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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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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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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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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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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 $$0 \to M(1) \... 0answers 13 views Free graded k[x] modules have homogeneous bases I was reading the article "Cary Webb. Decomposition of graded modules. Proceedings of the American Math- ematical Society, 94(4):565–571, 1985" where in the beginning "Free graded k[x] modules have ... 1answer 88 views Castelnuovo-Mumford regularity and exact sequence. In a question on MathOverflow it is said that: It is known that given a short exact sequence of finitely generated graded modules over a polynomial ring over a field:$$0 \to M'' \to M \to M' \to 0$... 0answers 39 views Reflexive Graded Module Let$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$is reflexive as ... 1answer 62 views Noetherian assumptions in basic properties of coherent sheaves of modules Using Hartshorne's definition of 'coherent sheaf': Proposition 5.11c Let$S$be a graded ring,$M$a graded$S$-module,$X=\operatorname{Proj} S$. Then$\tilde M$is a quasi-coherent$\mathscr O_X$... 0answers 87 views Cones of max-Spec Let$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$is a finitely ... 0answers 32 views Compute directly that the mapping cone of a homotopy equivalence is contractible Let's consider the category$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$is a homotopy equivalence then its ... 2answers 32 views Example of a graded module that is not a ring. I'm looking for an example of a graded module, that is not a ring. All the examples of graded modules that I have come across, like$k[x_1,x_2,\dots,x_n]$are all graded rings. Thanks in advance 2answers 45 views Support of a tail of a graded module. Suppose that$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$): $$\text{Supp}_R M=\... 0answers 21 views Finite generation and associated graded modules My question is as follows: Let R be a ring and let M a right R-module. Suppose that (F_i)_{i \geqslant 0} is a filtration of R and (M_i)_{i \geqslant 0} is a compatible filtration of ... 1answer 30 views When are minimal faithful modules over algebras unique? Let A be a unital associative algebra over \mathbb{C}. We say an A-module V is "minimal faithful" if (i) V is faithful and (ii) V does not have a proper submodule that is faithful. First ... 1answer 46 views Poincaré series and the Hilbert polynomial of A = A_0[X_1,\dots , X_s] [closed] Let A = A_0[X_1, \dots , X_s] be a polynomial ring in s variables over an Artin ring A_0. This is a graded ring, and can be regarded as a graded module over itself. 1. What are the ... 1answer 72 views Poincaré series and Hilbert polynomial of some graded modules [closed] Let k be a field, and let k[X, Y ] be the polynomial ring in two variables equipped with the usual grading such that \deg(X) = \deg(Y ) = 1. Consider the ideals I = (X, Y^2) and J = (X^2, Y^2)... 1answer 33 views where do elements go under multiplication in a graded module? Assume that M = \bigoplus_{n = 0}^\infty M_n is a graded A-module, where A = \bigoplus_{n = 0}^\infty A_n is a graded ring. We have by definition A_m M_n \subset M_{m + n}. Does this mean that ... 1answer 89 views Explanation of a proof about graded module structure Let \Bbb F be a field and M a finitely generated \Bbb F[x]-module. The structure theorem for modules over a PID says that$$ M\cong \Bbb F[x]^r\oplus\biggl(\bigoplus_{j=0}^s\Bbb F[x]/(f_j(x))\... 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$S(-a)$. ... 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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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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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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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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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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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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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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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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$M/x_nM$ 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 ...
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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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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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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 ...