EXTENSION AND TORSION FUNCTORS WITH RESPECT TO SERRE CLASSES

Document Type : Original Manuscript

Authors

Department of Mathematics, Payame Noor University (PNU), P.O. Box 19395- 4697, Tehran, Iran.

Abstract

In this paper we generalize the Rigidity Theorem and Zero Divisor Conjecture for an arbitrary Serre subcategory of modules. For this purpose, for any regular
M-sequence x1; :::; xn with respect to S we prove that if TorR 2 ((x1;:::;x R n); M) 2 S, then
TorR i ((x1;:::;x R n); M) 2 S, for all i ≥ 1. Also we show that if Extn R+2((x1;:::;x R n); M) 2 S,
then Exti R((x1;:::;x R n); M) 2 S, for all integers i ≥ 0 (i ̸= n).

Keywords


 1. M. Asgharzadeh and M. Tousi, Cohen Macaulayness with respect to serre classes, Illinois J. Math., 53 (2009), 67–85.
 
2. M. Auslander, Modules over unramified regular local rings, Illinois J. Math., 5 (1961), 631–647.
 
3. K. Bahmanpour, A complex of modules and its applications to local cohomology and extension functors, Math. Scand., 117 (2015), 150–160.
 
4. W. Bruns and J. Herzog, Chen-Macaulay Rings, Cambridge University Press, Cambridge, 1998.
 
5. S. Lichtenbaum, On the vanishing of Tor in regular local rings, Illinois J. Math., 10 (1966), 220–236.
 
6. C. Peskin and L. Szpiro, Dimension projective finie et cohomologie locale, Inst. Hautes tudes Sci. Publ. Math., 42 (1973), 47–119.
 
7. J. Rotman, An Introduction to Homological Algebra, Springer, New York, 2009.