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Register a new userOn Universal Deformation Rings for Gorenstein Algebras
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Jose A. Velez-Marulanda
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Let $mathbf{k}$ be an algebraically closed field, and let $Lambda$ be a finite dimensional $mathbf{k}$-algebra. We prove that if $Lambda$ is a Gorenstein algebra, then every finitely generated Cohen-Macaulay $Lambda$-module $V$ whose stable endomorphism ring is isomorphic to $mathbf{k}$ has a universal deformation ring $R(Lambda,V)$, which is a complete local commutative Noetherian $mathbf{k}$-algebra with residue field $mathbf{k}$, and which is also stable under taking syzygies. We investigate a particular non-self-injective Gorenstein algebra $Lambda_0$, which is of infinite global dimension and which has exactly three isomorphism classes of finitely generated indecomposable Cohen-Macaulay $Lambda_0$-modules $V$ whose stable endomorphism ring is isomorphic to $mathbf{k}$. We prove that in this situation, $R(Lambda_0,V)$ is isomorphic either to $mathbf{k}$ or to $mathbf{k}[[t]]/(t^2)$.
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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 finite dimensional $mathbf{k}$-algebra, and let $V$ be a finitely generated $Lambda$-module. F. M. Bleher and the third author previously proved that $V$ has a well-defined versal deformation ring $R(Lambda,V)$. If the stable endomorphism ring of $V$ is isomorphic to $mathbf{k}$, they also proved under the additional assumption that $Lambda$ is self-injective that $R(Lambda,V)$ is universal. In this paper, we prove instead that if $Lambda$ is arbitrary but $V$ is Gorenstein-projective then $R(Lambda,V)$ is also universal when the stable endomorphism ring of $V$ is isomorphic to $mathbf{k}$. Moreover, we show that singular equivalences of Morita type (as introduced by X. W. Chen and L. G. Sun) preserve the isomorphism classes of versal deformation rings of finitely generated Gorenstein-projective modules over Gorenstein algebras. We also provide examples. In particular, if $Lambda$ is a monomial algebra in which there is no overlap (as introduced by X. W. Chen, D. Shen and G. Zhou) we prove that every finitely generated indecomposable Gorenstein-projective $Lambda$-module has a universal deformation ring that is isomorphic to either $mathbf{k}$ or to $mathbf{k}[![t]!]/(t^2)$.
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.
Let $mathbf{k}$ be field of arbitrary characteristic and let $Lambda$ be a finite dimensional $mathbf{k}$-algebra. From results previously obtained by F.M Bleher and the author, it follows that if $V^bullet$ is an object of the bounded derived category $mathcal{D}^b(Lambdatextup{-mod})$ of $Lambda$, then $V^bullet$ has a well-defined versal deformation ring $R(Lambda, V^bullet)$, which is complete local commutative Noetherian $mathbf{k}$-algebra with residue field $mathbf{k}$, and which is universal provided that $textup{Hom}_{mathcal{D}^b(Lambdatextup{-mod})}(V^bullet, V^bullet)=mathbf{k}$. Let $mathcal{D}_textup{sg}(Lambdatextup{-mod})$ denote the singularity category of $Lambda$ and assume that $V^bullet$ is a bounded complex whose terms are all finitely generated Gorenstein projective left $Lambda$-modules. In this article we prove that if $textup{Hom}_{mathcal{D}_textup{sg}(Lambdatextup{-mod})}(V^bullet, V^bullet)=mathbf{k}$, then the versal deformation ring $R(Lambda, V^bullet)$ is universal. We also prove that certain singular equivalences of Morita type (as introduced by X. W. Chen and L. G. Sun) preserve the isomorphism class of versal deformation rings of bounded complexes whose terms are finitely generated Gorenstein projective $Lambda$-modules.
Let $mathbf{k}$ be an algebraically closed field of arbitrary characteristic, let $Lambda$ be a finite dimensional $mathbf{k}$-algebra and let $V$ be a $Lambda$-module with stable endomorphism ring isomorphic to $mathbf{k}$. If $Lambda$ is self-injective, then $V$ has a universal deformation ring $R(Lambda,V)$, which is a complete local commutative Noetherian $mathbf{k}$-algebra with residue field $mathbf{k}$. Moreover, if $Lambda$ is further a Frobenius $mathbf{k}$-algebra, then $R(Lambda,V)$ is stable under syzygies. We use these facts to determine the universal deformation rings of string $Lambda_{m,N}$-modules whose corresponding stable endomorphism ring is isomorphic to $mathbf{k}$, and which lie either in a connected component of the stable Auslander-Reiten quiver of $Lambda_{m,N}$ containing a module with endomorphism ring isomorphic to $mathbf{k}$ or in a periodic component containing only string $Lambda_{m,N}$-modules, where $mgeq 3$ and $Ngeq 1$ are integers, and $Lambda_{m,N}$ is a self-injective special biserial $mathbf{k}$-algebra.
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