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13
votes
3answers
1k views

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 $\mathrm{Hom}_i(M,N)$ ...
13
votes
2answers
328 views

“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 ...
2
votes
1answer
202 views

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 ...
11
votes
0answers
301 views

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 ...
4
votes
1answer
229 views

Finitely generated graded modules over $K[x]$

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 ...
3
votes
2answers
52 views

If $M_*$ and $N_*$ are graded modules over the *graded* ring $R_*$, what is the definition of $M_* \otimes_{R_*} N_*$?

Quick question (hopefully): What is the correct definition of a tensor product of two graded $R_*$-modules and/or graded $R_*$-algebras $M_*$ and $N_*$ over the graded ring $R_*$? $M_* \otimes_{R_*} ...
2
votes
2answers
176 views

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 ...
3
votes
1answer
271 views

Notation for free graded resolutions of graded modules?

I am now reading a paper about Castelnuovo-Mumford regularity and in this paper, there is a notation as following: Let $S=k[x_1,...,x_{n}]$, by Hilbert's syzygy theorem, if $N$ is a graded module ...
2
votes
1answer
62 views

Noetherian assumptions in basic properties of coherent sheaves of modules

Using Hartshorne's definition of 'coherent sheaf': Proposition 5.11c Let $S$ be a graded ring, $M$ a graded $S$-module, $X=\operatorname{Proj} S$. Then $\tilde M$ is a quasi-coherent $\mathscr O_X$...
1
vote
1answer
107 views

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,...
0
votes
1answer
59 views

Graded direct products can differ from direct products

Assume $R$ is a graded ring and the $M_i$ are graded modules. Then Bruns and Herzog define the graded direct product $^*\Pi M_i$ as the submodule of $\Pi M_i$ generated by the sequences $(x_i)$ with $...
0
votes
0answers
84 views

Decomposition of a graded module

In an article I am reading I found the following statement: If $D$ is a PID, then every finitely generated $D$-module is isomorphic to a direct sum of cyclic $D$-modules. That is, it decomposes ...