# 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

 $\mathrm{rad}_{A}(X,Y)=\{f\in\mathrm{Hom}_{A}(X,Y)\ |\ \forall_{g\in\mathrm{Hom% }_{A}(Y,X)}\ 1_{X}-gf\mbox{ is invertible in }\mathrm{End}_{A}(X)\}.$

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

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

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

 $\mathrm{rad}_{A}(X,Y)=\{f\in\mathrm{Hom}_{A}(X,Y)\ |\ f\mbox{ is not an % isomorphism}\}.$

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

Let $n\in\mathbb{N}$ and let $f\in\mathrm{Hom}_{A}(X,Y)$. Assume there is a sequence of $A$-modules $X=X_{0},X_{1},\ldots,X_{n-1},X_{n}=Y$ and for any $0\leq i\leq n-1$ we have an $A$-homomorphism $f_{i}\in\mathrm{rad}_{A}(X_{i},X_{i+1})$ such that $f=f_{n-1}f_{n-2}\cdots f_{1}f_{0}$. 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

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

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

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

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

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

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

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

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

 $\mathrm{rad}^{\infty}_{A}(X,Y)=\mathrm{rad}^{n}_{A}(X,Y).$
Title Jacobson radical of a module category and its power JacobsonRadicalOfAModuleCategoryAndItsPower 2013-03-22 19:02:23 2013-03-22 19:02:23 joking (16130) joking (16130) 6 joking (16130) Definition msc 16N20