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Intertwining numbers; the n -rowed shapes

Hyoung J. Ko, Kyoung J. Lee (2007)

Czechoslovak Mathematical Journal

A fairly old problem in modular representation theory is to determine the vanishing behavior of the H o m groups and higher E x t groups of Weyl modules and to compute the dimension of the / ( p ) -vector space H o m A ¯ r ( K ¯ λ , K ¯ μ ) for any partitions λ , μ of r , which is the intertwining number. K. Akin, D. A. Buchsbaum, and D. Flores solved this problem in the cases of partitions of length two and three. In this paper, we describe the vanishing behavior of the groups H o m A ¯ r ( K ¯ λ , K ¯ μ ) and provide a new formula for the intertwining number for any...

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 Spec ( R ) with depth ( R 𝔭 ) 1 , and G- dim R 𝔭 ( M 𝔭 ) depth ( R 𝔭 ) - 2 for 𝔭 Spec ( R ) with depth ( R 𝔭 ) 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 2 we give a characterization of n -Gorenstein rings via Gorenstein dimension of the dual of modules. Finally it is shown...

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