0
votes
1answer
45 views

Is a graded module over a graded ring zero when all of it's graded localizations at graded primes not containing the irrelevant ideal are zero?

Let $M$ be a graded module over an $\mathbb{N}$-graded ring $S$ and $S_+$ be the ideal of positive degree elements. Is it true that $M=0$ iff the homogeneous localization $M_{(\mathfrak p)}=0$ for ...
0
votes
0answers
66 views

Extending a lemma about Castelnouvo-Mumford regularity

I'm learning Castelnouvo-Mumford regularity of associated graded ring. There is a lemma: It's from "Castelnuovo-Mumford regularity postulation number and relation types" by Markus Brodmann and ...
2
votes
0answers
281 views

Multiplicity and regular sequences

We define multiplicity of a module $M$ of dimension $d>0$ as $$e(M) := \operatorname{lc} (P_M) (d-1)!,$$ where $P_M$ denotes the Hilbert polynomial of $M$ and $\operatorname{lc}(P_M)$ its leading ...
0
votes
0answers
40 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 ...
0
votes
1answer
82 views

Problem with Eisenbud's Lemma “Symmetry of Diagonalization”?

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, $S(M)$ ...
0
votes
0answers
97 views

Graded version of Grothendieck's Non-vanishing Theorem

Is there a graded version of the Grothendieck Non-vanishing Theorem? (Theorem 6.1.4 of the book Local Cohomology. An Algebraic Introduction with Geometric Applications written by M. P. Brodmann and ...
1
vote
1answer
57 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 ...
3
votes
1answer
84 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 ...
2
votes
0answers
69 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 $\mathfrak a$ ...
0
votes
0answers
119 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? ($R$ is Noetherian ring and $M$ finite $R$-module.)
1
vote
0answers
65 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 ...
4
votes
1answer
146 views

Dimension of graded module

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 ...
0
votes
1answer
68 views

Grading on the graded direct product

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 ...
0
votes
1answer
42 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 ...
2
votes
0answers
124 views

$\mathrm{Hom}$ and $^*\mathrm{Hom}$ for graded modules: Exercise 1.5.19(f) of Bruns-Herzog

Assume $R$ is a graded ring and $M$ and $N$ graded modules. Denote by $^*\mathrm{Hom}_R(M,N)$ the set of all homogeneous $R$-linear maps from $M$ to $N$. How can I prove that if $M$ is finitely ...
2
votes
1answer
96 views

Poincaré series of quotient module

I am trying to calculate the Poincaré series $P(M,t)$ with respect to the standard degree grading of the graded $\mathbb{C} [x,y,z,w]$-module $ M=\mathbb{C}[x,y,z,w]/I$, where $I = (x,w) \cap (z,w) ...
5
votes
1answer
101 views

Difference between graded ring and graded algebra

Wikipedia says that a graded $A$-algebra is just a graded $A$-module that is also a graded ring. Question: when one says then "finitely generated graded $A$-algebra", does one mean that every element ...
2
votes
1answer
129 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 ...
3
votes
1answer
66 views

Krull dimension and graded prime ideals

How can we show that $\dim R/p=0\Leftrightarrow p=(x_{1},\ldots,x_{n})\Leftrightarrow R/p\simeq\mathbb{K}$, where $R=\mathbb{K}[x_{1},\ldots,x_{n}]$ is considered graded with standard grading (i.e. ...
5
votes
0answers
76 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
60 views

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 ...
0
votes
0answers
74 views

Question about initial forms

I am working through Eisenbud's Commutative Algebra, and in Chapter $5$ he defines the following map. Say we have a filtration of modules ${\cal F}:M=M_0\supset M_1\supset\cdots$. Then for $f\in M$, ...
3
votes
1answer
203 views

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 ...
4
votes
0answers
206 views

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 ...
2
votes
1answer
209 views

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 ...
0
votes
1answer
150 views

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 ...
2
votes
1answer
97 views

Does the category of graded rings have limits?

Let $\mathfrak{C}$ be the category of ($\mathbb{Z}$)-graded-commutative rings. Does this category have limits in it? I am particulary interested in power series rings over a field. Is there a ...
3
votes
2answers
358 views

Localizing and taking degree zero commutes with tensor product

Let $S$ be a graded ring ($S_n=0$ for $n<0$), $f\in S$ a homogeneous element, and $M, N$ two graded $S$-modules. I'm trying to prove that $$(M\otimes_S N)_{(f)}\simeq ...
3
votes
3answers
398 views

Hilbert-Poincaré Series of Finite-Dimensional Graded Algebras

Suppose I have two finite-dimensional $\mathbb{Z}_{\geq 0}$-graded $\mathbb{C}$-algebras $A = \bigoplus_{k \geq 0} A_{k}$ and $B = \bigoplus_{k \geq 0} B_{k}$ with Hilbert-Poincaré series, $P_{A}(t) = ...
2
votes
1answer
336 views

Some questions on graded rings

At the moment I'm mostly interested in commutative graded rings (and in particular those graded over $\mathbb{N}$), but any comments or references about more general graded rings would also be ...
12
votes
2answers
739 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 $\mathrm{Hom}_i(M,N)$ ...
4
votes
2answers
169 views

Graded modules over $k[t,t^{-1}]$

If $R=k[t,t^{-1}]$ is a graded ring where $R_0=k$ is a field and $t\in R$ is a homogeneous element of positive degree which is transcendental over $k$, how can I prove that every graded $R$-module ...