![]() ![]() The sense in which more general symmetric algebras are $0$-Calabi-Yau is that the category $K^b(P_A)$ of perfect complexes (i.e., those isomorphic to bounded complexes of finitely generated projective modules) is $0$-Calabi-Yau. So the only examples are the semisimple symmetric $k$-algebras, which are precisely the separable $k$-algebras. Unless it is semisimple, a symmetric algebra has infinite global dimension, and the derived category $D^b(A)$ is not Calabi-Yau. Algebras with this stronger property are called "symmetric".Įdit: Actually, I answered in a hurry and misread exactly what you asked, so let me clarify. In order that $D(A)\otimes -$ is isomorphic to the identity functor (as functors from left $A$-modules to left $A$-modules) you need not only that $D(A)\cong A$ as left $A$-modules, but that they are isomorphic as $A$-bimodules. Almost correct, except that "Frobenius" is not sufficient.
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