Jacobson radical of a module category and its power

Assume that $k$ is a field and $A$ is a $k$-algebra. The category of (left) $A$-modules will be denoted by $\mathrm{Mod}(A)$ and ${\mathrm{Hom}}_A(X,Y)$ will denote the set of all $A$-homomorphisms between $A$-modules $X$ and $Y$. Of course ${\mathrm{Hom}}_A(X,Y)$ is an $A$-module itself and ${\mathrm{End}}_A(X)={\mathrm{Hom}}_A(X,X)$ is a $k$-algebra (even $A$-algebra) with composition as a multiplication.

Let $X$ and $Y$ be $A$-modules. Define

$${\mathrm{rad}}_A(X,Y)=\{f\in {\mathrm{Hom}}_A(X,Y)|{\forall }_{g\in {\mathrm{Hom}}_A(Y,X)}{\mathrm{\hspace{0.25em}1}}_X-gf\text{ is invertible in }{\mathrm{End}}_A(X)\}.$$

Definition. The Jacobson radical of a category $\mathrm{Mod}(A)$ is defined as a class

$$\mathrm{rad}\mathrm{Mod}(A)=\bigcup _{X,Y\in \mathrm{Mod}(A)}{\mathrm{rad}}_A(X,Y).\mathrm{\square }$$

Properties. $1)$ The Jacobson radical is an ideal in $\mathrm{Mod}(A)$, i.e. for any $X,Y,Z\in \mathrm{Mod}(A)$, for any $f\in {\mathrm{rad}}_A(X,Y)$, any $h\in {\mathrm{Hom}}_A(Y,Z)$ and any $g\in {\mathrm{Hom}}_A(Z,X)$ we have $hf\in {\mathrm{rad}}_A(X,Z)$ and $fg\in {\mathrm{rad}}_A(Z,X)$. Additionaly ${\mathrm{rad}}_A(X,Y)$ is an $A$-submodule of ${\mathrm{Hom}}_A(X,Y)$.

$2)$ For any $A$-module $X$ we have ${\mathrm{rad}}_A(X,X)=\mathrm{rad}({\mathrm{End}}_A(X))$, where on the right side we have the classical Jacobson radical.

$3)$ If $X,Y$ are both indecomposable $A$-modules such that both ${\mathrm{End}}_A(X)$ and ${\mathrm{End}}_A(Y)$ are local algebras (in particular, if $X$ and $Y$ are finite dimensional), then

$${\mathrm{rad}}_A(X,Y)=\{f\in {\mathrm{Hom}}_A(X,Y)|f\text{ is not an isomorphism}\}.$$

In particular, if $X$ and $Y$ are not isomorphic, then ${\mathrm{rad}}_A(X,Y)={\mathrm{Hom}}_A(X,Y)$. $\mathrm{\square }$

Let $n\in \mathbb{N}$ and let $f\in {\mathrm{Hom}}_A(X,Y)$. Assume there is a sequence of $A$-modules $X=X_0,X_1,\mathrm{\dots },X_{n-1},X_n=Y$ and for any $0\le i\le n-1$ we have an $A$-homomorphism $f_i\in {\mathrm{rad}}_A(X_i,X_{i+1})$ such that $f=f_{n-1}{f}_{n-2}\mathrm{\cdots }{f}_1{f}_0$. Then we will say that $f$ is $n$-factorizable through Jacobson radical.

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

$${\mathrm{rad}}^n\mathrm{Mod}(A)=\bigcup _{X,Y\in \mathrm{Mod}(A)}{\mathrm{rad}}_A^n(X,Y),$$

where ${\mathrm{rad}}_A^n(X,Y)$ is an $A$-submodule of ${\mathrm{Hom}}_A(X,Y)$ generated by all homomorphisms $n$-factorizable through Jacobson radical. Additionaly define

$${\mathrm{rad}}_A^{\mathrm{\infty }}(X,Y)=\bigcap _{n=1}^{\mathrm{\infty }}{\mathrm{rad}}_A^n(X,Y)\mathrm{\square }$$

Properties. $0)$ Obviously ${\mathrm{rad}}_A(X,Y)={\mathrm{rad}}_A^1(X,Y)$ and for any $n\in \mathbb{N}$ we have

$${\mathrm{rad}}_A^n(X,Y)\supseteq {\mathrm{rad}}_A^{\mathrm{\infty }}(X,Y).$$

$1)$ Of course each ${\mathrm{rad}}_A^n(X,Y)$ is an $A$-submodule of ${\mathrm{Hom}}_A(X,Y)$ and we have following sequence of inclusions:

$${\mathrm{Hom}}_A(X,Y)\supseteq {\mathrm{rad}}_A^1(X,Y)\supseteq {\mathrm{rad}}_A^2(X,Y)\supseteq {\mathrm{rad}}_A^3(X,Y)\supseteq \mathrm{\cdots }$$

$2)$ If both $X$ and $Y$ are finite dimensional, then there exists $n\in \mathbb{N}$ such that

$${\mathrm{rad}}_A^{\mathrm{\infty }}(X,Y)={\mathrm{rad}}_A^n(X,Y).$$

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