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2answers
27 views

Why in a graded ring $A$ finitely generated that's an algebra over a field $K$ every maximal ideal is a $K$-subspace?

Probably this question has already been asked, but I'm very bad in find old question and I searched for half an hour, so I'm asking it again. I suppose that's true beacuse my professor used this ...
2
votes
1answer
80 views

Definition of multiplicity

Q.1. Bruns_Herzog define multiplicity (in the case of graded rings and modules) as My question is that: why multiplicity for $d=0$ it is defined as $\ell(M)$? Is there a kind of ...
0
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0answers
20 views

About the meaning of associated graded ideal

Let $G$ be any multiplicative group (abelian or not). Suppose that $R$ is a $G$-graded ring, i.e., there exists a family of additive subgroup $\{R_g\}_{g\in G}$ such that $R=\bigoplus_{g\in G}R_g$ ...
1
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1answer
30 views

Basic question regarding a finitely generated graded $A$-algebra

Let $S = \oplus_{n \geq 0} S_n$ be a graded ring. Let $S$ be a finitely generated $A$-algebra, where $A = S_0$, a commutative ring with unity. Then there exists $t_1, .., t_M$ homogeneous elements of ...
0
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0answers
30 views

What is interesting (useful) about Multiplicity?

Multiplicity is defined at 4.1.5; Bruns_Herzog. People say it is an important invariant. I don't know what idea is behind this definition and What is interesting/useful about it. what important ...
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1answer
37 views

Do non-graded rings exist?

I've been reading about graded rings recently, and I was wondering if there exist commutative or non-commutative rings for which no non-trivial gradings exist? That is, a ring for which if $R ...
2
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0answers
11 views

First order differential calculus over algebra $A$

Let $A$ be a unital algebra and $d_u$ is universal differential given by $d_u(x)=1 \otimes x-x \otimes 1$. It is viewed as the map $A \to \Omega_u^1(A)$ where $\Omega_u^1(A)=\ker m$ is the kernel of ...
0
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1answer
19 views

When is a D-module holonomic?

Suppose that I have a small family of partial differential equations such that $y(x)$ must be a solution to $$ P_i(x,D) y(x)=0,\ \ \ \ \ \ i=1,...,m $$ for all $i=1,...,m$, where I am using ...
2
votes
1answer
59 views

If $I$ is a homogeneous ideal of $A$ contained in $A_+$, then $\sqrt{I} = \bigcap\limits_{I\subset P\in\text{Proj }A} P$?

EDIT: This is from an exercise of Vakil's Foundations of Algebraic Geometry. 4.5.H: Suppose $I$ is any homogeneous ideal of $S$ contained in $S_+$, and if $f$ is a homogeneous element of ...
0
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1answer
29 views

Factoring a homogeneous element in graded ring

Let $k$ be a field, and $A = k[w,x,y,x] / (wz-xy)$, which is an integral domain. I would like to show that if $h$ is a homogeneous element in $A$, not irreducible, then it factors into a product of ...
2
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1answer
35 views

Question on homogeneous ideal in a graded ring

In an $\Bbb{N}$-graded ring $R=\bigoplus_nR_n$, an element is called homogenous (of degree $n$) if it is contained in $R_n$. An ideal is called homogenous if it is generated by homogenous elements. ...
1
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1answer
33 views

Different definitions of graded rings

In Atiyah i recently read the definition of a graded ring as a ring that can be written as $R=\displaystyle \bigoplus_{i \geq 0}^{\infty}R_i$ where each $R_i$ is an abelian subgroup of $R$ (with the ...
2
votes
1answer
62 views

Non-isomorphic algebras with equal Hilbert-Poincaré series

Let $A,B$ be two finite-dimensional graded algebras and let $P_A(x),P_B(z)$ be theirs Poincaré series. Suppose now that $P_A(x)=P_B(z)$. Question. Is it implies that $A \cong B?$
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0answers
28 views

Without loss of generality $P_1, . . . , P_s$ contain all elements of $gr_I (R)$ of positive degree, and $P_{s+1}, . . . , P_r$ do not

In the $\underline {Proposition\ 8.5.7}$ of book: Integral Closure of Ideals, Rings, and Modules, (Irena Swanson and Craig Huneke), they say: "Without loss of generality $P_1, . . . , P_s$ contain all ...
1
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1answer
37 views

Is there a name for these sequences of subsets of a commutative ring resembling the definition of a graded algebra?

(I am experimenting with writing arrows backwards.) Let $R$ denote a commutative ring. Is there a term for those sequences $A : \mathcal{P}(R) \leftarrow \mathbb{N}$ satisfying the following ...
1
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0answers
46 views

Equivalent definitions of homogeneous ideal

I need to show that an ideal $I$ of a $\mathbb{Z}$-graded ring R is homogeneous iff for every element $f \in I$, all homogeneous components of $f$ are in $I$. $\Leftarrow$ implication is obviously. ...
0
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2answers
76 views

David Eisenbud, Hilbert theorem

I just started reading D. Eisenbud Commutative algebra with a view towards algebraic geometry and I wonder about a theorem on page 42: If $M$ is a finitely generated graded module over ...
0
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0answers
26 views

Is the notion of strongly graded algebra a Morita invariant?

Let $G$ be a group and $A$ be a ring. $A$ is a $G$-graded ring if $A=\oplus_{g\in G} A_g$ such that $A_gA_h \subset A_{gh}$ for all $g,h\in A$. Such a ring is said to be strongly graded if ...
0
votes
2answers
66 views

What is $S_d$ in algebraic geometry?

I'm trying to read algebraic geometry on my own by doing homeworks on course hold in 2003. One of the problem is the following: Let $k$ be a field, $S=k[T_0,\ldots,T_r]$, ...
1
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2answers
61 views

Show that $\operatorname{Hom}(S(-d),S)\cong S(d)$ where $S$ is polynomial ring?

As stated above, $S$ is polynomial ring, and since the polynomial ring is $S$ and $S(-d)$ are finite over $S$ as graded modules, we can say that $\operatorname{Hom}(S(-d),S)$ is also graded. My ...
4
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0answers
91 views

Flatness of homomorphisms of graded-commutative rings

Algebraic geometry offers some properties and criteria for homomorphism of commutative rings to be flat. What about homomorphisms of graded-commutative rings? You can define flatness as usual: $R \to ...
1
vote
2answers
45 views

In a $\mathbb{Z}$-graded ring with unity, $1$ is homogeneous of degree zero

Suppose I have a $\mathbb{Z}$-graded ring (commutative) $A$ with unity. I am sure that the unity $1 \in A$ has degree $0$. I was wondering how could one show that? (I am guessing we don't have ...
0
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1answer
67 views

Castelnuovo-Mumford regularity and postulation numbers

I have a problem about Castelnuovo-Mumford regularity. This is a proposition from Castelnuovo-Mumford regularity, relation types and postulation numbers by M. Brodmann and C. H. Linh. My ...
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1answer
38 views

Graded ring localization. Why is this function bijective? [duplicate]

From Hartshorne, Chapter II.2, Proposition 2.5(b). If $R$ is a graded ring and $\mathfrak a$ is a homogenous ideal, then the function defined as $$\phi(\mathfrak a) = \mathfrak aR_f\cap R_{(f)}$$ ...
1
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1answer
79 views

Quotient ring of a graded algebra with respect to a graded ideal

An algebra $A$ over $F$ is said to be a graded algebra if as a vector space over $F$, $A$ can be written in the form $$A=\bigoplus_{i=0}^\infty A_i$$ for subspaces $A_i$ of $A$ along with other ...
0
votes
1answer
60 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
1answer
59 views

A question on graded rings

For a ring $A$ and an ideal $\mathfrak{a}$ of $A$, Atiyah-Macdonald define $$A^*=\bigoplus_{n=0}^\infty \mathfrak{a}^n$$ and claim that it is a graded ring on p. 107 of their commutative algebra book. ...
4
votes
3answers
100 views

stalk of projective variety in terms of the coordinate ring

Let $X \subseteq \mathbb{P}^n$ be an embedded projective variety over some field $k$ with its corresponding homogeneous coordinate ring $R = k[X_0,\dots,X_n]/I(X)$. Let further $X = \bigcup_{i=0}^n ...
0
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0answers
31 views

Confused with a terminology used in a book Group Representation Vol 1 Part A by G Karpilovsky

I am confused with a terminology used in a book Group Representation Vol 1 Part A by G Karpilovsky (Chapter 10, page 441). Which is as follows: Let $A=\oplus_{g \in G} A_g$ be a $G$-graded ...
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0answers
33 views

Extension of graded algebra by a homogeneous ideal

If an algebra is graded by the group $G$: $A=\bigoplus\limits_{d \in G} A_d$ and contains a homogeneous ideal $I \subset A$, then we have the quotient $B:=A/I$ and canonical epimorphism $\nu:A ...
1
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0answers
55 views

question about gradation of a ring

I was reading Mumford's 'Red book on varieties and schemes', when I came across the following paragraph: I am confused about meaning of the phrase "We let $k(X)$ be the zeroth graded piece of the ...
2
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0answers
99 views

Computing generators of the positive component of a graded ring

Let $R$ be a sub-algebra of $\mathbb{Q}[X_1^{\pm 1}, \dots, X_n^{\pm 1}]$ given by finitely many generators, and let $\lambda$ be a linear form $\lambda : \mathbb{Z}^n \to \mathbb{Z}$. This defines a ...
0
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0answers
74 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 ...
0
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1answer
86 views

Castelnouvo-Mumford Regularity

I'm learning Castelnouvo-Mumford regularity of associated graded ring. As u see in 2 pics below, Lemma 3.3. $(A,\mathfrak{m})$ is a Noetherian ring local, $\dim(A)=1$; $\mathfrak{q}=(x)$ is a ...
2
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1answer
98 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 ...
1
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0answers
32 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
50 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
133 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
32 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
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0answers
37 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
70 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 ...
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0answers
114 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 ...
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0answers
73 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
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1answer
57 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
64 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
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1answer
85 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
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0answers
73 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
34 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 ...
3
votes
1answer
87 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
61 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 ...