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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 $\mathrm{Hom}_i(M,N)$ ...
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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 ...
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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 ...
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### 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 ...
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### 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 ...
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 $R_*$? $M_* \otimes_{R_*} ... 2answers 176 views ### Decomposition of finitely generated graded modules over PID I found this decomposition theorem used in a paper I'm reading, but it isn't referenced and I can't seem to find it in any of the books I have: Every graded module$M$over a graded PID decomposes ... 1answer 271 views ### Notation for free graded resolutions of graded modules? I am now reading a paper about Castelnuovo-Mumford regularity and in this paper, there is a notation as following: Let$S=k[x_1,...,x_{n}]$, by Hilbert's syzygy theorem, if$N$is a graded module ... 1answer 62 views ### Noetherian assumptions in basic properties of coherent sheaves of modules Using Hartshorne's definition of 'coherent sheaf': Proposition 5.11c Let$S$be a graded ring,$M$a graded$S$-module,$X=\operatorname{Proj} S$. Then$\tilde M$is a quasi-coherent$\mathscr O_X$... 1answer 107 views ### Generating connected module over a connected$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$be a commutative ring with unit,... 1answer 59 views ### Graded direct products can differ from direct products Assume$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)$with$...
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 ...