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تسجيل مستخدم جديدUniversal Deformation Rings of Finitely Generated Gorenstein-Projective Modules over Finite Dimensional Algebras
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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 $mathbf{k}[![t]!]/(t^2)$.
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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 catego
ry $mathcal{D}^b(Lambdatextup{-mod})$ 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, 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 endomorph
ism 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 $mathbf{k}[[t]]/(t^2)$.
Let $Lambda$ be a basic finite dimensional algebra over an algebraically closed field $mathbf{k}$, and let $widehat{Lambda}$ be the repetitive algebra of $Lambda$. In this article, we prove that if $widehat{V}$ is a left $widehat{Lambda}$-module with
finite dimension over $mathbf{k}$, then $widehat{V}$ has a well-defined versal deformation ring $R(widehat{Lambda},widehat{V})$, which is a local complete Noetherian commutative $mathbf{k}$-algebra whose residue field is also isomorphic to $mathbf{k}$. We also prove that $R(widehat{Lambda},widehat{V})$ is universal provided that $underline{mathrm{End}}_{widehat{Lambda}}(widehat{V})=mathbf{k}$ and that in this situation, $R(widehat{Lambda},widehat{V})$ is stable after taking syzygies. We apply the obtained results to finite dimensional modules over the repetitive algebra of the $2$-Kronecker algebra, which provides an alternative approach to the deformation theory of objects in the bounded derived category of coherent sheaves over $mathbb{P}^1_{mathbf{k}}$
Let $mathbf{k}$ be a field of arbitrary characteristic, let $Lambda$ be a finite dimensional $mathbf{k}$-algebra, and let $V$ be an indecomposable Gorenstein-projective $Lambda$-module with finite dimension over $mathbf{k}$. It follows that $V$ has a
well-defined versal deformation ring $R(Lambda, V)$, which is complete local commutative Noetherian $mathbf{k}$-algebra with residue field $mathbf{k}$, and which is universal provided that the stable endomorphism ring of $V$ is isomorphic to $mathbf{k}$. We prove that if $Lambda$ is a monomial algebra without overlaps, then $R(Lambda,V)$ is universal and isomorphic either to $mathbf{k}$ or to $mathbf{k}[[t]]/(t^2)$
Let $mathbf{k}$ be a fixed field of arbitrary characteristic, and let $Lambda$ be a finite dimensional $mathbf{k}$-algebra. Assume that $V$ is a left $Lambda$-module of finite dimension over $mathbf{k}$. F. M. Bleher and the author previously proved
that $V$ has a well-defined versal deformation ring $R(Lambda,V)$ which is a local complete commutative Noetherian ring with residue field isomorphic to $mathbf{k}$. Moreover, $R(Lambda,V)$ is universal if the endomorphism ring of $V$ is isomorphic to $mathbf{k}$. In this article we prove that if $Lambda$ is a basic connected cycle Nakayama algebra without simple modules and $V$ is a Gorenstein-projective left $Lambda$-module, then $R(Lambda,V)$ is universal. Moreover, we also prove that the universal deformation rings $R(Lambda,V)$ and $R(Lambda, Omega V)$ are isomorphic, where $Omega V$ denotes the first syzygy of $V$. This result extends the one obtained by F. M. Bleher and D. J. Wackwitz concerning universal deformation rings of finitely generated modules over self-injective Nakayama algebras. In addition, we also prove the following result concerning versal deformation rings of finitely generated modules over triangular matrix finite dimensional algebras. Let $Sigma=begin{pmatrix} Lambda & B0& Gammaend{pmatrix}$ be a triangular matrix finite dimensional Gorenstein $mathbf{k}$-algebra with $Gamma$ of finite global dimension and $B$ projective as a left $Lambda$-module. If $begin{pmatrix} VWend{pmatrix}_f$ is a finitely generated Gorenstein-projective left $Sigma$-module, then the versal deformation rings $Rleft(Sigma,begin{pmatrix} VWend{pmatrix}_fright)$ and $R(Lambda,V)$ are isomorphic.
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