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تسجيل مستخدم جديدUniversal deformation rings of string modules over a certain symmetric special biserial algebra
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تأليف
Jose A. Velez-Marulanda
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Let $k$ be an algebraically closed field, let $A$ be a finite dimensional $k$-algebra and let $V$ be a $A$-module with stable endomorphism ring isomorphic to $k$. If $A$ is self-injective then $V$ has a universal deformation ring $R(A,V)$, which is a complete local commutative Noetherian $k$-algebra with residue field $k$. Moreover, if $Lambda$ is also a Frobenius $k$-algebra then $R(A,V)$ is stable under syzygies. We use these facts to determine the universal deformation rings of string $Ar$-modules whose stable endomorphism ring isomorphic to $k$, where $Ar$ is a symmetric special biserial $k$-algebra that has quiver with relations depending on the four parameters $ bar{r}=(r_0,r_1,r_2,k)$ with $r_0,r_1,r_2geq 2$ and $kgeq 1$.
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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-injec
tive, 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.
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 ver
sal 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 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.
For a finite ring $R$, not necessarily commutative, we prove that the category of $text{VIC}(R)$-modules over a left Noetherian ring $mathbf{k}$ is locally Noetherian, generalizing a theorem of the authors that dealt with commutative $R$. As an appli
cation, we prove a very general twisted homology stability for $text{GL}_n(R)$ with $R$ a finite noncommutative ring.
Let k be an algebraically closed field of positive characteristic, and let W be the ring of infinite Witt vectors over k. Suppose G is a finite group and B is a block of kG of infinite tame representation type. We find all finitely generated kG-modul
es V that belong to B and whose endomorphism ring is isomorphic to k and determine the universal deformation ring R(G,V) for each of these modules.
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