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0
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1answer
21 views

Poincare series and the Hilbert polynomial of $A = A_0[X_1,\dots , X_s]$

Let $A = A_0[X_1, \dots , X_s]$ be a polynomial ring in $s$ variables over an Artin ring $A_0$. This is a graded ring, and can be regarded as a graded module over itself. 1. What are the ...
0
votes
1answer
35 views

Poincare series and Hilbert polynomial of some graded modules [on hold]

Let $k$ be a field, and let $k[X, Y ]$ be the polynomial ring in two variables equipped with the usual grading such that $\deg(X) = \deg(Y ) = 1$. Consider the ideals $I = (X, Y^2)$ and $J = (X^2, ...
0
votes
1answer
24 views

where do elements go under multiplication in a graded module?

Assume that $M = \bigoplus_{n = 0}^\infty M_n$ is a graded $A$-module, where $A = \bigoplus_{n = 0}^\infty A_n$ is a graded ring. We have by definition $A_m M_n \subset M_{m + n}$. Does this mean that ...
1
vote
1answer
62 views

Explanation of a proof about graded module structure

Let $\Bbb F$ be a field and $M$ a finitely generated $\Bbb F[x]$-module. The structure theorem for modules over a PID says that $$ M\cong \Bbb F[x]^r\oplus\biggl(\bigoplus_{j=0}^s\Bbb ...
-1
votes
1answer
29 views

What does the shifting of graded modules mean here?

For any graded module $M$ we denote $M(a)$ the module $M$ "shifted by $a$" so that $M(a)_d=M_{a+d}$. Thus for example the free $S$-module of rank $1$ generated by an element of degree $a$ is ...
3
votes
2answers
42 views

If $M_*$ and $N_*$ are graded modules over the *graded* ring $R_*$, what is the definition of $M_* \otimes_{R_*} N_*$?

Quick question (hopefully): What is the correct definition of a tensor product of two graded $R_*$-modules and/or graded $R_*$-algebras $M_*$ and $N_*$ over the graded ring $R_*$? $M_* \otimes_{R_*} ...
0
votes
0answers
67 views

Decomposition of a graded module

In an article I am reading I found the following statement: If $D$ is a PID, then every finitely generated $D$-module is isomorphic to a direct sum of cyclic $D$-modules. That is, it decomposes ...
2
votes
1answer
30 views

Shift of a simple graded module

I am trying to understand the simple graded modules over a graded ring $R$ (all the gradings over $\mathbb{Z}$). I know that there exists a bijection between the simple graded $R$-modules and simple ...
1
vote
1answer
39 views

Graded modules/algebras on commutative monoids?

In the book Algebra of the Bourbaki group they deal with graded modules/algebras which are graded on a commutative monoid. What is the need to require the commutativity condition? Thanks.
2
votes
1answer
23 views

Sign convention when commuting shifts and tensor product

In Markl, Schnider, and Stasheff's Operads in algebra, topology, and physics, they give the observation $\mathfrak{s}^{-1} \mathcal{E}nd_V \cong \mathcal{E}nd_{V[-1]}$ (lemma 3.16) as motivation for ...
0
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0answers
43 views

Question about the classification theorem for finitely generated graded $F[t]$-modules

I am in the beginnings on learning persistence homology, and as a start I'm studying Gunnar Carlsson's survey "Topology and Data". Theorem 2.10 states the following: "Suppose $M_{\star}$ is a ...
2
votes
0answers
69 views

Koszul sign convention and symmetric group action on the graded n-th tensor product

Let $V_\bullet = (V_k)_{k \in \mathbb{Z}}$ and $W_\bullet = (W_k)_{k \in \mathbb{Z}}$ be two graded vector spaces on 0 caracteristic field. We define the tensor product of $V_\bullet$ by $W_\bullet$ ...
4
votes
1answer
32 views

$M/x_nM$ finitely generated over $ k[x_1,…, x_{n-1}] $ as graded module

I'm trying to figure out the following: Let $M$ be a finitely generated graded module over $S=k[x_1,..., x_n]$ with standard grading. Let $K$ be the kernel of multiplication by $x_n$ in $M$. Then ...
2
votes
1answer
70 views

Minimal free resolution of the twisted cubic

This is exercise 13.15 in Harris' book "A First Course...". Let $X$ be the twisted cubic with ideal $I(X) = (XZ-Y^2,YW-Z^2,XZ-YW).$ Let $S(X)$ denote the homogeneous coordinate ring of $X$ and ...
0
votes
0answers
68 views

A regular sequence in a determinantal ring

Let $S=K[X_{ij}\colon 1\le i\le m, 1\le j \le n, m\le n]$ be a ring of polynomials with coefficients in a field, $X=(X_{ij})$ a matrix of indeterminates, $I$ the ideal of maximal minors and $R=S/I$. ...
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0answers
170 views

How to intuitively understand prolongations

This question is concerned with the algebraic side of the theory of prolongations as explained in this paper by V. Guillemin and S. Sternberg. Let me first introduce my notation. We're working with a ...
1
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0answers
20 views

Is the Hilbert series of graded module interpreted as a rational function meaningful everywhere?

Say we have a graded module $M$ over a field $k$. In my mind I have $M=k[x,y,z]/(y^2-xz)$ graded by rank. The Hilbert series of this is $\sum_{n=0}^{\infty} (2n+1)t^n$. You can represent this as a ...
0
votes
0answers
34 views

Quotient of graded rings by graded ideals are again graded rings

I want to prove the following statement. Let $A = \bigoplus_{n=0}^{\infty}A_n$ to be a graded $R$-algebra and $I$ a graded ideal of $A$. Let $(A/I)_n = (A_n + I)/I$ be the image of $A_n$ in $A/I$. ...
0
votes
1answer
17 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 ...
5
votes
1answer
208 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
1answer
202 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 ...
1
vote
0answers
45 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
29 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
48 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 ...
1
vote
1answer
209 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
68 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
2answers
84 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 ...
3
votes
0answers
62 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 ...
2
votes
2answers
104 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
72 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
101 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
87 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 ...
1
vote
0answers
114 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
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0answers
80 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
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0answers
323 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 ...
3
votes
0answers
98 views

Structure theorem of modules in the graded case [duplicate]

I ran into an exercise in D. Passman's book A Course in Ring Theory (page 130) asking me to prove the structure theorem of modules over PIDs in the graded case. Find all finitely generated graded ...
0
votes
2answers
198 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
41 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
2answers
143 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
126 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 ...
0
votes
0answers
108 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
90 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, $S(M)$ ...
2
votes
1answer
149 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 ...
4
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1answer
169 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 ...
3
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1answer
93 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
98 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
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0answers
135 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
0answers
107 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 ...
6
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1answer
137 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, ...
2
votes
1answer
227 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 ...