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Register a new userOn Deformations of Gorenstein-Projective Modules over Monomial Algebras with no Overlaps
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Jose A. Velez-Marulanda
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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 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 $mathbf{k}[[t]]/(t^2)$
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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.
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 $mathbf{k}[![t]!]/(t^2)$.
In this paper, we introduce the notions of Gorenstein projective $tau$-rigid modules, Gorenstein projective support $tau$-tilting modules and Gorenstein torsion pairs and give a Gorenstein analog to Adachi-Iyama-Reitens bijection theorem on support $tau$-tilting modules. More precisely, for an algebra $Lambda$, we show that there is a bijection between the set of Gorenstein projective $tau$-rigid pairs in $mod Lambda$ and the set of Gorenstein injective $tau^{-1}$-rigid pairs in $mod Lambda^{rm op}$. We prove that there is a bijection between the set of Gorenstein projective support $tau$-tilting modules and the set of functorially finite Gorenstein projective torsion classes. As an application, we introduce the notion of CM-$tau$-tilting finite algebras and show that $Lambda$ is CM-$tau$-tilting finite if and only if $Lambda^{rm {op}}$ is CM-$tau$-tilting finite.
Let $T=left( begin{array}{cc} R & M 0 & S end{array} right) $ be a triangular matrix ring with $R$ and $S$ rings and $_RM_S$ an $R$-$S$-bimodule. We describe Gorenstein projective modules over $T$. In particular, we refine a result of Enochs, Cort{e}s-Izurdiaga and Torrecillas [Gorenstein conditions over triangular matrix rings, J. Pure Appl. Algebra 218 (2014), no. 8, 1544-1554]. Also, we consider when the recollement of $mathbb{D}^b(T{text-} Mod)$ restricts to a recollement of its subcategory $mathbb{D}^b(T{text-} Mod)_{fgp}$ consisting of complexes with finite Gorenstein projective dimension. As applications, we obtain recollements of the stable category $underline{T{text-} GProj}$ and recollements of the Gorenstein defect category $mathbb{D}_{def}(T{text-} Mod)$.
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 $mathbb{P}^1_{mathbf{k}}$
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