Articles of graded modules

Generating connected module over a connected $K$-algebra

I have an algebraic question that I cannot solve. It is extracted from Adams and Margolis’ paper on modules over the Steenrod Algebra. Here is the problem : Let $K$ be a commutative ring with unit, $R$ a connected $K$-algebra (not commutative in my case), i.e., graded as $R \cong K \oplus R_1 \oplus R_2 […]

Does the usual filtration on graded objects satisfy a universal property?

Let $\mathsf{A}$ be an abelian category, such as $R \mathsf{Mod}$ for concreteness. We can think of the category of $\mathbb{Z}$-graded objects in $\mathsf{A}$ as the functor category $\mathsf{A}^\mathbb{Z}$ where $\mathbb{Z}$ is viewed as a discrete category. This works for $\mathbb{N}$-gradings as well. The category of filtered objects in $\mathsf{A}$ in the classical sense may (potentially) […]

Are projective modules “graded projective”?

Let $A^{\bullet}$ be a graded commutative algebra. Denote by $A^{\bullet}$-mod the category of graded modules over $A^{\bullet}$. Let $A$ be $A^{\bullet}$ considered as an algebra (we forgot grading). Finally let $A$-mod be category of modules over $A$. So we have an oblivion functor $$ Obl: A^{\bullet}-{\rm mod} \rightarrow A-{\rm mod}.$$ Consider $P^{\bullet} \in A^{\bullet}$-mod such […]

Decomposition of finitely generated graded modules over PID

I found this decomposition theorem used in a paper I’m reading, but it isn’t referenced and I can’t seem to find it in any of the books I have: Every graded module $M$ over a graded PID decomposes uniquely into the form $$({\bigoplus\limits_{i=1}^n \Sigma^{\alpha_i}D}) \oplus ({\bigoplus\limits_{j=1}^m \Sigma^{\gamma_j}D/d_jD})$$ where $d_j \in D$ are homogenous elements so […]

Finitely generated graded modules over $K$

I need some help on this exercise from A Course in Ring Theory by Donald S. Passman Find all finitely generated graded $K[x]$-modules up to abstract isomorphism. Remember, $K[x]$ is a principal ideal domain. The result is supposedly similar to the well-known structure theorem in the non-graded case. So let $M$ be a finitely generated […]

An elegant description for graded-module morphisms with non-zero zero component

In an example I have worked out for my work, I have constructed a category whose objects are graded $R$-modules (where $R$ is a graded ring), and with morphisms the usual morphisms quotient the following class of morphisms: $\Sigma=\left\lbrace f\in \hom_{\text{gr}R\text{-mod}}\left(A,B\right) \ | \ \ker\left(f\right)_0\neq 0, \ \mathrm{coker}\left(f\right)_0\neq 0\right\rbrace$ (by quotient I mean simply that […]

Finite free graded modules and the grading of their duals

Let $S$ be a $\mathbb{Z}$-graded ring and $F$ a $\mathbb{Z}$-graded module that is free of finite rank $n$. Then we can write $F = \oplus_{i=1}^n S(\nu_i)$, where $S(\nu_i)$ is a graded ring isomorphic to $S$ given by $S(\nu_i)_k = S_{k+\nu_i}$. Question 1: the ring isomorphism $S(\nu_i) \rightarrow S$ given by $x \mapsto x$ is not […]

“Graded free” is stronger than “graded and free”?

This topic suggested me the following question: If $R$ is a commutative graded ring and $F$ a graded $R$-module which is free, then can we conclude that $F$ has a homogeneous basis (that is, a basis consisting of homogeneous elements)? In general the answer is negative, and such an example can be found in Nastasescu, […]

Homomorphisms of graded modules

Let $M$ and $N$ be graded $R$-modules (with $R$ a graded ring). $\varphi:M\rightarrow N$ is a homogeneous homomorphism of degree $i$ if $\varphi(M_n)\subset N_{n+i}$. Denote by $\mathrm{Hom}_i(M,N)$ the group of homogeneous homomorphisms of degree $i$. We define $^*\mathrm{Hom}_R(M,N)=\bigoplus_{i\in\mathbb{Z}}\mathrm{Hom}_i(M,N)$. This is a (graded) $R$-submodule of $\mathrm{Hom}_R(M,N)$. How can I prove that these two modules are equal […]