An excellent pure ring of prime characteristic has a big tight closure test element
Author:
Rodney Y. Sharp
Journal:
Trans. Amer. Math. Soc. 362 (2010), 54555481
MSC (2010):
Primary 13A35, 16S36, 13D45, 13E05, 13E10, 13H10; Secondary 13J10
Published electronically:
May 3, 2010
MathSciNet review:
2657687
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Abstract: In two recent papers, the author has developed a theory of graded annihilators of left modules over the Frobenius skew polynomial ring over a commutative Noetherian ring of prime characteristic , and has shown that this theory is relevant to the theory of test elements in tight closure theory. One result of that work was that, if is local and the module structure on the injective envelope of the simple module can be extended to a structure as a torsionfree left module over the Frobenius skew polynomial ring, then is pure and has a tight closure test element. One of the central results of this paper is the converse, namely that, if is pure, then has a structure as a torsionfree left module over the Frobenius skew polynomial ring; a corollary is that every pure local ring of prime characteristic, even if it is not excellent, has a tight closure test element. These results are then used, along with embedding theorems for modules over the Frobenius skew polynomial ring, to show that every excellent (not necessarily local) pure ring of characteristic must have a socalled `big' test element.
 1.
M. Blickle, The intersection homology module in finite characteristic, Ph.D. dissertation, University of Michigan, Ann Arbor, 2001.
 2.
M.
P. Brodmann and R.
Y. Sharp, Local cohomology: an algebraic introduction with
geometric applications, Cambridge Studies in Advanced Mathematics,
vol. 60, Cambridge University Press, Cambridge, 1998. MR 1613627
(99h:13020)
 3.
Richard
Fedder, 𝐹purity and rational
singularity, Trans. Amer. Math. Soc.
278 (1983), no. 2,
461–480. MR
701505 (84h:13031), http://dx.doi.org/10.1090/S00029947198307015050
 4.
Robin
Hartshorne and Robert
Speiser, Local cohomological dimension in characteristic
𝑝, Ann. of Math. (2) 105 (1977), no. 1,
45–79. MR
0441962 (56 #353)
 5.
Melvin
Hochster and Craig
Huneke, Tight closure, invariant theory, and
the BriançonSkoda theorem, J. Amer.
Math. Soc. 3 (1990), no. 1, 31–116. MR 1017784
(91g:13010), http://dx.doi.org/10.1090/S08940347199010177846
 6.
Melvin
Hochster and Craig
Huneke, 𝐹regularity, test elements,
and smooth base change, Trans. Amer. Math.
Soc. 346 (1994), no. 1, 1–62. MR 1273534
(95d:13007), http://dx.doi.org/10.1090/S0002994719941273534X
 7.
Melvin
Hochster and Joel
L. Roberts, Rings of invariants of reductive groups acting on
regular rings are CohenMacaulay, Advances in Math.
13 (1974), 115–175. MR 0347810
(50 #311)
 8.
Craig
Huneke, Tight closure and its applications, CBMS Regional
Conference Series in Mathematics, vol. 88, Published for the
Conference Board of the Mathematical Sciences, Washington, DC; by the
American Mathematical Society, Providence, RI, 1996. With an appendix by
Melvin Hochster. MR 1377268
(96m:13001)
 9.
Craig
L. Huneke and Rodney
Y. Sharp, Bass numbers of local cohomology
modules, Trans. Amer. Math. Soc.
339 (1993), no. 2,
765–779. MR 1124167
(93m:13008), http://dx.doi.org/10.1090/S00029947199311241676
 10.
Mordechai
Katzman, Parametertestideals of CohenMacaulay rings,
Compos. Math. 144 (2008), no. 4, 933–948. MR 2441251
(2009d:13030), http://dx.doi.org/10.1112/S0010437X07003417
 11.
Mordechai
Katzman and Rodney
Y. Sharp, Uniform behaviour of the Frobenius closures of ideals
generated by regular sequences, J. Algebra 295
(2006), no. 1, 231–246. MR 2188859
(2006i:13007), http://dx.doi.org/10.1016/j.jalgebra.2005.01.025
 12.
Gennady
Lyubeznik, 𝐹modules: applications to local cohomology and
𝐷modules in characteristic 𝑝>0, J. Reine Angew.
Math. 491 (1997), 65–130. MR 1476089
(99c:13005), http://dx.doi.org/10.1515/crll.1997.491.65
 13.
Gennady
Lyubeznik and Karen
E. Smith, On the commutation of the test ideal
with localization and completion, Trans. Amer.
Math. Soc. 353 (2001), no. 8, 3149–3180 (electronic). MR 1828602
(2002f:13010), http://dx.doi.org/10.1090/S0002994701026435
 14.
Hideyuki
Matsumura, Commutative ring theory, Cambridge Studies in
Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge,
1986. Translated from the Japanese by M. Reid. MR 879273
(88h:13001)
 15.
Rodney
Y. Sharp, Graded annihilators of modules over
the Frobenius skew polynomial ring, and tight closure, Trans. Amer. Math. Soc. 359 (2007), no. 9, 4237–4258 (electronic). MR 2309183
(2008b:13006), http://dx.doi.org/10.1090/S000299470704247X
 16.
Rodney
Y. Sharp, Graded annihilators and tight closure test ideals,
J. Algebra 322 (2009), no. 9, 3410–3426. MR 2567428
(2010k:13009), http://dx.doi.org/10.1016/j.jalgebra.2008.08.007
 17.
Rodney
Y. Sharp and Nicole
Nossem, Ideals in a perfect closure, linear
growth of primary decompositions, and tight closure, Trans. Amer. Math. Soc. 356 (2004), no. 9, 3687–3720 (electronic). MR 2055750
(2005a:13009), http://dx.doi.org/10.1090/S0002994704034208
 18.
D.
W. Sharpe and P.
Vámos, Injective modules, Cambridge University Press,
LondonNew York, 1972. Cambridge Tracts in Mathematics and Mathematical
Physics, No. 62. MR 0360706
(50 #13153)
 19.
Karen
E. Smith, 𝐹rational rings have rational
singularities, Amer. J. Math. 119 (1997), no. 1,
159–180. MR 1428062
(97k:13004)
 1.
 M. Blickle, The intersection homology module in finite characteristic, Ph.D. dissertation, University of Michigan, Ann Arbor, 2001.
 2.
 M. P. Brodmann and R. Y. Sharp, Local cohomology: An algebraic introduction with geometric applications, Cambridge Studies in Advanced Mathematics 60, Cambridge University Press, 1998. MR 1613627 (99h:13020)
 3.
 R. Fedder, purity and rational singularity, Transactions Amer. Math. Soc. 278 (1983) 461480. MR 701505 (84h:13031)
 4.
 R. Hartshorne and R. Speiser, Local cohomological dimension in characteristic , Annals of Math. 105 (1977) 4579. MR 0441962 (56:353)
 5.
 M. Hochster and C. Huneke, Tight closure, invariant theory and the BriançonSkoda Theorem, J. Amer. Math. Soc. 3 (1990) 31116. MR 1017784 (91g:13010)
 6.
 M. Hochster and C. Huneke, regularity, test elements, and smooth base change, Transactions Amer. Math. Soc. 346 (1994) 162. MR 1273534 (95d:13007)
 7.
 M. Hochster and J. L. Roberts, Rings of invariants of reductive groups acting on regular rings are CohenMacaulay, Advances in Math. 13 (1974) 115175. MR 0347810 (50:311)
 8.
 C. Huneke, Tight closure and its applications, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics 88, American Mathematical Society, Providence, 1996. MR 1377268 (96m:13001)
 9.
 C. Huneke and R. Y. Sharp, Bass numbers of local cohomology modules, Transactions Amer. Math. Soc. 339 (1993) 765779. MR 1124167 (93m:13008)
 10.
 M. Katzman, Parametertestideals of CohenMacaulay rings, Compositio Math. 144 (2008) 933948. MR 2441251 (2009d:13030)
 11.
 M. Katzman and R. Y. Sharp, Uniform behaviour of the Frobenius closures of ideals generated by regular sequences, J. Algebra 295 (2006) 231246. MR 2188859 (2006i:13007)
 12.
 G. Lyubeznik, modules: Applications to local cohomology and modules in characteristic , J. reine angew. Math. 491 (1997) 65130. MR 1476089 (99c:13005)
 13.
 G. Lyubeznik and K. E. Smith, On the commutation of the test ideal with localization and completion, Transactions Amer. Math. Soc. 353 (2001) 31493180. MR 1828602 (2002f:13010)
 14.
 H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, 1986. MR 879273 (88h:13001)
 15.
 R. Y. Sharp, Graded annihilators of modules over the Frobenius skew polynomial ring, and tight closure, Transactions Amer. Math. Soc. 359 (2007) 42374258. MR 2309183 (2008b:13006)
 16.
 R. Y. Sharp, Graded annihilators and tight closure test ideals, J. Algebra 322 (2009) 34103426. MR 2567428
 17.
 R. Y. Sharp and N. Nossem, Ideals in a perfect closure, linear growth of primary decompositions, and tight closure, Transactions Amer. Math. Soc. 356 (2004) 36873720. MR 2055750 (2005a:13009)
 18.
 D. W. Sharpe and P. Vámos, Injective modules, Cambridge Tracts in Mathematics and Mathematical Physics 62, Cambridge University Press, 1972. MR 0360706 (50:13153)
 19.
 K. E. Smith, rational rings have rational singularities, Amer. J. Math. 119 (1997) 159180. MR 1428062 (97k:13004)
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Additional Information
Rodney Y. Sharp
Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield, S3 7RH, United Kingdom
Email:
R.Y.Sharp@sheffield.ac.uk
DOI:
http://dx.doi.org/10.1090/S0002994710051664
PII:
S 00029947(10)051664
Keywords:
Commutative Noetherian ring,
prime characteristic,
Frobenius homomorphism,
tight closure,
test element,
$F$pure ring,
Frobenius skew polynomial ring
Received by editor(s):
December 18, 2008
Published electronically:
May 3, 2010
Article copyright:
© Copyright 2010
American Mathematical Society
