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2
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1answer
43 views

Maximal homogeneous ideals of a graded $k$-algebra.

Let $k$ be an algebraically closed field, and let $A$ be a finitely generated commutative $k$-algebra. Given any maximal ideal $\mathfrak{m}\subset A$, we can form the quotient to obtain a map $A\to ...
0
votes
1answer
40 views

Grading on fractional ideal of Rees ring

Let $R=k[x_1,\ldots,x_n]$, $m=\langle x_1,\ldots,x_n\rangle$ and $I$ be an $m$-primary ideal. Let $R(I)$ be the Rees ring of $I$. I am trying to understand the following: $(R(I):mR(I))/R(I)$ is a ...
1
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0answers
21 views

maximal chains of graded prime ideals

In Theorem 1.5.8 [Bruns,Herzog - Cohen-Macaulay-Rings] it is proved that for a noetherian graded ring $R$, a finitely generated graded $R$-module $M$ and any chain $\mathfrak{p}_0 \subsetneq ...
1
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0answers
23 views

Krull dimension and Hilbert polynomial of graded rings

Let $P \subseteq \mathbb{R}^n$ be a polytope with integral vertices, and $Q := \mathbb{R}_+ (P \times \{1\}) \cap \mathbb{Z}^{n+1}$ be the monoid of all lattice points of the cone generated by $P$ in ...
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2answers
71 views

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?
0
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1answer
24 views

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 ...
2
votes
0answers
32 views

Natural examples of multiplicatively-graded rings

A ring $A$ is multiplicatively graded if, as an additive group, it decomposes as $$A=\bigoplus_{n\geq 1} A_n$$ and the product satisfies $A_n \times A_m \rightarrow A_{nm}$. A natural example of ...
1
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2answers
57 views

Graded tensor algebra

Given a finite dimensional $\mathbb R$-vectorspace $V$ we can make $$ T(V) := \bigoplus_{n=0}^\infty V^{\otimes n}. $$ Here $V^{\otimes n} = V \otimes \cdots \otimes V$. An element of $T(V)$ looks ...
1
vote
1answer
89 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 ...
0
votes
0answers
65 views

Grade of a graded ideal, Bruns-Herzog, Exercise 1.5.21

The first part is easy: $\operatorname{grade} I = \dim S - 1 =3$. But I can't prove the second. Can you help please?
0
votes
1answer
25 views

Is every minimal set of generators for a homogeneous ideal composed by homogeneous elements?

An ideal $\mathfrak a$ of a graded ring $A$ is said to be homogeneous if I can find a set of homogeneous generators for $\mathfrak a$. Is it true that every minimal set of generators for a ...
0
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1answer
39 views

Associated Graded Module

I'm learning about Associated Graded Ring and Associated Graded Module. I got definition from Wiki: My questions: What's $I^nM$?, it's mean: $I^nM=am, a\in I, m\in M$? With above define, so: ...
3
votes
1answer
79 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
63 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 $\mathfrak a$ ...
1
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0answers
28 views

Grading and commutators

If $R$ is a unital associtaitve commutative ring, then can any $R$-algebra $A$ may be filtered as $A_0:=A$ and $A_{i+1}:=(A_i,A_i)$ where $(-,-)$ is the commutator of $A$ with respect to $A$'s ...
2
votes
1answer
64 views

Projecting an affine hypersurface away from a point in its projective closure is never a finite map?

Let $X\subset \mathbb{A}_k^r$ be an irreducible hypersurface defined by a polynomial $g$, where $k$ is an algebraically closed field. Embed $\mathbb{A}^r\hookrightarrow\mathbb{P}^r$ in the usual way. ...
2
votes
1answer
56 views

Why is this projective curve in $\mathbf{P}^3_k$ nonsingular?

Consider $C$ in $\mathbf{P}^3_k = \mathrm{Proj}[x_0,...,x_3]$ defined by $$x_0x_3 - x_1^2 = 0$$ and $$x_0^2 + x_2^2 - x_3^2 = 0$$ where $k$ is an algebraically closed field. Why is this curve ...
1
vote
1answer
130 views

Height unmixed homogeneous ideal and a non-zero divisor

Let $R=k[x_1,\ldots,x_n]$ be a standard graded polynomial over field $k$ and $I$ an unmixed homogeneous ideal of $R$. Let $x\in R$ be an $R/I$-regular element. Can we conclude that $x+I$ is an ...
1
vote
1answer
68 views

Socle degrees and last shift of free resolution

I have seen in several references that the degrees of the socle of an Artinian graded algebra $k[x_1,\ldots,x_d]/I$ can be computed by looking at the shifts of the end of its graded free resolution. ...
2
votes
0answers
47 views

Relation between closed subschemes and saturated ideals

Let $A=\mathbb{C}[x_0,x_1,\dots,x_n]$ and $X=\operatorname{Proj}A$. For any homogeneous ideal $I\subset A$, define the saturation $I^{\rm sat}:=\{f\in A\mid (x_0,\dots,x_n)^mf\subset I$ for some ...
0
votes
0answers
118 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.)
0
votes
1answer
208 views

Homogeneous and maximal ideal in a $\mathbb Z$-graded ring

Is Exercise 2.8 from Marley's notes on "GRADED RINGS AND MODULES" true? Exercise 2.8: Let $R$ be a graded ring and $M$ a homogeneous maximal ideal of $R$. Prove that $M =…⊕R_{-1}⨁m_0⨁R_1⨁…$, ...
7
votes
1answer
63 views

Why is the topology on $\operatorname{Proj} B$ induced from that on $\operatorname{Spec}(B)?$

In the proof of Lemma $3.36$ in Algebraic Geometry and Arithmetic Curves, it is stated that, if $B=\oplus_{d\ge0}B_d$ is a graded algebra over a ring $A,$ and if $I$ is an ideal of $B,$ then ...
5
votes
1answer
129 views

Exercise 4.5.E a) in Ravi Vakil's Foundations of Algebraic Geometry.

Hi! I am following the hint given in Exercise 4.5.E in Vakil's Foundations of Algebraic Geometry, but I am stuck trying to prove that if $a_1,a_2 \in Q_i$, then $a_1^2 + 2a_1 a_2 + a_2^2 \in Q_{2i}$. ...
1
vote
1answer
47 views

Are there homogeneous elements with two distinct grades?

In a graded ring $B=\bigoplus_{d\ge 0} B_d$, the element $0$ is homogeneous with grade $d$ for every $d\ge 0$, in fact since every $B_d$ is an additive subgroup of $B$, then it must contain $0$. Can ...
0
votes
0answers
23 views

Quotients of $\mathbb{Z}_2$ graded rings

This is a follow-up of Grading of the quotient module $M/N$ in a special case of $\mathbb{Z}_2$ graduation. Unfortunately, the answers there are not very helpful to me. So I'm asking whether the ...
1
vote
1answer
76 views

Rees ring associated to an ideal

I am reading Atiyah-Macdonald, the chapter on completions. Let $A$ be a ring (not graded), and let $\mathfrak{a}$ be an ideal of $A$. Then we can form a graded ring ...
1
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0answers
42 views

Construction of projective space of a graded ring

Let $S=k[x_0,x_1,x_2]$, $k$ an algebraically closed field and $ S_d=k[x_0^d,x_0^{d-1}x_1,\ldots,x_2^{d-1}x_1, x_2^d]$. $\mathbb{P}S_d $ is identified with the projective space $\mathbb{P}^{N_d} $, ...
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0answers
62 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 ...
0
votes
1answer
65 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
39 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 ...
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0answers
112 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 ...
5
votes
1answer
97 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 ...
5
votes
0answers
203 views

Irreducible homogeneous ideals

I have the following question: Let $I$ be a homogeneous ideal. Is it true that $I$ is irreducible if and only if it can't be written as the intersection of two homogeneous ideals? So, is it ...
1
vote
1answer
121 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
64 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. ...
3
votes
1answer
80 views

Associated graded ring of a Fermat cubic

Let $R$ be a graded Fermat cubic, i.e. $R$ is a graded ring given by $$ R=\mathbb{C}[x,y,z]/(x^3+y^3+z^3), $$ with a standard grading $\operatorname{deg}(x)= ...
1
vote
2answers
53 views

Doubts about graded algebra.

In the last days I've been studying the tensor algebra $T(V)$ of a vector space $V$ over the field $K$ and I've realised that what I'm not understanding hasn't to do with tensor products, but rather ...
5
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0answers
74 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
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0answers
34 views

Minimal spectrum of graded rings

Let $R$ be a left Noetherian ($\mathbb N$-)graded ring and let $R_0$ be its $0$-th component. When $R$ is commutative it is well-known, and easy to prove, that the minimal prime ideals of $R$ are ...
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0answers
38 views

Graded comodule isomorphism

Let $R$ be a semisimple ring. Let $C$ be a graded, connected $R$-coring (i.e. coalgebra in the monoidal category of $R$-bimodules), $C=\oplus_{n\geq 0} C_n$, such that $C_0 \simeq R$. Denote by $C_+$ ...
1
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1answer
43 views

Doubt in the definition of graded rings

My doubt is quite simple, I didn't understand what is the operation $\cdot$ in this definition: Thanks
4
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1answer
88 views

What does it mean for the coordinate ring of an affine variety to be graded?

My question is relatively simple, assume that $X$ is an affine variety such that its coordinate ring $A:=\Bbbk[X]:=H^0(\mathcal O_X,X)$ is $\Lambda$-graded for some monoid $\Lambda$. Now if ...
2
votes
1answer
308 views

Hartshorne's proof of Proposition 2.5, Chapter II of his book Algebraic Geometry [duplicate]

Let $S = \sum_{n\ge 0} S_n$ be a graded commutative ring. Let $f$ be a homogeneous element of $S$ of degree $> 0$. Let $D_+(f) = \{\mathfrak{p} \in\operatorname{Proj} S\mid f \notin ...
6
votes
1answer
174 views

Are minimal prime ideals in a graded ring graded?

Let $A=\oplus A_i$ be a graded ring. Let $\mathfrak p$ be a minimal prime in $A$. Is $\mathfrak p$ a graded ideal? Intuitively, this means the irreducible components of a projective variety are ...
6
votes
2answers
186 views

Question on Noetherian/Artinian properties of a graded ring

Let $R$ be a non-negatively graded Noetherian ring such that $R_{0}$ is Artinian and $R_{+}$ is a nilpotent ideal. Prove that $R$ is Artinian. Give an example to show that this is false if the ...
3
votes
1answer
120 views

Equivalent definition of a graded ring

Is this a equivalent definition for a graded ring? Let $R$ be a ring. We say that $R$ is graded if there are subgroups $R_n, n\in \mathbb{Z}$ of $R$ such that given $x\in R$, there are $x_n \in R_n$ ...
4
votes
1answer
179 views

Definition of graded rings

So, there are two types of definitions of graded rings (I will consider only commutative rings) that I have seen: 1) A ring $R$ is called a graded ring if $R$ has a direct sum decomposition $R = ...
2
votes
2answers
348 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 ...
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2answers
350 views

The bijection between homogeneous prime ideals of $S_f$ and prime ideals of $(S_f)_0$

It is well-known that if $S$ is a graded ring, and $f$ is a homogeneous element of positive degree, then there is a bijection between the homogeneous prime ideals of the localization $S_f$ and the ...