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Register a new userAn explicit construction of simple-minded systems over self-injective Nakayama algebras
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Recently, we obtained in [7] a new characterization for an orthogonal system to be a simple-minded system in the stable module category of any representation-finite self-injective algebra. In this paper, we apply this result to give an explicit construction of simple-minded systems over self-injective Nakayama algebras.
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Let $A$ be a representation-finite self-injective algebra over an algebraically closed field $k$. We give a new characterization for an orthogonal system in the stable module category $A$-$stmod$ to be a simple-minded system. As a by-product, we show that every Nakayama-stable orthogonal system in $A$-$stmod$ extends to a simple-minded system.
Let $A$ be a finite-dimensional self-injective algebra over an algebraically closed field, $mathcal{C}$ a stably quasi-serial component (i.e. its stable part is a tube) of rank $n$ of the Auslander-Reiten quiver of $A$, and $mathcal{S}$ be a simple-minded system of the stable module category $stmod{A}$. We show that the intersection $mathcal{S}capmathcal{C}$ is of size strictly less than $n$, and consists only of modules with quasi-length strictly less than $n$. In particular, all modules in the homogeneous tubes of the Auslander-Reiten quiver of $A$ cannot be in any simple-minded system.
We determine the derived representation type of Nakayama algebras and prove that a derived tame Nakayama algebra without simple projective module is gentle or derived equivalent to some skewed-gentle algebra, and as a consequence, we determine its singularity category.
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 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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