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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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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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### 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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### 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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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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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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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### 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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### 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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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 $R_*$? $M_* \otimes_{R_*} ... 1answer 64 views ### Shift of a simple graded module I am trying to understand the simple graded modules over a graded ring$R$(all the gradings over$\mathbb{Z}$). I know that there exists a bijection between the simple graded$R$-modules and simple ... 1answer 61 views ### Graded modules/algebras on commutative monoids? In the book Algebra of the Bourbaki group they deal with graded modules/algebras which are graded on a commutative monoid. What is the need to require the commutativity condition? Thanks. 2answers 127 views ### Questions on (subring/ submodule) of a graded (ring/ module) I have a question which seems a bit silly... If we have$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$... 1answer 28 views ### Sign convention when commuting shifts and tensor product In Markl, Schnider, and Stasheff's Operads in algebra, topology, and physics, they give the observation$\mathfrak{s}^{-1} \mathcal{E}nd_V \cong \mathcal{E}nd_{V[-1]}$(lemma 3.16) as motivation for ... 1answer 189 views ### Is the module of homomorphisms between graded modules also a graded module? If$M,N$are graded modules over a graded ring$R$, then is$\operatorname{Hom}_{R}(M,N)$also a graded module and how? 1answer 35 views ###$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$. Then ... 0answers 119 views ### Koszul sign convention and symmetric group action on the graded n-th tensor product Let$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$... 1answer 109 views ### Minimal free resolution of the twisted cubic This is exercise 13.15 in Harris' book "A First Course...". Let$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$... 0answers 178 views ### How to intuitively understand prolongations This question is concerned with the algebraic side of the theory of prolongations as explained in this paper by V. Guillemin and S. Sternberg. Let me first introduce my notation. We're working with a ... 1answer 274 views ### Definition of multiplicity Q.1. Bruns_Herzog define multiplicity (in the case of graded rings and modules) as My question is that: why multiplicity for$d=0$it is defined as$\ell(M)$? Is there a kind of ... 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 $$\... 1answer 219 views ### Classification of finitely generated multigraded modules over K[x_1,\ldots,x_n]? Let K be a field and R=K[x_1,\ldots,x_n]=\bigoplus_{a\in\mathbb{N}^n}Kx^a the multigraded polynomial ring. Have finitely-generated multigraded R-modules been classified? Are they of the ... 2answers 176 views ### Decomposition of finitely generated graded modules over PID I found this decomposition theorem used in a paper I'm reading, but it isn't referenced and I can't seem to find it in any of the books I have: Every graded module M over a graded PID decomposes ... 0answers 57 views ### Hilbert series of exterior algebra? Let the exterior polynomial algebra A=\Lambda_K[x_1,\ldots,x_n] have grading \deg x_i=d_i. Is there some nice formula for the Hilbert-Poincare series HP_A=\sum_k\dim_K \!A_k\,t^k\in\mathbb{Z}[[t]]... 1answer 40 views ### When is a D-module holonomic? Suppose that I have a small family of partial differential equations such that y(x) must be a solution to$$ P_i(x,D) y(x)=0,\ \ \ \ \ \ i=1,...,m$$for all$i=1,...,m$, where I am using multi-... 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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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 ...
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 $A \cong B?$