Geodesic
VANISHING OF (CO)HOMOLOGY OVER DEFORMATIONS OF COHEN-MACAULAY LOCAL RINGS OF MINIMAL MULTIPLICITY
Glasgow mathematical journal, Tome 61 (2019) no. 3, pp. 705-725

Voir la notice de l'article provenant de la source Cambridge University Press

Let R be a d-dimensional Cohen–Macaulay (CM) local ring of minimal multiplicity. Set S := R/(f), where f := f1,. . .,fc is an R-regular sequence. Suppose M and N are maximal CM S-modules. It is shown that if ExtSi(M, N) = 0 for some (d + c + 1) consecutive values of i ⩾ 2, then ExtSi(M, N) = 0 for all i ⩾ 1. Moreover, if this holds true, then either projdimR(M) or injdimR(N) is finite. In addition, a counterpart of this result for Tor-modules is provided. Furthermore, we give a number of necessary and sufficient conditions for a CM local ring of minimal multiplicity to be regular or Gorenstein. These conditions are based on vanishing of certain Exts or Tors involving homomorphic images of syzygy modules of the residue field. 
GHOSH, DIPANKAR; PUTHENPURAKAL, TONY J. VANISHING OF (CO)HOMOLOGY OVER DEFORMATIONS OF COHEN-MACAULAY LOCAL RINGS OF MINIMAL MULTIPLICITY. Glasgow mathematical journal, Tome 61 (2019) no. 3, pp. 705-725. doi: 10.1017/S0017089518000459
@article{10_1017_S0017089518000459,
     author = {GHOSH, DIPANKAR and PUTHENPURAKAL, TONY J.},
     title = {VANISHING {OF} {(CO)HOMOLOGY} {OVER} {DEFORMATIONS} {OF} {COHEN-MACAULAY} {LOCAL} {RINGS} {OF} {MINIMAL} {MULTIPLICITY}},
     journal = {Glasgow mathematical journal},
     pages = {705--725},
     year = {2019},
     volume = {61},
     number = {3},
     doi = {10.1017/S0017089518000459},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089518000459/}
}
TY  - JOUR
AU  - GHOSH, DIPANKAR
AU  - PUTHENPURAKAL, TONY J.
TI  - VANISHING OF (CO)HOMOLOGY OVER DEFORMATIONS OF COHEN-MACAULAY LOCAL RINGS OF MINIMAL MULTIPLICITY
JO  - Glasgow mathematical journal
PY  - 2019
SP  - 705
EP  - 725
VL  - 61
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089518000459/
DO  - 10.1017/S0017089518000459
ID  - 10_1017_S0017089518000459
ER  - 
%0 Journal Article
%A GHOSH, DIPANKAR
%A PUTHENPURAKAL, TONY J.
%T VANISHING OF (CO)HOMOLOGY OVER DEFORMATIONS OF COHEN-MACAULAY LOCAL RINGS OF MINIMAL MULTIPLICITY
%J Glasgow mathematical journal
%D 2019
%P 705-725
%V 61
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089518000459/
%R 10.1017/S0017089518000459
%F 10_1017_S0017089518000459

[1] Abhyankar, S. S., Local rings of high embedding dimension, Amer. J. Math. 89 (1967), 1073–1077. Google Scholar | DOI

[2] Auslander, M., Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631–647. Google Scholar | DOI

[3] Auslander, M. and Buchweitz, R.-O., The homological theory of maximal Cohen-Macaulay approximations, Mém. Soc. Math. France 38 (1989), 5–37. Google Scholar | DOI

[4] Avramov, L. L., Modules with extremal resolutions, Math. Res. Lett. 3 (1996), 319–328. Google Scholar | DOI

[5] Avramov, L. L., Infinite free resolutions, Six lectures on commutative algebra, Bellaterra 1996, Progr. Math. 166, (Birkhäuser, Basel, 1998), 1–118. Google Scholar

[6] Avramov, L. L. and Buchweitz, R.-O.. Support varieties and cohomology over complete intersections, Invent. Math. 142 (2000), 285–318. Google Scholar | DOI

[7] Avramov, L. L., Buchweitz, R.-O. and Şega, L. M., Extensions of a dualizing complex by its ring: Commutative versions of a conjecture of Tachikawa, J. Pure Appl. Algebra 201 (2005), 218–239. Google Scholar | DOI

[8] Brennan, J. P., Herzog, J. and Ulrich, B., Maximally generated Cohen-Macaulay modules, Math. Scand. 61 (1987), 181–203. Google Scholar | DOI

[9] Bruns, W. and Herzog, J., Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics 39, Revised Edition, (Cambridge University Press, Cambridge, 1998). Google Scholar | DOI

[10] Celikbas, O., Dao, H. and Takahashi, R., Modules that detect finite homological dimensions, Kyoto J. Math. 54 (2014), 295–310. Google Scholar | DOI

[11] Dutta, S. P., Syzygies and homological conjectures, in Commutative algebra, Berkeley, CA, 1987, Math. Sci. Res. Inst. Publ., vol. 15 (Springer, New York, 1989) 139–156. Google Scholar | DOI

[12] Eisenbud, D., Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics vol. 150 (Springer-Verlag, New York, 1995). Google Scholar

[13] Ghosh, D., Some criteria for regular and Gorenstein local rings via syzygy modules, J. Algebra Appl. Available at: . Google Scholar | DOI

[14] Ghosh, D., Gupta, A. and Puthenpurakal, T. J., Characterizations of regular local rings via syzygy modules of the residue field, J. Commut. Algebra. Available at: https://projecteuclid.org/euclid.jca/1491379239. Google Scholar

[15] Gulliksen, T. H., A change of ring theorem with applications to Poincaré series and intersection multiplicity, Math. Scand. 34 (1974), 167–183. Google Scholar | DOI

[16] Gulliksen, T. H., On the deviations of a local ring, Math. Scand. 47 (1980), 5–20. Google Scholar | DOI

[17] Heitmann, R., A counterexample to the rigidity conjecture for rings, Bull. Amer. Math. Soc. 29 (1993), 94–97. Google Scholar | DOI

[18] Herzog, J., Ulrich, B. and Backelin, J., Linear maximal Cohen-Macaulay modules over strict complete intersections, J. Pure Appl. Algebra 71 (1991), 187–202. Google Scholar | DOI

[19] Huckaba, S. and Marley, T., Hilbert coefficients and the depths of associated graded rings, J. London Math. Soc. 56 (1997), 64–76. Google Scholar | DOI

[20] Huneke, C. and Jorgensen, D. A., Symmetry in the vanishing of Ext over Gorenstein rings, Math. Scand. 93 (2003), 161–184. Google Scholar | DOI

[21] Huneke, C. and Wiegand, R., Tensor products of modules, rigidity and local cohomology, Math. Scand. 81 (1997), 161–183. Google Scholar | DOI

[22] Jorgensen, D. A., A generalization of the Auslander-Buchsbaum formula, J. Pure Appl. Algebra 144 (1999), 145–155. Google Scholar | DOI

[23] Jorgensen, D. A., Complexity and Tor on a complete intersection, J. Algebra 211 (1999), 578–598. Google Scholar | DOI

[24] Jorgensen, D. A. and Leuschke, G. J., On the growth of the Betti sequence of the canonical module, Math. Z. 256 (2007), 647–659. Google Scholar | DOI

[25] Jorgensen, D. A. and Şega, L. M., Nonvanishing cohomology and classes of Gorenstein rings, Adv. Math. 188 (2004), 470–490. Google Scholar | DOI

[26] Lam, T. Y., A First Course in Noncommutative Rings, 2nd Edition (Springer-Verlag, New York, 2001). Google Scholar | DOI

[27] Lichtenbaum, S., On the vanishing of Tor in regular local rings, Illinois J. Math. 10 (1966), 220–226. Google Scholar | DOI

[28] Martsinkovsky, A., A remarkable property of the (co) syzygy modules of the residue field of a nonregular local ring, J. Pure Appl. Algebra 110 (1996), 9–13. Google Scholar | DOI

[29] Matsumura, H., Commutative Ring Theory (Cambridge University Press, Cambridge, 1986). Google Scholar

[30] Murthy, M. P., Modules over regular local rings, Illinois J. Math. 7 (1963), 558–565. Google Scholar | DOI

[31] Nasseh, S. and Takahashi, R., Local rings with quasi-decomposable maximal ideal, Math. Proc. Cambridge Philos. Soc. Available at: . Google Scholar | DOI

[32] Puthenpurakal, T. J., Hilbert-coefficients of a Cohen-Macaulay module, J. Algebra 264 (2003), 82–97. Google Scholar | DOI

[33] Rotman, J. J., An introduction to homological algebra, 2nd Edition, Universitext (Springer, New York, 2009). Google Scholar | DOI

[34] Takahashi, R., Syzygy modules with semidualizing or G-projective summands, J. Algebra 295 (2006), 179–194. Google Scholar | DOI

[35] Tate, J., Homology of noetherian rings and local rings, Illinois J. Math. 1 (1957), 14–27. Google Scholar | DOI

Cité par Sources :