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4
votes
1answer
56 views

Looking for a proof of a result given as a remark in a question on Mathoverflow.

In a question on MathOverflow it is said that: It is known that given a short exact sequence of finitely generated graded modules over a polynomial ring over a field:$$0 \to M'' \to M \to M' \to ...
1
vote
0answers
32 views

Reflexive Graded Module

Let $R=k[x_1,\dots,x_d]$ be a polynomial ring and $M=M_0\oplus M_1\oplus M_2\oplus\cdots$ be a graded $R$-module. Is it true that $M$ is reflexive as an $R$-module if and only if $M_i$ is reflexive as ...
3
votes
0answers
80 views

Cones of max-Spec

Let $k$ be an algebraically closed field and $R=k\oplus R_1\oplus R_2\oplus \ldots$ be a graded commutative ring that's finitely generated by elements of positive degree. If $M$ is a finitely ...
2
votes
0answers
25 views

Compute directly that the mapping cone of a homotopy equivalence is contractible

Let's consider the category $Ch_R$ of cochain complexes of modules over a commutative ring $R$. I'm trying to prove that if the chain map $\phi:M\rightarrow N$ is a homotopy equivalence then its ...
0
votes
2answers
28 views

Example of a graded module that is not a ring.

I'm looking for an example of a graded module, that is not a ring. All the examples of graded modules that I have come across, like $k[x_1,x_2,\dots,x_n]$ are all graded rings. Thanks in advance
1
vote
2answers
42 views

Support of a tail of a graded module.

Suppose that $R$ is a non-negatively, graded commutative ring. I have been trying to decide if the following is true for a graded $R$-module $M$ (not necessarily finite over $R$): $$\text{Supp}_R ...
0
votes
0answers
17 views

Finite generation and associated graded modules

My question is as follows: Let $R$ be a ring and let $M$ a right $R$-module. Suppose that $(F_i)_{i \geqslant 0}$ is a filtration of $R$ and $(M_i)_{i \geqslant 0}$ is a compatible filtration of ...
0
votes
1answer
103 views

Errata with Eisenbud's Lemma “Symmetry of Diagonalization” proof.

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
1answer
25 views

When are minimal faithful modules over algebras unique?

Let $A$ be a unital associative algebra over $\mathbb{C}$. We say an $A$-module $V$ is "minimal faithful" if (i) $V$ is faithful and (ii) $V$ does not have a proper submodule that is faithful. First ...
0
votes
1answer
61 views

Poincare series and Hilbert polynomial of some graded modules [closed]

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, ...
13
votes
2answers
320 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 ...
1
vote
1answer
76 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
vote
1answer
35 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
0answers
46 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 ...
1
vote
0answers
25 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 ...
4
votes
1answer
209 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
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
0answers
75 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
240 views

Notation for free graded resolutions of graded modules?

I am now reading a paper about Castelnuovo-Mumford regularity and in this paper, there is a notation as following: Let $S=k[x_1,...,x_{n}]$, by Hilbert's syzygy theorem, if $N$ is a graded module ...
0
votes
1answer
30 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
votes
1answer
38 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
45 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_*} ...
2
votes
1answer
52 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
47 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.
0
votes
2answers
120 views

Questions on (subring/ submodule) of a graded (ring/ module)

I have a question which seems a bit silly... If we have $R$ a graded ring, does it follow that every subring of $R$ is also graded? Because I have a problem here as such: I have a graded ring $R$ ...
2
votes
1answer
25 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 ...
2
votes
1answer
180 views

Is the module of homomorphisms between graded modules also a graded module?

If $M,N$ are graded modules over a graded ring $R$, then is $\operatorname{Hom}_{R}(M,N)$ also a graded module and how?
4
votes
1answer
34 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
0answers
95 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 $W_\bullet$ ...
0
votes
0answers
74 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 $R=S/I$. ...
2
votes
1answer
95 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 ...
7
votes
0answers
174 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 ...
2
votes
1answer
238 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
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
215 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
165 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
172 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
55 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
36 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
214 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
49 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
76 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 $A \cong B?$
2
votes
1answer
136 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
173 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
85 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
67 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
136 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
116 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
75 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
104 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 ...