An analogue of adjoint ideals and PLT singularities in mixed characteristic
Authors:
Linquan Ma, Karl Schwede, Kevin Tucker, Joe Waldron and Jakub Witaszek
Journal:
J. Algebraic Geom. 31 (2022), 497-559
DOI:
https://doi.org/10.1090/jag/797
Published electronically:
May 5, 2022
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We use the framework of perfectoid big Cohen-Macaulay (BCM) algebras to define a class of singularities for pairs in mixed characteristic, which we call purely BCM-regular singularities, and a corresponding adjoint ideal. We prove that these satisfy adjunction and inversion of adjunction with respect to the notion of BCM-regularity and the BCM test ideal defined by the first two authors. We compare them with the existing equal characteristic purely log terminal (PLT) and purely $F$-regular singularities and adjoint ideals. As an application, we obtain a uniform version of the Briançon-Skoda theorem in mixed characteristic. We also use our theory to prove that two-dimensional Kawamata log terminal singularities are BCM-regular if the residue characteristic $p>5$, which implies an inversion of adjunction for three-dimensional PLT pairs of residue characteristic $p>5$. In particular, divisorial centers of PLT pairs in dimension three are normal when $p > 5$. Furthermore, in Appendix A we provide a streamlined construction of perfectoid big Cohen-Macaulay algebras and show new functoriality properties for them using the perfectoidization functor of Bhatt and Scholze.
References
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- Raymond C. Heitmann and Linquan Ma, Extended plus closure in complete local rings, J. Algebra 571 (2021), 134–150. MR 4200713, DOI 10.1016/j.jalgebra.2018.10.006
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- Melvin Hochster, Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc. 231 (1977), no. 2, 463–488. MR 463152, DOI 10.1090/S0002-9947-1977-0463152-5
- Melvin Hochster, Big Cohen-Macaulay algebras in dimension three via Heitmann’s theorem, J. Algebra 254 (2002), no. 2, 395–408. MR 1933876, DOI 10.1016/S0021-8693(02)00086-8
- M. Hochster, Foundations of tight closure theory, lecture notes from a course taught at the University of Michigan, Fall 2007.
- Eero Hyry and Karen E. Smith, On a non-vanishing conjecture of Kawamata and the core of an ideal, Amer. J. Math. 125 (2003), no. 6, 1349–1410. MR 2018664, DOI 10.1353/ajm.2003.0041
- János Kollár, Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge, 2013. With a collaboration of Sándor Kovács. MR 3057950, DOI 10.1017/CBO9781139547895
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- Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279. MR 276239, DOI 10.1007/BF02684604
- Joseph Lipman, Cohen-Macaulayness in graded algebras, Math. Res. Lett. 1 (1994), no. 2, 149–157. MR 1266753, DOI 10.4310/MRL.1994.v1.n2.a2
- Linquan Ma and Karl Schwede, Perfectoid multiplier/test ideals in regular rings and bounds on symbolic powers, Invent. Math. 214 (2018), no. 2, 913–955. MR 3867632, DOI 10.1007/s00222-018-0813-1
- Linquan Ma and Karl Schwede, Singularities in mixed characteristic via perfectoid big Cohen-Macaulay algebras, Duke Math. J. 170 (2021), no. 13, 2815–2890. MR 4312190, DOI 10.1215/00127094-2020-0082
- Fabrice Orgogozo, Exposé IV. Le théorème de Cohen-Gabber, Astérisque 363-364 (2014), 51–68 (French). Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents. MR 3329773
- J. B. Sancho de Salas, Blowing-up morphisms with Cohen-Macaulay associated graded rings, Géométrie algébrique et applications, I (La Rábida, 1984) Travaux en Cours, vol. 22, Hermann, Paris, 1987, pp. 201–209. MR 907914
- Kenta Sato and Shunsuke Takagi, General hyperplane sections of threefolds in positive characteristic, J. Inst. Math. Jussieu 19 (2020), no. 2, 647–661. MR 4079156, DOI 10.1017/s1474748018000166
- Karl Schwede and Karen E. Smith, Globally $F$-regular and log Fano varieties, Adv. Math. 224 (2010), no. 3, 863–894. MR 2628797, DOI 10.1016/j.aim.2009.12.020
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- K. E. Smith, Tight closure of parameter ideals, Invent. Math. 115 (1994), no. 1, 41–60. MR 1248078, DOI 10.1007/BF01231753
- Karen E. Smith, $F$-rational rings have rational singularities, Amer. J. Math. 119 (1997), no. 1, 159–180. MR 1428062, DOI 10.1353/ajm.1997.0007
- The Stacks Project authors, Stacks Project.
- Shunsuke Takagi, An interpretation of multiplier ideals via tight closure, J. Algebraic Geom. 13 (2004), no. 2, 393–415. MR 2047704, DOI 10.1090/S1056-3911-03-00366-7
- Shunsuke Takagi, A characteristic $p$ analogue of plt singularities and adjoint ideals, Math. Z. 259 (2008), no. 2, 321–341. MR 2390084, DOI 10.1007/s00209-007-0227-z
- Shunsuke Takagi, Adjoint ideals along closed subvarieties of higher codimension, J. Reine Angew. Math. 641 (2010), 145–162. MR 2643928, DOI 10.1515/CRELLE.2010.031
- Shunsuke Takagi, Adjoint ideals and a correspondence between log canonicity and $F$-purity, Algebra Number Theory 7 (2013), no. 4, 917–942. MR 3095231, DOI 10.2140/ant.2013.7.917
- T. Takamatsu and S. Yoshikawa, Minimal model program for semi-stable threefolds in mixed characteristic, arXiv:2012.07324, 2020.
- Hiromu Tanaka, Minimal model program for excellent surfaces, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 1, 345–376 (English, with English and French summaries). MR 3795482, DOI 10.5802/aif.3163
References
- Yves André, La conjecture du facteur direct, Publ. Math. Inst. Hautes Études Sci. 127 (2018), 71–93 (French, with French summary). MR 3814651, DOI 10.1007/s10240-017-0097-9
- Yves André, Le lemme d’Abhyankar perfectoide, Publ. Math. Inst. Hautes Études Sci. 127 (2018), 1–70 (French, with French summary). MR 3814650, DOI 10.1007/s10240-017-0096-x
- Yves André, Weak functoriality of Cohen-Macaulay algebras, J. Amer. Math. Soc. 33 (2020), no. 2, 363–380. MR 4073864, DOI 10.1090/jams/937
- Luchezar L. Avramov, Hans-Bjørn Foxby, and Bernd Herzog, Structure of local homomorphisms, J. Algebra 164 (1994), no. 1, 124–145. MR 1268330, DOI 10.1006/jabr.1994.1057
- B. Bhatt, Lecture notes on prismatic cohomology, Available at http://www-personal.umich.edu/~bhattb/teaching/prismatic-columbia/, 2019.
- B. Bhatt, Cohen–Macaulayness of absolute integral closures, arXiv:2008.08070, 2020.
- Bhargav Bhatt, Srikanth B. Iyengar, and Linquan Ma, Regular rings and perfect(oid) algebras, Comm. Algebra 47 (2019), no. 6, 2367–2383. MR 3957103, DOI 10.1080/00927872.2018.1524009
- B. Bhatt, L. Ma, Z. Patakfalvi, K. Schwede, K. Tucker, J. Waldron, and J. Witaszek, Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic, arXiv:2012.15801, 2020.
- Bhargav Bhatt, Matthew Morrow, and Peter Scholze, Integral $p$-adic Hodge theory, Publ. Math. Inst. Hautes Études Sci. 128 (2018), 219–397. MR 3905467, DOI 10.1007/s10240-019-00102-z
- B. Bhatt and P. Scholze, Prisms and prismatic cohomology, arXiv:1905.08229, 2019.
- Manuel Blickle and Karl Schwede, $p^{-1}$-linear maps in algebra and geometry, Commutative algebra, Springer, New York, 2013, pp. 123–205. MR 3051373, DOI 10.1007/978-1-4614-5292-8_5
- Manuel Blickle, Karl Schwede, and Kevin Tucker, $F$-singularities via alterations, Amer. J. Math. 137 (2015), no. 1, 61–109. MR 3318087, DOI 10.1353/ajm.2015.0000
- Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
- J. Carvajal-Rojas, Finite torsors over strongly F-regular singularities, arXiv:1710.06887, 2017.
- Javier Carvajal-Rojas, Linquan Ma, Thomas Polstra, Karl Schwede, and Kevin Tucker, Covers of rational double points in mixed characteristic, J. Singul. 23 (2021), 127–150. MR 4292622, DOI 10.5427/jsing
- J. Carvajal-Rojas and A. Stäbler, On the behavior of $F$-signatures, splitting primes, and test modules under finite covers, arXiv:1904.10382, 2019.
- Paolo Cascini, Yoshinori Gongyo, and Karl Schwede, Uniform bounds for strongly $F$-regular surfaces, Trans. Amer. Math. Soc. 368 (2016), no. 8, 5547–5563. MR 3458390, DOI 10.1090/tran/6515
- K. Cesnavicius and P. Scholze, Purity for flat cohomology, arXiv:1912.10932.
- Vincent Cossart and Olivier Piltant, Resolution of singularities of arithmetical threefolds, J. Algebra 529 (2019), 268–535. MR 3942183, DOI 10.1016/j.jalgebra.2019.02.017
- Omprokash Das, On strongly $F$-regular inversion of adjunction, J. Algebra 434 (2015), 207–226. MR 3342393, DOI 10.1016/j.jalgebra.2015.03.025
- Geoffrey D. Dietz, Big Cohen-Macaulay algebras and seeds, Trans. Amer. Math. Soc. 359 (2007), no. 12, 5959–5989. MR 2336312, DOI 10.1090/S0002-9947-07-04252-3
- O. Gabber, Observations made after the MSRI workshop on homological conjectures, https://www.msri.org/workshops/842/schedules/23854/documents/3322/assets/31362, 2018.
- Ofer Gabber and Lorenzo Ramero, Almost ring theory, Lecture Notes in Mathematics, vol. 1800, Springer-Verlag, Berlin, 2003. MR 2004652, DOI 10.1007/b10047
- Raymond C. Heitmann and Linquan Ma, Extended plus closure in complete local rings, J. Algebra 571 (2021), 134–150. MR 4200713, DOI 10.1016/j.jalgebra.2018.10.006
- Raymond Heitmann and Linquan Ma, Big Cohen-Macaulay algebras and the vanishing conjecture for maps of Tor in mixed characteristic, Algebra Number Theory 12 (2018), no. 7, 1659–1674. MR 3871506, DOI 10.2140/ant.2018.12.1659
- Raymond C. Heitmann, The étale locus in complete local rings 150 years with Roger and Sylvia Wiegand, Commutative algebra—150 years with Roger and Sylvia Wiegand, Contemp. Math., vol. 773, Amer. Math. Soc., [Providence], RI, [2021] ©2021, pp. 49–62. MR 4321390, DOI 10.1090/conm/773/15532
- Melvin Hochster, Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc. 231 (1977), no. 2, 463–488. MR 463152, DOI 10.2307/1997914
- Melvin Hochster, Big Cohen-Macaulay algebras in dimension three via Heitmann’s theorem, J. Algebra 254 (2002), no. 2, 395–408. MR 1933876, DOI 10.1016/S0021-8693(02)00086-8
- M. Hochster, Foundations of tight closure theory, lecture notes from a course taught at the University of Michigan, Fall 2007.
- Eero Hyry and Karen E. Smith, On a non-vanishing conjecture of Kawamata and the core of an ideal, Amer. J. Math. 125 (2003), no. 6, 1349–1410. MR 2018664
- János Kollár, Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge, 2013. With a collaboration of Sándor Kovács. MR 3057950, DOI 10.1017/CBO9781139547895
- J. Kollár and 14 coauthors: Flips and abundance for algebraic threefolds, Société Mathématique de France, Paris, 1992, Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Astérisque (1992), no. 211. MR 1225842
- Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279. MR 276239
- Joseph Lipman, Cohen-Macaulayness in graded algebras, Math. Res. Lett. 1 (1994), no. 2, 149–157. MR 1266753, DOI 10.4310/MRL.1994.v1.n2.a2
- Linquan Ma and Karl Schwede, Perfectoid multiplier/test ideals in regular rings and bounds on symbolic powers, Invent. Math. 214 (2018), no. 2, 913–955. MR 3867632, DOI 10.1007/s00222-018-0813-1
- Linquan Ma and Karl Schwede, Singularities in mixed characteristic via perfectoid big Cohen-Macaulay algebras, Duke Math. J. 170 (2021), no. 13, 2815–2890. MR 4312190, DOI 10.1215/00127094-2020-0082
- Fabrice Orgogozo, Exposé IV. Le théorème de Cohen-Gabber, Astérisque 363-364 (2014), 51–68 (French). Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents. MR 3329773
- J. B. Sancho de Salas, Blowing-up morphisms with Cohen-Macaulay associated graded rings, Géométrie algébrique et applications, I (La Rábida, 1984) Travaux en Cours, vol. 22, Hermann, Paris, 1987, pp. 201–209. MR 907914
- Kenta Sato and Shunsuke Takagi, General hyperplane sections of threefolds in positive characteristic, J. Inst. Math. Jussieu 19 (2020), no. 2, 647–661. MR 4079156, DOI 10.1017/s1474748018000166
- Karl Schwede and Karen E. Smith, Globally $F$-regular and log Fano varieties, Adv. Math. 224 (2010), no. 3, 863–894. MR 2628797, DOI 10.1016/j.aim.2009.12.020
- V. V. Shokurov, Three-dimensional log perestroikas, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 1, 105–203 (Russian); English transl., Russian Acad. Sci. Izv. Math. 40 (1993), no. 1, 95–202. MR 1162635, DOI 10.1070/IM1993v040n01ABEH001862
- K. E. Smith, Tight closure of parameter ideals, Invent. Math. 115 (1994), no. 1, 41–60. MR 1248078, DOI 10.1007/BF01231753
- Karen E. Smith, $F$-rational rings have rational singularities, Amer. J. Math. 119 (1997), no. 1, 159–180. MR 1428062
- The Stacks Project authors, Stacks Project.
- Shunsuke Takagi, An interpretation of multiplier ideals via tight closure, J. Algebraic Geom. 13 (2004), no. 2, 393–415. MR 2047704, DOI 10.1090/S1056-3911-03-00366-7
- Shunsuke Takagi, A characteristic $p$ analogue of plt singularities and adjoint ideals, Math. Z. 259 (2008), no. 2, 321–341. MR 2390084, DOI 10.1007/s00209-007-0227-z
- Shunsuke Takagi, Adjoint ideals along closed subvarieties of higher codimension, J. Reine Angew. Math. 641 (2010), 145–162. MR 2643928, DOI 10.1515/CRELLE.2010.031
- Shunsuke Takagi, Adjoint ideals and a correspondence between log canonicity and $F$-purity, Algebra Number Theory 7 (2013), no. 4, 917–942. MR 3095231, DOI 10.2140/ant.2013.7.917
- T. Takamatsu and S. Yoshikawa, Minimal model program for semi-stable threefolds in mixed characteristic, arXiv:2012.07324, 2020.
- Hiromu Tanaka, Minimal model program for excellent surfaces, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 1, 345–376 (English, with English and French summaries). MR 3795482
Additional Information
Linquan Ma
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
MR Author ID:
1050700
Email:
ma326@purdue.edu
Karl Schwede
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
MR Author ID:
773868
Email:
schwede@math.utah.edu
Kevin Tucker
Affiliation:
Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois 60607
MR Author ID:
888804
ORCID:
0000-0003-1255-2665
Email:
kftucker@uic.edu
Joe Waldron
Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544
MR Author ID:
1213375
Email:
jaw8@princeton.edu
Jakub Witaszek
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
MR Author ID:
1110594
ORCID:
0000-0002-3728-4305
Email:
jakubw@umich.edu
Received by editor(s):
June 25, 2020
Published electronically:
May 5, 2022
Additional Notes:
The first author was supported by NSF Grant DMS #1901672, NSF FRG Grant #1952366, and a fellowship from the Sloan Foundation. The second author was supported by NSF CAREER Grant DMS #1252860/1501102, NSF FRG Grant #1952522, and NSF Grants #1801849 and #2101800. The third author was supported by NSF Grants DMS #1602070 and #1707661, and by a fellowship from the Sloan Foundation. The fourth author was supported by the Simons Foundation under Grant No. 850684. The fifth author was supported by the National Science Foundation under Grant No. DMS-1638352 at the Institute for Advanced Study in Princeton as well as by NSF Grant DMS-2101897. This material was partially based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2019 semester. The authors also worked on this while attending an AIM SQUARE in June 2019.
Article copyright:
© Copyright 2022
University Press, Inc.
