Homological algebra in degree zero.
Słowa kluczowe:
Stabilization of additive functors, zeroth derived functor, torsion, Bass torsion, cotorsion, Tate homologyAbstrakt
The term "homological algebra in degree zero" refers, in the narrow sense of the word, to calculation of the zeroth derived functor of an additive functor between abelian categories. Most people do not realize that this is an interesting problem because the zeroth right derived functor of a Hom functor is the same Hom functor, and the same can be said about the zeroth left derived functor of the tensor product.
Bibliografia
M. Auslander. Coherent functors. In Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), pages 189–231. Springer, New York, 1966.
M. Auslander. Isolated Singularities and Existence of Almost Split Sequences. Representation Theory II 1178 (1984), 194–241.
M. Auslander and M. Bridger. Stable module theory. Memoirs of the American Mathematical Society, No. 94. American Mathematical Society, Providence, R.I., 1969.
H. Bass. Finitistic dimension and a homological generalization of semi-primary rings. Trans. Amer. Math. Soc., 95:466–488, 1960.
H. Cartan and S. Eilenberg. Homological algebra. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1999. With an appendix by David A. Buchsbaum, Reprint of the 1956 original.
S. Dean and J. Russell. The Auslander-Gruson-Jensen recollement. J. Algebra, 511:440–454, 2018.
L. Gruson and C. U. Jensen. Dimensions cohomologiques reli.ees aux foncteurs lim <.(..). In Paul Dubreil and Marie-Paule Malliavin Algebra Seminar, 33rd Year (Paris, 1980), volume 867 of Lecture Notes in Math., pages 234–294. Springer, Berlin-New York, 1981.
A. Martsinkovsky. 1-torsion of finite modules over semiperfect rings. J. Algebra, 324(10):2595–2607, 2010.
A. Martsinkovsky and J. Russell. Injective stabilization of additive functors. I. Preliminaries. J. Algebra 530:429-469, 2019.
A. Martsinkovsky and J. Russell. Injective stabilization of additive functors. II. (Co)torsion and the Auslander- Gruson-Jensen functor, arXiv:1701.00151, 22 pp. 2018.
A. Martsinkovsky and J. Russell. Injective stabilization of additive functors. III. Asymptotic stabilization of the tensor product, arXiv:1701.00268, 29 pp. 2018.
A. Martsinkovsky and J. R. Strooker. Linkage of modules. J. Algebra, 271(2):587–626, 2004.
