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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
votes
1answer
7 views

Example of non interchangeability of the order of taking graded moduls with respect to three filtrations

Assume we have an R-Module A, with R a commutative ring and three descending filtrations F,G,H on this. We can take the associated graded module with respect to any of these, say F, by setting ...
3
votes
1answer
106 views

Classification of finitely generated multigraded modules over $K[x_1,\ldots,x_n]$?

Let $K$ be a field and $R=K[x_1,\ldots,x_n]=\bigoplus_{a\in\mathbb{N}^n}Kx^a$ the multigraded polynomial ring. Have finitely-generated multigraded $R$-modules been classified? Are they of the ...
2
votes
2answers
106 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 ...
2
votes
1answer
102 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 ...
1
vote
0answers
24 views

Hilbert series of exterior algebra?

Let the exterior polynomial algebra $A=\Lambda_K[x_1,\ldots,x_n]$ have grading $\deg x_i=d_i$. Is there some nice formula for the Hilbert-Poincare series $HP_A=\sum_k\dim_K ...
0
votes
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 ...
1
vote
1answer
153 views

Bruns Herzog Theorem 9.2.1

In Theorem 9.2.1 of Bruns-Herzog Cohen-Macaulay rings it is proven that if $x_1,...,x_n$ is a regular sequence on an $R$-module $M$, and $t \ge 0$, then $x_1^t x_2^t \dotsm x_n^t \notin ...
2
votes
1answer
44 views

Is grading unique?

It must be a very basic question, but I just can't figure out... Let $P$ be a graded $A$-module ($A$ is a commutative associative with unity). Can $P$ have two different direct decompositions, that ...
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 $A \cong B?$
0
votes
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 ...
2
votes
1answer
98 views

About Betti Numbers

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 ...
7
votes
1answer
124 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 ...
0
votes
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 ...
2
votes
0answers
40 views

Homology of Derivations of a dgca algebra

Let $(A,d)$ be a differential graded commutative and associative algebra. A derivation on $A$ is a linear endomorphism $L: A \to A$, that satsfies $L(ab)= L(a)b+ aL(b)$. More general a derivation of ...
0
votes
0answers
130 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 $M$ ...
2
votes
2answers
56 views

Are projective modules “graded projective”?

Let $A^{\bullet}$ be a graded commutative algebra. Denote by $A^{\bullet}$-mod the category of graded modules over $A^{\bullet}$. Let $A$ be $A^{\bullet}$ considered as an algebra (we forgot grading). ...
1
vote
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
votes
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 ...
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 ...
2
votes
0answers
304 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 ...
1
vote
0answers
87 views

Projective covers of graded modules.

I want to prove that there exist projective covers in the category of graded modules over an algebra. I am fairly new to "this" kind of mathematic and don't really know where to start: I found the ...
0
votes
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 ...
0
votes
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 ...
3
votes
1answer
115 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 The result is supposely similar to the well-known structure theorem in the non-graded case. So let $M$ be a ...
0
votes
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 ...
0
votes
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
votes
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 ...
9
votes
0answers
237 views

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 ...
0
votes
1answer
88 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
66 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
0answers
27 views

Splitting of Abelian Groups in Graded Modules

Suppose I have a graded module $M=\oplus M_i$ over a graded ring $R=\oplus R_i$. Assume $M_i\cong A \oplus B$, where A and B are abelian groups. Since $M$ is a graded ring, we know that $R_k(A\oplus ...
3
votes
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
votes
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 $\mathfrak a$ ...
2
votes
0answers
146 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 ...
12
votes
3answers
875 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 $\mathrm{Hom}_i(M,N)$ ...
1
vote
0answers
81 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
172 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 ...
5
votes
1answer
104 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, ...
0
votes
1answer
82 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
47 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 ...
1
vote
1answer
290 views

Direct sum and direct product of graded modules

Let $M$, $N$ be $R$-graded modules, say: $$M= \bigoplus_{i\in\mathbb Z} M_{i}, N= \bigoplus_{j\in\mathbb Z} N_{j}.$$ Then $$\operatorname{Hom}(M,N) \cong \prod_{i\in\mathbb Z} \bigoplus_{j\in\mathbb ...
2
votes
1answer
144 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 ...
5
votes
2answers
181 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 ...
2
votes
1answer
109 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) ...
2
votes
1answer
234 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 ...
2
votes
1answer
141 views

Should Hom(M,N) be also graded ??

If $M,N$ are graded modules over a graded ring $R$, then $\operatorname{Hom}_{R}(M,N)$ is also a graded module and how ?
3
votes
1answer
81 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. ...
2
votes
3answers
112 views

$0$-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 ...
5
votes
0answers
81 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 ...