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Register a new userUniversal deformation rings for a class of self-injective special biserial algebras
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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-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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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 $kgeq 1$.
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 $mathbf{k}[[t]]/(t^2)$.
We show that the irreducible components of any moduli space of semistable representations of a special biserial algebra are always isomorphic to products of projective spaces of various dimensions. This is done by showing that irreducible components of varieties of representations of special biserial algebras are isomorphic to irreducible components of products of varieties of circular complexes, and therefore normal, allowing us to apply recent results of the second and third authors on moduli spaces.
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 $Lambda$ 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.
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