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Register a new userReduction techniques of singular equivalences
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Authors
Yongyun Qin
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It is shown that a singular equivalence induced by tensoring with a suitable complex of bimodules defines a singular equivalence of Morita type with level, in the sense of Wang. This result is applied to homological ideals and idempotents to produce new reduction techniques for testing the properties of syzygy-finite and injectives generation of finite dimensional algebras over a field.
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Auslander-Reiten conjecture, which says that an Artin algebra does not have any non-projective generator with vanishing self-extensions in all positive degrees, is shown to be invariant under certain singular equivalences induced by adjoint pairs, which occur often in matrix algebras, recollements and change of rings. Accordingly, several reduction methods are established to study this conjecture.
Let $Lambda$ be a finite-dimensional algebra over a fixed algebraically closed field $mathbf{k}$ of arbitrary characteristic, and let $V$ be a finitely generated $Lambda$-module. It follows from results previously obtained by F.M. Bleher and the third author that $V$ has a well-defined versal deformation ring $R(Lambda, V)$, which is a complete local commutative Noetherian $mathbf{k}$-algebra with residue field $mathbf{k}$. The third author also proved that if $Lambda$ is a Gorenstein $mathbf{k}$-algebra and $V$ is a Cohen-Macaulay $Lambda$-module whose stable endomorphism ring is isomorphic to $mathbf{k}$, then $R(Lambda, V)$ is universal. In this article we prove that the isomorphism class of a versal deformation ring is preserved under singular equivalence of Morita type between Gorenstein $mathbf{k}$-algebras.
Let $mathbf{k}$ be a field of arbitrary characteristic, let $Lambda$ be a Gorenstein $mathbf{k}$-algebra, and let $V$ be an indecomposable finitely generated non-projective Gorenstein-projective left $Lambda$-module whose stable endomorphism ring is isomorphic to $mathbf{k}$. In this article, we prove that the universal deformation rings $R(Lambda,V)$ and $R(Lambda,Omega_Lambda V)$ are isomorphic, where $Omega_Lambda V$ denotes the first syzygy of $V$ as a left $Lambda$-module. We also prove the following result. Assume that $Gamma$ is another Gorenstein $mathbf{k}$-algebra such that there exists $ell geq 0$ and a pair of bimodules $({_Gamma}X_Lambda, {_Lambda}Y_Gamma)$ that induces a singular equivalence of Morita type with level $ell$ (as introduced by Z. Wang). Then the left $Gamma$-module $Xotimes_Lambda V$ is also Gorenstein-projective and the universal deformation rings $R(Gamma, Xotimes_Lambda V)$ and $R(Lambda, V)$ are isomorphic.
We apply the Auslander-Buchweitz approximation theory to show that the Iyama and Yoshinos subfactor triangulated category can be realized as a triangulated quotient. Applications of this realization go in three directions. Firstly, we recover both a result of Iyama and Yang and a result of the third author. Secondly, we extend the classical Buchweitzs triangle equivalence from Iwanaga-Gorenstein rings to Noetherian rings. Finally, we obtain the converse of Buchweitzs triangle equivalence and a result of Beligiannis, and give characterizations for Iwanaga-Gorenstein rings and Gorenstein algebras
It is well-known that derived equivalences preserve tensor products and trivial extensions. We disprove both constructions for stable equivalences of Morita type.
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