Tilting theory of noetherian algebras

For a ring $\Lambda$ with a Krull-Schmidt homotopy category, we study mutation theory on $2$-term silting complexes. As a consequence, mutation works when $\Lambda$ is a complete noetherian algebra, that is, a module-finite algebra over a commutative complete local noetherian ring $R$. By using results on $2$-term silting complexes of such noetherian algebras and $\tau$-tilting theory of Artin algebras, we study torsion classes of the module category of a noetherian algebra. When $R$ has Krull dimension one, the set of torsion classes of $\Lambda$ is decomposed into subsets so that each subset bijectively corresponds to a certain subset of the set of torsion classes of Artin algebras.Comment: 26 page

エネルギー変換デバイス用ペロブスカイト関連酸化物の機械的特性

Tohoku University川田達也課

Highest weight representations and Kac determinants for a class of conformal Galilei algebras with central extension

We investigate the representations of a class of conformal Galilei algebras in one spatial dimension with central extension. This is done by explicitly constructing all singular vectors within the Verma modules, proving their completeness and then deducing irreducibility of the associated highest weight quotient modules. A resulting classification of infinite dimensional irreducible modules is presented. It is also shown that a formula for the Kac determinant is deduced from our construction of singular vectors. Thus we prove a conjecture of Dobrev, Doebner and Mrugalla for the case of the Schrodinger algebra.Comment: 24 page

Classifying torsion pairs of Noetherian algebras

For a commutative Noetherian ring $R$ and a module-finite $R$-algebra $\Lambda$, we study the set $\mathsf{tors} \Lambda$ (respectively, $\mathsf{torf}\Lambda$) of torsion (respectively, torsionfree) classes of the category of finitely generated $\Lambda$-modules. We construct a bijection from $\mathsf{torf}\Lambda$ to $\prod_{\mathfrak{p}} \mathsf{torf}(\Lambda\otimes_R \kappa(\mathfrak{p}))$, and an embedding $\Phi$ from $\mathsf{tors} \Lambda$ to $\mathbb{T}_R(\Lambda):=\prod_{\mathfrak{p}} \mathsf{tors}(\Lambda\otimes_R \kappa(\mathfrak{p}))$, where $\mathfrak{p}$ runs all prime ideals of $R$. When $\Lambda=R$, these give classifications of torsionfree classes, torsion classes and Serre subcategories of $\mathsf{mod} R$ due to Takahashi, Stanley-Wang and Gabriel. To give a description of $\mathrm{Im} \Phi$, we introduce the notion of compatible elements in $\mathbb{T}_R(\Lambda)$, and prove that all elements in $\mathrm{Im} \Phi$ are compatible. We give a sufficient condition on $(R, \Lambda)$ such that all compatible elements belong to $\mathrm{Im} \Phi$ (we call $(R, \Lambda)$ compatible in this case). For example, if $R$ is semi-local and $\dim R \leq 1$, then $(R, \Lambda)$ is compatible. We also give a sufficient condition in terms of silting $\Lambda$-modules. As an application, for a Dynkin quiver $Q$, $(R, RQ)$ is compatible and we have a poset isomorphism $\mathsf{tors} RQ \simeq \mathsf{Hom}_{\rm poset}(\mathsf{Spec} R, \mathfrak{C}_Q)$ for the Cambrian lattice $\mathfrak{C}_Q$ of $Q$.Comment: 33 pages, one subsection was moved to Appendix, some propositions were added in Section

Improving Rewriting Induction Approach for Proving Ground Confluence

In (Aoto&Toyama, FSCD 2016), a method to prove ground confluence of many-sorted term rewriting systems based on rewriting induction is given. In this paper, we give several methods that add wider flexibility to the rewriting induction approach for proving ground confluence. Firstly, we give a method to deal with the case in which suitable rules are not presented in the input system. Our idea is to construct additional rewrite rules that supplement or replace existing rules in order to obtain a set of rules that is adequate for applying rewriting induction. Secondly, we give a method to deal with non-orientable constructor rules. This is accomplished by extending the inference system of rewriting induction and giving a sufficient criterion for the correctness of the system. Thirdly, we give a method to deal with disproving ground confluence. The presented methods are implemented in our ground confluence prover AGCP and experiments are reported. Our experiments reveal the presented methods are effective to deal with problems for which state-of-the-art ground confluence provers can not handle