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### Graded rings and modules - which book to refer to?

Could you recommend a book with a good treatment of the theory of graded rings and modules?
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### Sign convention for derivatives in a $\mathbb{Z}_2$ graded space

Suppose $V=V_0\oplus\theta V_1$ is a $\mathbb{Z}_2$ graded super vector space. Note: Since $\theta^2=0$, this implies $\theta\mathrm{d}\theta=-\mathrm{d}\theta\cdot\theta$. However, I wish to know ...
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### 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 ...
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I'm studying the book 'The Geometry of Syzygies' of David Eisenbud, but I'm having problem with the following step, in page 7 he says the we have a free resolution to the set of ten points in $\mathbb{... 0answers 122 views ### Suggest a good book or reference on graded modules over polynomial rings I am looking for reference books or papers on graded modules over the polynomial ring$k[x_0, \ldots, x_n]$. Any good commutative algebra text like Eisenbud's Commutative Algebra already contains a ... 1answer 106 views ### Errata with Eisenbud's Lemma “Symmetry of Diagonalization” proof. In his proof of Lemma A2.5 in his book Commutative Algebra with a View towards Algebraic Geometry, Prof. Eisenbud writes something like this: Let R be a commutative ring,$M$an$R$-module,$S(M)$... 1answer 197 views ### Tor for graded modules over a graded ring I am confused about how this Tor is defined. Suppose$R$is a graded ring,$M,N$graded modules over$R$. What is$\operatorname{Tor}_{st}^R(M,N)$? I am confused about the subscripts. I realize ... 1answer 229 views ### Finitely generated graded modules over$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]$is a principal ... 1answer 96 views ### Why we can consider both modules as modules over$R_{(p)}$? (Bruns and Herzog, Theorem 1.5.9) I'm reading Bruns-Herzog's book Cohen Macaulay rings and have a probably elementary question. Why we may consider both modules as modules over$R_{(p)}$in this theorem? ... i know that ... 0answers 107 views ### Is the length of the composition series of a free module identical to the number of its bases? Let$A_0$be an Artinian ring,$M$a free$A_0$-module. Then, is the length of the composition series of$M$identical to the number of its bases? It seems to me that it is not. If$\mathfrak a$is ... 0answers 136 views ### When is$\operatorname{gr}_I (M)$finite? When is$\operatorname{gr}_I (R)$(I mean associated graded ring of$I$) finite? When is$\operatorname{gr}_I (M)$finite? if 1)$R$is non-Noetherian ring , 2)$R$is Noetherian ring and$M$is ... 0answers 112 views ### 'Finitely generated modules' versus 'Finitely generated algebra' I'm reading Commutative Algebra by Bourbaki, and I'm on chapter III of the book. I really think that the author made a typo there, but I'm not so sure. It's Proposition 1, and its Corollary on page ... 1answer 154 views ### holonomic D-modules I am trying to develop an intuition about holonomic D-modules and find the literature formidable (I study physics). My question is, given a linear differential operator in n-variables,$x=(x_1,...,...
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Let $R$ be a Noetherian positively graded ring and $M$ a finite graded $R$-module. Prove that $\dim M = \sup\{\dim M_p: p\in\operatorname{Supp} M \text{ graded}\}$. This is the Exercise 1.5.25 in ...
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This question is related to this one. Probably it's obvious but could you tell me what is the grading on the graded direct product? I was thinking about $^*\Pi M^i=\oplus_j(\Pi_i M^i_j)$ where ...
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### $0$-th exterior power, empty product of modules and their tensor product

$\def\finiteprod#1#2{#1_{1}\times#1_{2}\times\dots\times#1_{#2}}$In Lang's algebra he defines the tensor product as a universal object in the category of multilinear maps from $\finiteprod En$ where ...
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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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### 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,...
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### on exactness of the functors $M \mapsto \hat{M}$ and $M \mapsto \hat{A}\otimes_{A}M$

if $A$is a Noetherian ring, $M$ a finitely generated module,$I$ is an ideal of $A$, and $\hat{A}$ is the $I-adic$ completion of $A$, then we know $\hat{A}\otimes_{A}M\cong\hat{M}$. Also on Atiyah&...
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### Proof details of Theorem 11.1 in Atiyah-Macdonald

I have some trouble filling in the details of this proof from Atiyah-Macdonald. In this result, the authors assume what follows: 1) $A = \oplus_{n=0}^\infty A_n$ is a Noetherian graded ring, and ...
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### Reference request: Tensor product of DG-modules

Let $A$ be a (skew-) commutative DG-algebra, and let $M,N$ be two DG $A$-modules. I am looking for a reference which describes the functor $-\otimes_A -$ and its basic properties (associtivity, ...
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### Split-Lemma for chain complexes

Suppose $k$ is a field and $A$, $B$ and $C$ are chain complexes of $k$-vector spaces, i.e., objects in $\mathbf{Ch}(k\text{-}\mathbf{Vect})$. Is there are chain complex version of the split lemma, i.e....
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### What does $p+q=k$ mean in the index of summation?

I need help solving something I don't understand. OK so the problem is this: $$H^k(X,C)=\bigoplus_{p+q=k} H^{p,q}(X),$$ What does the $\;p+q=k\;$ mean? Thank you anybody that helps! :)
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### Questions on (subring/ submodule) of a graded (ring/ module)

I have a question which seems a bit silly... If we have $R$ a graded ring, does it follow that every subring of $R$ is also graded? Because I have a problem here as such: I have a graded ring $R$ ...
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### Path Algebra for Categories

For a while I had been thinking that the path algebra of a quiver $Q$ over a commutative ring $R$ is the same as the "category ring" $R[P]$ (analogous to "group ring", "monoid ring", "semigroup ring", ...
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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 ...
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### Construction of graded rings and modules

In Algebraic Geometry and Homological Algebra - as far as I know - we often consider graded rings and modules so as to encode more information, say, some sort of (computational) complexity. For ...
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### Compute Hilbert function of a monomial ideal

I'd like to know whether there exist easy methods that compute the Hilbert function of a graded $k$-algebra, without computer programs. My homework asks to me to compute the Hilbert function of $R/I$...
### Isomorphism of ($\mathbb{Z}/{(n)}$-graded) Rings
Let $A=\bigoplus_{d=0}^n A_g$ and $B=\bigoplus_{d=0}^n B_h$ be $\mathbb{Z}/{(n)}$-graded rings. In particular, we assume $A_n\ne 0$ and $B_n\ne 0$. Let $\phi:A\to B$ be an isomorphism of rings. My ...