Finiteness aspects of Gorenstein homological dimensions

Samir Bouchiba. "Finiteness aspects of Gorenstein homological dimensions." Colloquium Mathematicae 131.2 (2013): 171-193. <http://eudml.org/doc/283666>.

@article{SamirBouchiba2013,
abstract = {We present an alternative way of measuring the Gorenstein projective (resp., injective) dimension of modules via a new type of complete projective (resp., injective) resolutions. As an application, we easily recover well known theorems such as the Auslander-Bridger formula. Our approach allows us to relate the Gorenstein global dimension of a ring R to the cohomological invariants silp(R) and spli(R) introduced by Gedrich and Gruenberg by proving that leftG-gldim(R) = maxleftsilp(R), leftspli(R), recovering a recent theorem of [I. Emmanouil, J. Algebra 372 (2012), 376-396]. Moreover, this formula permits to recover the main theorem of [D. Bennis and N. Mahdou, Proc. Amer. Math. Soc. 138 (2010), 461-465]. Furthermore, we prove that, in the setting of a left and right Noetherian ring, the Gorenstein global dimension is left-right symmetric, generalizing a theorem of Enochs and Jenda. Finally, using recent work of I. Emmanouil and O. Talelli, we compute the Gorenstein global dimension for various types of rings such as commutative ℵ₀-Noetherian rings and group rings.},
author = {Samir Bouchiba},
journal = {Colloquium Mathematicae},
keywords = {Gorenstein projective dimension; Gorenstein injective dimension; Gorenstein global dimension; Gorenstein weak global dimension},
language = {eng},
number = {2},
pages = {171-193},
title = {Finiteness aspects of Gorenstein homological dimensions},
url = {http://eudml.org/doc/283666},
volume = {131},
year = {2013},
}

TY - JOUR
AU - Samir Bouchiba
TI - Finiteness aspects of Gorenstein homological dimensions
JO - Colloquium Mathematicae
PY - 2013
VL - 131
IS - 2
SP - 171
EP - 193
AB - We present an alternative way of measuring the Gorenstein projective (resp., injective) dimension of modules via a new type of complete projective (resp., injective) resolutions. As an application, we easily recover well known theorems such as the Auslander-Bridger formula. Our approach allows us to relate the Gorenstein global dimension of a ring R to the cohomological invariants silp(R) and spli(R) introduced by Gedrich and Gruenberg by proving that leftG-gldim(R) = maxleftsilp(R), leftspli(R), recovering a recent theorem of [I. Emmanouil, J. Algebra 372 (2012), 376-396]. Moreover, this formula permits to recover the main theorem of [D. Bennis and N. Mahdou, Proc. Amer. Math. Soc. 138 (2010), 461-465]. Furthermore, we prove that, in the setting of a left and right Noetherian ring, the Gorenstein global dimension is left-right symmetric, generalizing a theorem of Enochs and Jenda. Finally, using recent work of I. Emmanouil and O. Talelli, we compute the Gorenstein global dimension for various types of rings such as commutative ℵ₀-Noetherian rings and group rings.
LA - eng
KW - Gorenstein projective dimension; Gorenstein injective dimension; Gorenstein global dimension; Gorenstein weak global dimension
UR - http://eudml.org/doc/283666
ER -