze, Prisms and prismatic cohomology, Ann. of Math. (2) 196 (2022), no. 3, 1135–1275. 10.4007/annals.2022.196.3.5Search in Google Scholar

[13] M. Blickle, The intersection homology 𝐷-module in finite characteristic, Math. Ann. 328 (2004), no. 3, 425–450. 10.1007/s00208-003-0492-zSearch in Google Scholar

[14] M. Blickle and G. Böckle, Cartier modules: Finiteness results, J. reine angew. Math. 661 (2011), 85–123. 10.1515/CRELLE.2011.087Search in Google Scholar

[15] M. Blickle and R. Bondu, Local cohomology multiplicities in terms of étale cohomology, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 7, 2239–2256. 10.5802/aif.2160Search in Google Scholar

[16] G. Böckle and R. Pink, Cohomological theory of crystals over function fields, EMS Tracts Math. 9, European Mathematical Society, Zürich 2009. 10.4171/074Search in Google Scholar

[17] N. Bridgland, On the de Rham homology of affine varieties in characteristic 0, J. Algebra 643 (2024), 203–223. 10.1016/j.jalgebra.2023.09.028Search in Google Scholar

[18] J.-L. Brylinski, Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques, Géométrie et analyse microlocales, Astérisque 140–141, Société Mathématique de France, Paris (1986), 3–134. Search in Google Scholar

[19] R. Cass, Perverse F p -sheaves on the affine Grassmannian, J. reine angew. Math. 785 (2022), 219–272. 10.1515/crelle-2021-0089Search in Google Scholar

[20] R. Cass and J. Lourenço, Mod 𝑝 sheaves on Witt flags, preprint (2025), https://arxiv.org/abs/2503.01796. Search in Google Scholar

[21] B. Conrad, Grothendieck duality and base change, Lecture Notes in Math. 1750, Springer, Berlin 2000. 10.1007/b75857Search in Google Scholar

[22] M. Emerton and M. Kisin, The Riemann–Hilbert correspondence for unit 𝐹-crystals, Astérisque 293, Société Mathématique de France, Paris 2004. 10.1515/9783110198133.2.677Search in Google Scholar

[23] O. Gabber, Notes on some 𝑡-structures, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter, Berlin (2004), 711–734. 10.1515/9783110198133.2.711Search in Google Scholar

[24] R. García López and C. Sabbah, Topological computation of local cohomology multiplicities, Collect. Math. 49 (1998), 317–324. Search in Google Scholar

[25] M. Goresky and R. MacPherson, Intersection homology theory, Topology 19 (1980), no. 2, 135–162. 10.1016/0040-9383(80)90003-8Search in Google Scholar

[26] M. Goresky and R. MacPherson, Intersection homology. II, Invent. Math. 72 (1983), no. 1, 77–129. 10.1007/BF01389130Search in Google Scholar

[27] R. Hartshorne, On the De Rham cohomology of algebraic varieties, Publ. Math. Inst. Hautes Études Sci. 45 (1975), 5–99. 10.1007/BF02684298Search in Google Scholar

[28] R. Hartshorne and R. Speiser, Local cohomological dimension in characteristic 𝑝, Ann. of Math. (2) 105 (1977), no. 1, 45–79. 10.2307/1971025Search in Google Scholar

[29] M. Hochster and C. Huneke, Infinite integral extensions and big Cohen–Macaulay algebras, Ann. of Math. (2) 135 (1992), no. 1, 53–89. 10.2307/2946563Search in Google Scholar

[30] R. Hotta, K. Takeuchi and T. Tanisaki, 𝐷-modules, perverse sheaves, and representation theory, Progr. Math. 236, Birkhäuser, Boston 2008. 10.1007/978-0-8176-4523-6Search in Google Scholar

[31] C. L. Huneke and R. Y. Sharp, Bass numbers of local cohomology modules, Trans. Amer. Math. Soc. 339 (1993), no. 2, 765–779. 10.1090/S0002-9947-1993-1124167-6Search in Google Scholar

[32] G. Kuperberg, Diffeomorphic Kähler manifolds with different Hodge numbers, Mathoverflow (2010), https://mathoverflow.net/q/42746. Search in Google Scholar

[33] G. Lyubeznik, Finiteness properties of local cohomology modules (an application of 𝐷-modules to commutative algebra), Invent. Math. 113 (1993), no. 1, 41–55. 10.1007/BF01244301Search in Google Scholar

[34] G. Lyubeznik, 𝐹-modules: Applications to local cohomology and 𝐷-modules in characteristic p > 0 , J. reine angew. Math. 491 (1997), 65–130. 10.1515/crll.1997.491.65Search in Google Scholar

[35] G. Lyubeznik, A partial survey of local cohomology, Local cohomology and its applications (Guanajuato 1999), Lecture Notes Pure Appl. Math. 226, Dekker, New York (2002), 121–154. Search in Google Scholar

[36] G. Lyubeznik, On the vanishing of local cohomology in characteristic p > 0 , Compos. Math. 142 (2006), no. 1, 207–221. 10.1112/S0010437X05001533Search in Google Scholar

[37] G. Lyubeznik, A. K. Singh and U. Walther, Local cohomology modules supported at determinantal ideals, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 11, 2545–2578. 10.4171/jems/648Search in Google Scholar

[38] L. Ma and W. Zhang, Eulerian graded 𝒟-modules, Math. Res. Lett. 21 (2014), no. 1, 149–167. 10.4310/MRL.2014.v21.n1.a13Search in Google Scholar

[39] M. Mustaţă and M. Popa, Hodge filtration on local cohomology, Du Bois complex and local cohomological dimension, Forum Math. Pi 10 (2022), Paper No. e22. 10.1017/fmp.2022.15Search in Google Scholar

[40] A. Ogus, Local cohomological dimension of algebraic varieties, Ann. of Math. (2) 98 (1973), 327–365. 10.2307/1970785Search in Google Scholar

[41] T. Reichelt, M. Saito and U. Walther, Dependence of Lyubeznik numbers of cones of projective schemes on projective embeddings, Selecta Math. (N. S.) 27 (2021), no. 1, Paper No. 6. 10.1007/s00029-020-00612-3Search in Google Scholar

[42] T. Reichelt, M. Saito and U. Walther, Topological calculation of local cohomological dimension, J. Singul. 26 (2023), 13–22. 10.5427/jsing.2023.26bSearch in Google Scholar

[43] A. Sannai and A. K. Singh, Galois extensions, plus closure, and maps on local cohomology, Adv. Math. 229 (2012), no. 3, 1847–1861. 10.1016/j.aim.2011.12.021Search in Google Scholar

[44] T. Schedlmeier, A Riemann–Hilbert correspondence for Cartier crystals, preprint (2018), https://arxiv.org/abs/1812.00256. Search in Google Scholar

[45] K. E. Smith, Tight closure of parameter ideals, Invent. Math. 115 (1994), no. 1, 41–60. 10.1007/BF01231753Search in Google Scholar

[46] K. E. Smith, Tight closure and graded integral extensions, J. Algebra 175 (1995), no. 2, 568–574. 10.1006/jabr.1995.1201Search in Google Scholar

[47] A. Stäbler, Intermediate extensions of perverse constructible F p -sheaves commute with smooth pullbacks, J. Number Theory 197 (2019), 317–340. 10.1016/j.jnt.2018.09.001Search in Google Scholar

[48] N. Switala, Lyubeznik numbers for nonsingular projective varieties, Bull. Lond. Math. Soc. 47 (2015), no. 1, 1–6. 10.1112/blms/bdu089Search in Google Scholar

[49] N. Switala, On the de Rham homology and cohomology of a complete local ring in equicharacteristic zero, Compos. Math. 153 (2017), no. 10, 2075–2146. 10.1112/S0010437X17007345Search in Google Scholar

[50] B. Wang, Lyubeznik numbers of irreducible projective varieties depend on the embedding, Proc. Amer. Math. Soc. 148 (2020), no. 5, 2091–2096. 10.1090/proc/14889Search in Google Scholar

[51] W. Zhang, Lyubeznik numbers of projective schemes, Adv. Math. 228 (2011), no. 1, 575–616. 10.1016/j.aim.2011.05.020Search in Google Scholar

[52] Y. Zhang, Graded 𝐹-modules and local cohomology, Bull. Lond. Math. Soc. 44 (2012), no. 4, 758–762. 10.1112/blms/bds008Search in Google Scholar

[53] The Stacks Project Authors, Stacks project, https://stacks.math.columbia.edu. Search in Google Scholar