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Roberto C. Soto

Publications and source records attributed to Roberto C. Soto.

3 recordsLinked to original sources
arXiv2022

Universal deformation rings of modules for generalized Brauer tree algebras of polynomial growth

Let $k$ be an arbitrary field, $Λ$ be a $k$-algebra and $V$ be a $Λ$-module. When it exists, the universal deformation ring $R(Λ,V)$ of $V$ is a $k$-algebra whose local homomorphisms to $R$ parametrize the lifts of $V$ up to $R\otimes_k Λ$, where $R$ is any complete, local commutative Noetherian $k$-algebra with residue field $k$. Symmetric special biserial algebras, which coincide with Brauer graph algebras, can be viewed as generalizing the blocks of finite type $p$-modular group algebras. Bleher and Wackwitz classified the universal deformation rings for all modules for symmetric special biserial algebras with finite representation type. In this paper, we begin to address the tame case. Specifically, let $Λ$ be any 1-domestic, symmetric special biserial algebra. By viewing $Λ$ as generalized Brauer tree algebras and making use of a derived equivalence, we classify the universal deformation rings for those $Λ$-modules $V$ with stable endomorphism ring isomorphic to $k$. The latter is a natural condition, since it guarantees the existence of the universal deformation ring $R(Λ,V)$.

math.RT↗
arXiv2017

Universal deformation rings, endo-trivial modules, and semidihedral and generalized quaternion 2-groups

Let $k$ be a field of characteristic $p>0$, and let $W$ be a complete discrete valuation ring of characteristic $0$ that has $k$ as its residue field. Suppose $G$ is a finite group and $G^{\mathrm{ab},p}$ is its maximal abelian $p$-quotient group. We prove that every endo-trivial $kG$-module $V$ has a universal deformation ring that is isomorphic to the group ring $WG^{\mathrm{ab},p}$. In particular, this gives a positive answer to a question raised by Bleher and Chinburg for all endo-trivial modules. Moreover, we show that the universal deformation of $V$ over $WG^{\mathrm{ab},p}$ is uniquely determined by any lift of $V$ over $W$. In the case when $p=2$ and $G=\mathrm{D}$ is a $2$-group that is either semidihedral or generalized quaternion, we give an explicit description of the universal deformation of every indecomposable endo-trivial $k\mathrm{D}$-module $V$.

math.GR↗