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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),...

Finiteness properties of local cohomology modules (an application of D-modules to Commutative Algebra).

Gennady Lyubeznik (1993)

Inventiones mathematicae

Fragments of almost ring theory

Lorenzo Ramero (2000)

Rendiconti del Seminario Matematico della Università di Padova

Frobenius et cohomologie locale

Jean-François Boutot (1974/1975)

Séminaire Bourbaki

Gaussian and Prüfer conditions in bi-amalgamated algebras

Najib Mahdou, Moutu Abdou Salam Moutui (2020)

Czechoslovak Mathematical Journal

Let $f : A \to B$ and $g : A \to C$ be two ring homomorphisms and let $J$ and $J^{'}$ be ideals of $B$ and $C$ , respectively, such that $f^{- 1} (J) = g^{- 1} (J^{'})$ . In this paper, we investigate the transfer of the notions of Gaussian and Prüfer rings to the bi-amalgamation of $A$ with $(B, C)$ along $(J, J^{'})$ with respect to $(f, g)$ (denoted by $A ⋈^{f, g} (J, J^{'})),$ introduced and studied by S. Kabbaj, K. Louartiti and M. Tamekkante in 2013. Our results recover well known results on amalgamations in C. A. Finocchiaro (2014) and generate new original examples of rings possessing these properties.

G-Dimension.

Siamak Yassemi (1995)

Mathematica Scandinavica

G-dimension over local homomorphisms with respect to a semi-dualizing complex

Wu Dejun (2014)

Czechoslovak Mathematical Journal

We study the G-dimension over local ring homomorphisms with respect to a semi-dualizing complex. Some results that track the behavior of Gorenstein properties over local ring homomorphisms under composition and decomposition are given. As an application, we characterize a dualizing complex for $R$ in terms of the finiteness of the G-dimension over local ring homomorphisms with respect to a semi-dualizing complex.

Generalized tilting modules over ring extension

Zhen Zhang (2019)

Czechoslovak Mathematical Journal

Let $Γ$ be a ring extension of $R$ . We show the left $Γ$ -module $U = Γ \otimes_{R} C$ with the endmorphism ring End $_{Γ} U = Δ$ is a generalized tilting module when $_{R} C$ is a generalized tilting module under some conditions.

Gorenstein Modules and Related Modules.

Hans-Bjorn Foxby (1972)

Mathematica Scandinavica

Homological dimension of algebras of invariants.

V.L. Popov (1983)

Journal für die reine und angewandte Mathematik

Homological Dimension of Pullbacks.

Susana Scrivanti (1992)

Mathematica Scandinavica

Isomorphims between complexes with applications to the homological theory of modules.

Hans-Bjorn Foxby (1977)

Mathematica Scandinavica

$k$ -torsionless modules with finite Gorenstein dimension

Maryam Salimi, Elham Tavasoli, Siamak Yassemi (2012)

Czechoslovak Mathematical Journal

Let $R$ be a commutative Noetherian ring. It is shown that the finitely generated $R$ -module $M$ with finite Gorenstein dimension is reflexive if and only if $M_{𝔭}$ is reflexive for $𝔭 \in Spec (R)$ with $depth (R_{𝔭}) \leq 1$ , and ${G- dim}_{R_{𝔭}} (M_{𝔭}) \leq depth (R_{𝔭}) - 2$ for $𝔭 \in Spec (R)$ with $depth (R_{𝔭}) \geq 2$ . This gives a generalization of Serre and Samuel’s results on reflexive modules over a regular local ring and a generalization of a recent result due to Belshoff. In addition, for $n \geq 2$ we give a characterization of $n$ -Gorenstein rings via Gorenstein dimension of the dual of modules. Finally it is shown...

Klassen von Moduln über Dedekind-Ringen und Satz von Stein-Serre

Martin Huber (1976)

Commentarii mathematici Helvetici

Length Formulas for the Local Cohomology of Exterior Powers.

Winfried Bruns, Udo Vetter (1986)

Mathematische Zeitschrift

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