Invariants of linkage of modules

Authors

  • Tony J. Puthenpurakal

DOI:

https://doi.org/10.7146/math.scand.a-125992

Abstract

Let $(A,\mathfrak{m})$ be a Gorenstein local ring and let $M$, $N$ be two Cohen-Macaulay $A$-modules with $M$ linked to $N$ via a Gorenstein ideal $\mathfrak{q}$. Let $L$ be another finitely generated $A$-module. We show that $\mathrm{Ext}^i_A(L,M) = 0 $ for all $i \gg 0$ if and only if $\mathrm{Tor}^A_i(L,N) = 0$ for all $i \gg 0$. If $D$ is a Cohen-Macaulay module then we show that $\mathrm{Ext}^i_A(M, D) = 0 $ for all $i \gg 0$ if and only if $\mathrm{Ext}^i_A(D^\dagger , N) = 0$ for all $i \gg 0$, where $D^\dagger = \mathrm{Ext}^r_A(D,A)$ and $r = \mathrm{codim}(D)$. As a consequence we get that $\mathrm{Ext}^i_A(M, M) = 0 $ for all $i \gg 0$ if and only if $\mathrm{Ext}^i_A(N, N) = 0$ for all $i \gg 0$. We also show that $\mathrm{End}_A(M)/\mathrm{rad}\,\mathrm{End}_A(M) \cong (\mathrm{End}_A(N)/\mathrm{rad}\,\mathrm{End}_A(N))^{\mathrm{op}}$. We also give a negative answer to a question of Martsinkovsky and Strooker.

References

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

Avramov, L. L., Modules of finite virtual projective dimension, Invent. math textbf 96 (1989), no. 1, 71–101. https://doi.org/10.1007/BF01393971

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

Avramov, L. L., and Buchweitz, R-O, Support varieties and cohomology over complete intersections, Invent. Math. 142 (2000), no. 2, 285–318. https://doi.org/10.1007/s002220000090

Bruns, W., and Herzog, J., Cohen-Macaulay Rings, revised edition, Cambridge Stud. Adv. Math., vol. 39, Cambridge University Press, Cambridge, (1993).

Decker, W., Greuel, G.-M., Pfister, G., and Schönemann, H., newblock textsc Singular 4-0-2 — A computer algebra system for polynomial computations. newblock http://www.singular.uni-kl.de (2015).

Eisenbud, D., Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. textbf 260 (1980), no. 1, 35–64. https://doi.org/10.2307/1999875

Huneke, C., Linkage and the Koszul homology of ideals, Amer. J. Math. 104 (1982), no. 5, 1043–1062. https://doi.org/10.2307/2374083

Huneke, C., Numerical invariants of liaison classes, Invent. Math. 75 (1984), no. 2, 301–325. https://doi.org/10.1007/BF01388567

Huneke, C., Hyman Bass and ubiquity: Gorenstein rings, Algebra, K-theory, groups, and education (New York, 1997), 55–78, Contemp. Math., 243, Amer. Math. Soc., Providence, RI, 1999, arXiv: math/0209199. https://doi.org/10.1090/conm/243/03686

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

Matsumura, H., Commutative ring theory, Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge, 1989.

Peskine, C. and Szpiro, L., Liaison des variétés algébriques. I, Invent. Math. 26 (1974) 271–302. https://doi.org/10.1007/BF01425554

Published

2021-08-31

How to Cite

Puthenpurakal, T. J. . (2021). Invariants of linkage of modules: Array. MATHEMATICA SCANDINAVICA, 127(2), 223–242. https://doi.org/10.7146/math.scand.a-125992

Issue

Section

Articles