Cohen-Macaulay homological dimensions

  • Parviz Sahandi
  • Tirdad Sharif
  • Siamak Yassemi

Abstract

We introduce new homological dimensions, namely the Cohen-Macaulay projective, injective and flat dimensions for homologically bounded complexes. Among other things we show that (a) these invariants characterize the Cohen-Macaulay property for local rings, (b) Cohen-Macaulay flat dimension fits between the Gorenstein flat dimension and the large restricted flat dimension, and (c) Cohen-Macaulay injective dimension fits between the Gorenstein injective dimension and the Chouinard invariant.

References

Auslander, M. and Bridger, M., Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969.

Auslander, M. and Buchsbaum, D. A., Homological dimension in local rings, Trans. Amer. Math. Soc. 85 (1957), 390–405. https://doi.org/10.2307/1992937

Avramov, L. L., Gasharov, V. N., and Peeva, I. V., Complete intersection dimension, Inst. Hautes Études Sci. Publ. Math. (1997), no. 86, 67–114.

Brodmann, M. P. and Sharp, R. Y., Local cohomology: An algebraic introduction with geometric applications, second ed., Cambridge Studies in Advanced Mathematics, vol. 136, Cambridge University Press, Cambridge, 2013.

Bruns, W. and Herzog, J., Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993.

Chouinard, II, L. G., On finite weak and injective dimension, Proc. Amer. Math. Soc. 60 (1976), 57–60. https://doi.org/10.2307/2041111

Christensen, L. W., Gorenstein dimensions, Lecture Notes in Mathematics, vol. 1747, Springer-Verlag, Berlin, 2000. https://doi.org/10.1007/BFb0103980

Christensen, L. W., Foxby, H.-B., and Frankild, A., Restricted homological dimensions and Cohen-Macaulayness, J. Algebra 251 (2002), no. 1, 479–502. https://doi.org/10.1006/jabr.2001.9115

Christensen, L. W., Frankild, A., and Holm, H., On Gorenstein projective, injective and flat dimensions—a functorial description with applications, J. Algebra 302 (2006), no. 1, 231–279. https://doi.org/10.1016/j.jalgebra.2005.12.007

Christensen, L. W. and Holm, H., Ascent properties of Auslander categories, Canad. J. Math. 61 (2009), no. 1, 76–108. https://doi.org/10.4153/CJM-2009-004-x

Christensen, L. W. and Sather-Wagstaff, S., Transfer of Gorenstein dimensions along ring homomorphisms, J. Pure Appl. Algebra 214 (2010), no. 6, 982–989. https://doi.org/10.1016/j.jpaa.2009.09.007

Enochs, E. E. and Jenda, O. M. G., Gorenstein injective and projective modules, Math. Z. 220 (1995), no. 4, 611–633. https://doi.org/10.1007/BF02572634

Enochs, E. E., Jenda, O. M. G., and Torrecillas, B., Gorenstein flat modules, Nanjing Daxue Xuebao Shuxue Bannian Kan 10 (1993), no. 1, 1–9.

Ferrand, D. and Raynaud, M., Fibres formelles d'un anneau local noethérien, Ann. Sci. École Norm. Sup. (4) 3 (1970), 295–311.

Foxby, H.-B. and Frankild, A. J., Cyclic modules of finite Gorenstein injective dimension and Gorenstein rings, Illinois J. Math. 51 (2007), no. 1, 67–82.

Gerko, A. A., On homological dimensions, Mat. Sb. 192 (2001), no. 8, 79–94. https://doi.org/10.1070/SM2001v192n08ABEH000587

Holm, H. and Jørgensen, P., Cohen-Macaulay homological dimensions, Rend. Semin. Mat. Univ. Padova 117 (2007), 87–112.

Iyengar, S., Depth for complexes, and intersection theorems, Math. Z. 230 (1999), no. 3, 545–567. https://doi.org/10.1007/PL00004705

Iyengar, S. and Sather-Wagstaff, S., G-dimension over local homomorphisms. Applications to the Frobenius endomorphism, Illinois J. Math. 48 (2004), no. 1, 241–272.

Sahandi, P., Sharif, T., and Yassemi, S., Homological flat dimensions, eprint arXiv:0709.4078 [math.AC], 2007.

Sahandi, P., Sharif, T., and Yassemi, S., Depth formula via complete intersection flat dimension, Comm. Algebra 39 (2011), no. 11, 4002–4013. https://doi.org/10.1080/00927872.2010.514875

Sahandi, P., Sharif, T., and Yassemi, S., Complete intersection flat dimension and the intersection theorem, Algebra Colloq. 19 (2012), no. Special Issue 1, 1161–1166. https://doi.org/10.1142/S1005386712000934

Sather-Wagstaff, S., Complete intersection dimensions and Foxby classes, J. Pure Appl. Algebra 212 (2008), no. 12, 2594–2611. https://doi.org/10.1016/j.jpaa.2008.04.005

Serre, J.-P., Sur la dimension homologique des anneaux et des modules noethériens, in “Proceedings of the international symposium on algebraic number theory, Tokyo & Nikko, 1955”, Science Council of Japan, Tokyo, 1956, pp. 175–189.

Sharif, T. and Yassemi, S., Depth formulas, restricted tor-dimension under base change, Rocky Mountain J. Math. 34 (2004), no. 3, 1131–1146. https://doi.org/10.1216/rmjm/1181069847

Yassemi, S., G-dimension, Math. Scand. 77 (1995), no. 2, 161–174. https://doi.org/10.7146/math.scand.a-12557
Published
2020-05-06
How to Cite
Sahandi, P., Sharif, T., & Yassemi, S. (2020). Cohen-Macaulay homological dimensions. MATHEMATICA SCANDINAVICA, 126(2), 189-208. https://doi.org/10.7146/math.scand.a-119382
Section
Articles