Abstract
Let \(\mathcal {C}\) be a triangulated category and \(\mathcal {E}\) a proper class of triangles. We show that Tate cohomology in triangulated category is balanced, i.e. there is an isomorphism \(\widehat {\mathcal {E}\text {xt}}^{i}_{\mathcal {P}}(A, B)\cong \widehat {\mathcal {E}\text {xt}}^{i}_{\mathcal {I}}(A, B)\) for any integer \(i\in \mathbb {Z}\), where the first cohomology group is computed by complete \(\mathcal {E}\)-projective resolution for \(A\in \mathcal {C}\) and the second one is computed by complete \(\mathcal {E}\)-injective coresolution for \(B\in \mathcal {C}\). This improves the theorem proposed by J. Asadollahi and Sh. Salarian [J. Algebra 299, 480-502 (2006)].
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Ren, W., Liu, Z. Balance of Tate cohomology in triangulated categories. Appl Categor Struct 23, 819–828 (2015). https://doi.org/10.1007/s10485-014-9381-8
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DOI: https://doi.org/10.1007/s10485-014-9381-8