A function on the set of isomorphism classes in the stable category of maximal Cohen-Macaulay modules over a Gorenstein ring: with applications to liaison theory


  • Tony J. Puthenpurakal




Let $(A,\mathfrak{m})$ be a Gorenstein local ring of dimension $d \geq 1$. Let $\operatorname{\underline{CM}}(A)$ be the stable category of maximal Cohen-Macauley $A$-modules and let $\operatorname{\underline{ICM}}(A)$ denote the set of isomorphism classes in $\operatorname{\underline{CM}}(A)$. We define a function $\xi \colon \operatorname{\underline{ICM}}(A) \to \mathbb{Z}$ which behaves well with respect to exact triangles in $\operatorname{\underline{CM}}(A)$. We then apply this to (Gorenstein) liaison theory. We prove that if $\dim A \geq 2$ and $A$ is not regular then the even liaison classes of $\{\,\mathfrak{m}^n \mid n\geq 1 \,\}$ is an infinite set. We also prove that if $A$ is Henselian with finite representation type with $A/\mathfrak{m}$ uncountable then for each $m \geq 1$ the set $\mathcal {C}_m = \{\, I \mid I \text { is a codim $2$ CM-ideal with } e_0(A/I) \leq m \,\}$ is contained in finitely many even liaison classes $L_1,\dots ,L_r$ (here $r$ may depend on $m$).


Auslander, M. and Buchweitz, R.-O., The homological theory of maximal Cohen-Macaulay approximations, Mém. Soc. Math. France (N.S.), Colloque en l'honneur de Pierre Samuel (Orsay, 1987) (1989), no. 38, 5–37.

Bruns, W. and Herzog, J., Cohen-Macaulay rings, revised ed., Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1998.

Buchweitz, R.-O., Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings, preprint Univ. Hannover, http://hdl.handle.net/1807/16682, 1986.

Herzog, J., Ringe mit nur endlich vielen Isomorphieklassen von maximalen, unzerlegbaren Cohen-Macaulay-Moduln, Math. Ann. 233 (1978), no. 1, 21–34. https://doi.org/10.1007/BF01351494

Herzog, J. and Kühl, M., Maximal Cohen-Macaulay modules over Gorenstein rings and Bourbaki-sequences, Commutative algebra and combinatorics (Kyoto, 1985), Adv. Stud. Pure Math., vol. 11, North-Holland, Amsterdam, 1987, pp. 65--92.

Kleppe, J. O., Migliore, J. C., Miró-Roig, R., Nagel, U., and Peterson, C., Gorenstein liaison, complete intersection liaison invariants and unobstructedness, Mem. Amer. Math. Soc. 154 (2001), no. 732. https://doi.org/10.1090/memo/0732

Knörrer, H., Cohen-Macaulay modules on hypersurface singularities. I, Invent. Math. 88 (1987), no. 1, 153–164. https://doi.org/10.1007/BF01405095

Martsinkovsky, A. and Strooker, J. R., Linkage of modules, J. Algebra 271 (2004), no. 2, 587–626. https://doi.org/10.1016/j.jalgebra.2003.07.020

Matsumura, H., Commutative ring theory, second ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989.

Polini, C. and Ulrich, B., Linkage and reduction numbers, Math. Ann. 310 (1998), no. 4, 631–651. https://doi.org/10.1007/s002080050163

Puthenpurakal, T. J., Hilbert-coefficients of a Cohen-Macaulay module, J. Algebra 264 (2003), no. 1, 82–97. https://doi.org/10.1016/S0021-8693(03)00231-X

Puthenpurakal, T. J., The Hilbert function of a maximal Cohen-Macaulay module, Math. Z. 251 (2005), no. 3, 551–573. https://doi.org/10.1007/s00209-005-0822-9

Takahashi, R., Direct summands of syzygy modules of the residue class field, Nagoya Math. J. 189 (2008), 1–25.

Wang, H.-J., Links of symbolic powers of prime ideals, Math. Z. 256 (2007), no. 4, 749–756. https://doi.org/10.1007/s00209-006-0099-7




How to Cite

Puthenpurakal, T. J. (2017). A function on the set of isomorphism classes in the stable category of maximal Cohen-Macaulay modules over a Gorenstein ring: with applications to liaison theory. MATHEMATICA SCANDINAVICA, 120(2), 161–180. https://doi.org/10.7146/math.scand.a-25728