Dualizing modules for orders and Artin algebras
Let \(R\) be a one-dimensional complete local Gorenstein domain, and let \(\Lambda,\Gamma\) be \(R\)-orders in semisimple algebras over the quotient field of \(R\). A faithfully balanced \((\Lambda,\Gamma)\)-bimodule \(\omega\) is said to be a Cohen-Macaulay module if it is self-orthogonal and of finite (left and right) injective dimension. Such a bimodule \(\omega\) is said to be a strong cotilting module over \(\Lambda\) if the category \({\mathcal I}^\infty(\Lambda):=\{M\in\Lambda\text{-mod}:\text{id}_\Lambda M<\infty\}\) coincides with the category \(\widehat{\text{add}}_\Lambda\omega\) of modules \(M\in\Lambda\text{-mod}\) which admit an exact sequence \[ 0\to G_n\to\cdots\to G_0\to M\to 0 \] with \(G_i\in\text{add}_\Lambda\omega\). The author provides a new criterion for strong Cohen-Macaulay modules. Namely, he shows that a Cohen-Macaulay module \(\omega\) is (left) strong if and only if \(\text{Ext}^i_\Gamma(S,\omega_\Gamma^*)\not=0\) for each simple \(\Gamma\)-module \(S\). A left and right strong Cohen-Macaulay module \(\omega\) is said to be a dualizing module. Taking \(\omega\) modulo a non-zero divisor in \(R\), one gets a dualizing bimodule over Artinian algebras. Using his criterion, the author obtains examples of dualizing modules which are not isomorphic to \(_\Lambda\Lambda_\Lambda\) or \(\Lambda^*\).
scientific article; zbMATH DE number 1783445