On the nilpotency index of the radical of a module category (original) (raw)
Related papers
On the Radical of the Module Category of an Endomorphism Algebra
Algebras and Representation Theory
Given a finite dimensional algebra A over an algebraically closed field we study the relationship between the powers of the radical of a morphism in the module category of the algebra A and the induced morphism in the module category of the endomorphism algebra of a tilting A-module. We compare the nilpotency indices of the radical of the mentioned module categories. We find an upper bound for the nilpotency index of the radical of the module category of iterated tilted algebras of Dynkin type.
A Note on the Radical of a Module Category
Communications in Algebra, 2013
We characterize the finiteness of the representation type of an artin algebra in terms of the behavior of the projective covers and the injective envelopes of the simple modules with respect to the infinite radical of the module category. In case the algebra is representation-finite, we show that the nilpotency of the radical of the module category is the maximal depth of the composites of these maps, which is independent from the maximal length of the indecomposable modules.
On the radical of Cluster tilted algebras
2020
We determine the minimal lower bound nnn, with ngeq1n \geq 1ngeq1, where the nnn-th power of the radical of the module category of a representation-finite cluster tilted algebra vanishes. We give such a bound in terms of the number of vertices of the underline quiver. Consequently, we get the nilpotency index of the radical of the module category for representation-finite self-injective cluster tilted algebras. We also study the non-zero composition of mmm, mge2m \ge 2mge2, irreducible morphisms between indecomposable modules in representation-finite cluster tilted algebras lying in the (m+1)(m+1)(m+1)-th power of the radical of their module category.
Quiver theories and formulae for nilpotent orbits of Exceptional algebras
Journal of High Energy Physics, 2017
We treat the topic of the closures of the nilpotent orbits of the Lie algebras of Exceptional groups through their descriptions as moduli spaces, in terms of Hilbert series and the highest weight generating functions for their representation content. We extend the set of known Coulomb branch quiver theory constructions for Exceptional group minimal nilpotent orbits, or reduced single instanton moduli spaces, to include all orbits of Characteristic Height 2, drawing on extended Dynkin diagrams and the unitary monopole formula. We also present a representation theoretic formula, based on localisation methods, for the normal nilpotent orbits of the Lie algebras of any Classical or Exceptional group. We analyse lower dimensioned Exceptional group nilpotent orbits in terms of Hilbert series and the Highest Weight Generating functions for their decompositions into characters of irreducible representations and/or Hall Littlewood polynomials. We investigate the relationships between the moduli spaces describing different nilpotent orbits and propose candidates for the constructions of some non-normal nilpotent orbits of Exceptional algebras.
The radical of an n-cluster tilting subcategory
arXiv (Cornell University), 2022
Let Λ be an artin algebra and C be a functorially finite subcategory of mod Λ which contains Λ or DΛ. We use the concept of the infinite radical of C and show that C has an additive generator if and only if rad ∞ C vanishes. In this case we describe the morphisms in powers of the radical of C in terms of its irreducible morphisms. Moreover, under a mild assumption, we prove that C is of finite representation type if and only if any family of monomorphisms (epimorphisms) between indecomposable objects in C is noetherian (conoetherian). Also, by using injective envelopes, projective covers, left C-approximations and right C-approximations of simple Λ-modules, we give other criteria to describe whether C is of finite representation type. In addition, we give a nilpotency index of the radical of C which is independent from the maximal length of indecomposable Λ-modules in C.
On modules which satisfy the radical formula
Turkish Journal of Mathematics, 2013
In this paper, the authors prove that every representable module over a commutative ring with identity satisfies the radical formula. With this result, they extend the class of modules satisfying the radical formula from that of Artinian modules to a larger one. They conclude their work by giving a description of the radical of a submodule of a representable module.
Journal of the London Mathematical Society, 1991
Let Q be a quiver without oriented cycles and let a be a dimension vector such that G^(a) has an open orbit on the representation space R(a). We find representations {S t } and corresponding polynomials {P s J on R(a), which generate the semi-invariants and are algebraically independent. 0. Notation and generalities Let Q be a quiver with vertex set Q Q and arrow set Q v Typically, we shall write veQ o , aeQ 1} and a has a tail, ta, and head, ha; a ta ha We fix an algebraically closed field k. A representation R of Q is a family of finite-dimensional vector spaces {R(v): v e Q o }, together with linear maps R(a):R(ta)->R(ha). The dimension vector of /? is the function, dim/?, defined by dim R(v) = dimi?(y); it lies in F, the space of integer-valued functions on Q o. If 7? and 5 are representations then a morphism :R->S is a collection of linear maps (v):R(v)->S(v) such that, for all atQ x , </>(ta)S(a) = R(a)(ha). The collection of morphisms is written Horn (R, S). With these definitions, the category of representations of Q, namely Rep (Q), may be seen to be an abelian category. We assume throughout this paper that there are no oriented cycles; that is, there is no sequence of arrows a 1 ,a 2 ,...,a n such that ha t = ta i+1 and ha n = ta v A path p = a 1 a 2 ...a n is a sequence of arrows such that ha { = ta i+1 for / = 1 to n-1. We define tp = ta 1 and hp = ha n. The trivial path from v to v has head and tail v. We define [v, w] to be the vector space on the basis of paths from v to w. This is finite dimensional given our assumption that there are no oriented cycles. We define a representation P v by P v (w) = [v, w], and P v (a): [v, ta]-> [v, ha] is defined by composition with a. One checks that Horn (P v , R) ~ R(v), so that Horn (/*",-) is an exact functor which implies that P v is a projective representation. The set {P v } is a complete set of indecomposable projective representations up to isomorphism. Similarly, we define the indecomposable injective representations of Q by I v (w) = D[w, v]; here D V for a vector space V is defined to be Hom fc (K, k). One checks that Horn (R,I V) ^ DR(v). Note that Horn (P v , P w) ~ [v, w] ^ DIJv) ~ Horn (/", / J. So there is a natural equivalence between the category of projective and the category of injective representations of Q. If we define A = © P v , A has a natural algebra structure, the path algebra of Q, and representations of Q are modules for A. One can show that the above equivalence between the category of projectives and injectives is induced by the functor D Horn (, A).
Kothe's upper nil radical for modules
Let M be a left R-module. In this paper a generalization of the notion of an s-system of rings to modules is given. Let N be a submodule of M. Define mathcalS(N):=minM:,mboxeverysmbox−systemcontainingmmboxmeetsN˜\mathcal{S}(N):=\{ {m\in M}:\, \mbox{every } s\mbox{-system containing } m \mbox{ meets}~N \}mathcalS(N):=minM:,mboxeverysmbox−systemcontainingmmboxmeetsN˜ . It is shown that mathcalS(N)\mathcal{S}(N)mathcalS(N) is equal to the intersection of all s-prime submodules of M containing N. We define mathcalN(RM)=mathcalS(0)\mathcal{N}({}_{R}M) = \mathcal{S}(0)mathcalN(RM)=mathcalS(0) . This is called (Köthe’s) upper nil radical of M. We show that if R is a commutative ring, then mathcalN(RM)=mathopmathrmradnolimitsR(M)\mathcal{N}({}_{R}M) = {\mathop{\mathrm{rad}}\nolimits}_{R}(M)mathcalN(RM)=mathopmathrmradnolimitsR(M) where mathopmathrmradnolimitsR(M){\mathop{\mathrm{rad}}\nolimits}_{R}(M)mathopmathrmradnolimitsR(M) denotes the prime radical of M. We also show that if R is a left Artinian ring, then mathopmathrmradnolimitsR(M)=mathcalN(RM)=mathopmathrmRadnolimits,(M)=mathopmathrmJacnolimits,(R)M{\mathop{\mathrm{rad}}\nolimits}_{R}(M)=\mathcal{N}({}_{R}M)= {\mathop{\mathrm{Rad}}\nolimits}\, (M)= {\mathop{\mathrm{Jac}}\nolimits}\, (R)MmathopmathrmradnolimitsR(M)=mathcalN(RM)=mathopmathrmRadnolimits,(M)=mathopmathrmJacnolimits,(R)M where mathopmathrmRadnolimits,(M){\mathop{\mathrm{Rad}}\nolimits}\, (M)mathopmathrmRadnolimits,(M) denotes the Jacobson radical of M and mathopmathrmJacnolimits,(R){\mathop{\mathrm{Jac}}\nolimits}\, (R)mathopmathrmJacnolimits,(R) the Jacobson radical of the ring R. Furthermore, we show that the class of all s-prime modules forms a special class of modules.