We introduce the notion of Burch submodules and weakly m-full modules over a local ring (R,m) study their properties. One our main results shows that satisfy 2-Tor rigid test also show (R,m), submodule M finitely generated R-module X, such either M=mX or M(⊆mX) is in 1-Tor rigid, module provided X faithful (and X∕M has finite length when m-full). As an application, we give some new class rings ...

Tensegrity frameworks are defined on a set of points in Rd and consist of bars, cables, and struts, which provide upper and/or lower bounds for the distance between their endpoints. The graph of the framework, in which edges are labeled as bars, cables, and struts, is called a tensegrity graph. It is said to be strongly rigid in Rd if every generic realization in Rd as a tensegrity framework is...

A differential operator D commuting with an S-action is said to be rigid if the non-constant Fourier coefficients of kerD and coker D are the same. Somewhat surprisingly, the study of rigid differential operators turns out to be closely related to the problem of defining Chern numbers on singular varieties. This relationship comes into play when we make use of the rigidity properties of the com...

We define the class of rigid Frobenius algebras in a (non-semisimple) modular category and prove that their categories local modules are, again, modular. This generalizes previous work Kirillov Ostrik (Adv Math 171(2):183–227, 2002) semisimple setup. Examples non-semisimple via modules, as well connections to authors’ prior on relative monoidal centers, are provided. In particular, we classify ...

A direction-length framework is a pair (G, p), where G = (V ;D,L) is a ‘mixed’ graph whose edges are labeled as ‘direction’ or ‘length’ edges, and p is a map from V to R for some d. The label of an edge uv represents a direction or length constraint between p(u) and p(v). Let G be obtained from G by adding, for each length edge e of G, a direction edge with the same end vertices as e. We show t...

in this errata, we reconsider and modify two propositions and their corollaries which were written on epi-retractable and co-epi-retractable modules.

We show that the quotient of a Hom-finite triangulated category C by the kernel of the functor HomC(T, −), where T is a rigid object, is preabelian. We further show that the class of regular morphisms in the quotient admit a calculus of left and right fractions. It follows that the Gabriel-Zisman localisation of the quotient at the class of regular morphisms is abelian. We show that it is equiv...

Rigid monoidal 1-categories are ubiquitous throughout quantum algebra and low-dimensional topology. We study a generalization of this notion, namely rigid algebras in an arbitrary 2-category. Examples include G-graded fusion 1-categories, G-crossed 1-categories. explore the properties 2-categories modules bimodules over algebra, by giving criterion for existence right left adjoints. Then, we co...

Let $V\subseteq A$ be a conformal inclusion of vertex operator algebras and let $\mathcal{C}$ category grading-restricted generalized $V$-modules that admits the algebraic braided tensor structure Huang-Lepowsky-Zhang. We give conditions under which inherits semisimplicity from $A$-modules in $\mathcal{C}$, vice versa. The most important condition is $A$ rigid $V$-module with non-zero categoric...