t-Structures and cotilting modules over commutative noetherian rings |
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Authors: | Lidia Angeleri Hügel Manuel Saorín |
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Institution: | 1. Università degli Studi di Verona, Strada Le Grazie 15-Ca’ Vignal 2, 37134?, Verona, Italy 2. Departamento de Matemáticas, Universidad de Murcia, 30100?, Espinardo, MU, Spain
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Abstract: | For a commutative noetherian ring \(R\) , we establish a bijection between the resolving subcategories consisting of finitely generated \(R\) -modules of finite projective dimension and the compactly generated t-structures in the unbounded derived category \(\mathcal {D}(R)\) that contain \(R1]\) in their heart. Under this bijection, the t-structures \((\mathcal U,\mathcal V)\) such that the aisle \(\mathcal U\) consists of objects with homology concentrated in degrees \(<n\) correspond to the \(n\) -cotilting classes in \({{\mathrm{Mod}\text {-}R}}\) . As a consequence of these results, we prove that the little finitistic dimension findim \(R\) of \(R\) equals an integer \(n\) if and only if the direct sum \(\bigoplus _{k=0}^n E_k(R)\) of the first \(n+1\) terms in a minimal injective coresolution \(0\rightarrow R\rightarrow E_0(R)\rightarrow E_1(R)\rightarrow \cdots \) of \(R\) is an injective cogenerator of \({{\mathrm{Mod}\text {-}R}}\) . |
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