Jacobson radical of a module category and its power

Assume that k is a field and A is a k-algebraPlanetmathPlanetmathPlanetmath. The categoryMathworldPlanetmath of (left) A-modules will be denoted by Mod(A) and HomA(X,Y) will denote the set of all A-homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath between A-modules X and Y. Of course HomA(X,Y) is an A-module itself and EndA(X)=HomA(X,X) is a k-algebra (even A-algebra) with composition as a multiplicationPlanetmathPlanetmath.

Let X and Y be A-modules. Define

radA(X,Y)={fHomA(X,Y)|gHomA(Y,X) 1X-gf is invertible in EndA(X)}.

Definition. The Jacobson radicalMathworldPlanetmath of a category Mod(A) is defined as a class


Properties. 1) The Jacobson radical is an ideal in Mod(A), i.e. for any X,Y,ZMod(A), for any fradA(X,Y), any hHomA(Y,Z) and any gHomA(Z,X) we have hfradA(X,Z) and fgradA(Z,X). Additionaly radA(X,Y) is an A-submoduleMathworldPlanetmath of HomA(X,Y).

2) For any A-module X we have radA(X,X)=rad(EndA(X)), where on the right side we have the classical Jacobson radical.

3) If X,Y are both indecomposableMathworldPlanetmathPlanetmath A-modules such that both EndA(X) and EndA(Y) are local algebras (in particular, if X and Y are finite dimensional), then

radA(X,Y)={fHomA(X,Y)|f is not an isomorphism}.

In particular, if X and Y are not isomorphic, then radA(X,Y)=HomA(X,Y).

Let n and let fHomA(X,Y). Assume there is a sequenceMathworldPlanetmathPlanetmath of A-modules X=X0,X1,,Xn-1,Xn=Y and for any 0in-1 we have an A-homomorphism firadA(Xi,Xi+1) such that f=fn-1fn-2f1f0. Then we will say that f is n-factorizable through Jacobson radical.

Definition. The n-th power of a Jacobson radical of a category Mod(A) is defined as a class


where radAn(X,Y) is an A-submodule of HomA(X,Y) generated by all homomorphisms n-factorizable through Jacobson radical. Additionaly define


Properties. 0) Obviously radA(X,Y)=radA1(X,Y) and for any n we have


1) Of course each radAn(X,Y) is an A-submodule of HomA(X,Y) and we have following sequence of inclusions:


2) If both X and Y are finite dimensional, then there exists n such that

Title Jacobson radical of a module category and its power
Canonical name JacobsonRadicalOfAModuleCategoryAndItsPower
Date of creation 2013-03-22 19:02:23
Last modified on 2013-03-22 19:02:23
Owner joking (16130)
Last modified by joking (16130)
Numerical id 6
Author joking (16130)
Entry type Definition
Classification msc 16N20