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Product notation for algebras

See the definition of "Graded G-algebra" on this page: https://ncatlab.org/nlab/show/crossed+G-algebra What is meant by the notation $L_gL_h\subseteq L_{gh}$ in condition (i)? In particular the ...
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1answer
197 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 ...
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0answers
10 views

Inverse of super matrices

I want to know that how does the inverse of a super matrix can be define?Is this inverse unique? If it is not can we find some equivalent relation that make this inverse unique up to equivalent class?
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1answer
88 views

Castelnuovo-Mumford regularity and exact sequence.

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 0$...
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1answer
46 views

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

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 ...
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1answer
72 views

Poincaré 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, Y^2)...
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1answer
104 views

Question concerning modules over a Clifford algebra

Let $R$be a commutative ring with unit element, and $M$ be an $R-$module. Let $f:M \times M \to R$ be a nondegenerate symmetric bilinear quadratic form, and $C(f)$ be the corresponding Clifford ...
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1answer
29 views

Exact sequence of graded modules and localization

I know that a sequence of modules is exact iff the localization at each prime ideal is exact What happens in the case we are working with graded modules? Can we say that a sequence is exact iff the ...
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1answer
62 views

Noetherian assumptions in basic properties of coherent sheaves of modules

Using Hartshorne's definition of 'coherent sheaf': Proposition 5.11c Let $S$ be a graded ring, $M$ a graded $S$-module, $X=\operatorname{Proj} S$. Then $\tilde M$ is a quasi-coherent $\mathscr O_X$...
3
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1answer
88 views

Can the sheaf associated to an indecomposable graded module be decomposable?

Let $R$ be a graded commutative ring such that $X=\operatorname{Proj}(R)$ is a smooth projective variety. Let $M$ be a finite graded module over $R$ and let $\mathcal{F}=\widetilde{M}$ be the ...
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1answer
18 views

Sum of Hilbert functions of a finite exact sequence of finitely generated graded modules

Let $A = \bigoplus_{n\geq 0} A_n$ be a graded ring that is generated as an $A_0$-algebra by a finite collection of elements of $A_1$, where $A_0$ is artinian. I wish to show that if $$ 0 \to M(1) \...
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0answers
13 views

Free graded k[x] modules have homogeneous bases

I was reading the article "Cary Webb. Decomposition of graded modules. Proceedings of the American Math- ematical Society, 94(4):565–571, 1985" where in the beginning "Free graded k[x] modules have ...
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0answers
39 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
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0answers
87 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 ...
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0answers
32 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 ...
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2answers
32 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
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2answers
45 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 M=\...
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0answers
21 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 $...
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1answer
106 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, $S(M)$ ...
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1answer
30 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 ...
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2answers
328 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 ...
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1answer
89 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 F[x]/(f_j(x))\...
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0answers
48 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 ...
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0answers
26 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 ...
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1answer
229 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 ...
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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 ...
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0answers
84 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 ...
3
votes
1answer
270 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 ...
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1answer
33 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 ...
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1answer
39 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 $S(-a)$. ...
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2answers
52 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_*} ...
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1answer
64 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 ...
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1answer
61 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.
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2answers
127 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 $R$ ...
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1answer
28 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 ...
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1answer
189 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?
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1answer
35 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
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0answers
119 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 $W_\bullet$ ...
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votes
1answer
109 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 $S$...
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0answers
178 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 ...
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1answer
274 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 ...
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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 $$\...
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1answer
219 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 ...
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2answers
176 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 ...
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0answers
57 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 \!A_k\,t^k\in\mathbb{Z}[[t]]...
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1answer
40 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 multi-...
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1answer
230 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 (x_1^{t+1},......
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1answer
51 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
79 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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1answer
152 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 $\mathbb{...