# Tagged Questions

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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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Let $M$ and $N$ be graded $R$-modules (with $R$ a graded ring). $\varphi:M\rightarrow N$ is a homogeneous homomorphism of degree $i$ if $\varphi(M_n)\subset N_{n+i}$. Denote by $\mathrm{Hom}_i(M,N)$ ...
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### Path Algebra for Categories

For a while I had been thinking that the path algebra of a quiver $Q$ over a commutative ring $R$ is the same as the "category ring" $R[P]$ (analogous to "group ring", "monoid ring", "semigroup ring", ...
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### An elegant description for graded-module morphisms with non-zero zero component

In an example I have worked out for my work, I have constructed a category whose objects are graded $R$-modules (where $R$ is a graded ring), and with morphisms the usual morphisms quotient the ...
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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 ...
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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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### Graded modules over $k[t,t^{-1}]$

If $R=k[t,t^{-1}]$ is a graded ring where $R_0=k$ is a field and $t\in R$ is a homogeneous element of positive degree which is transcendental over $k$, how can I prove that every graded $R$-module is ...
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### 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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### Notes on dga's with a look towards Rational Homotopy Theory?

What is the best set of notes that give an introduction to differential graded algebras, preferably with ample examples and calculations, for someone that is ultimately interested in doing ...
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### Proof details of Theorem 11.1 in Atiyah-Macdonald

I have some trouble filling in the details of this proof from Atiyah-Macdonald. In this result, the authors assume what follows: 1) $A = \oplus_{n=0}^\infty A_n$ is a Noetherian graded ring, and ...
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### Blow-up along an ideal sheaf

Let $k^2=\operatorname{Spec} \; k[x,y]$ where $k$ is an algebraically closed field. Let $\mathcal{I}$ be the ideal sheaf defined by $(x,y)$. Then $$Bl_{\mathcal{I}}k^2$$ is covered by two open ...
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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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### 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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### 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 ...
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### 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). ...
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### Some questions on graded rings

At the moment I'm mostly interested in commutative graded rings (and in particular those graded over $\mathbb{N}$), but any comments or references about more general graded rings would also be ...
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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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### 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 ...
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How can we show that $\dim R/p=0\Leftrightarrow p=(x_{1},\ldots,x_{n})\Leftrightarrow R/p\simeq\mathbb{K}$, where $R=\mathbb{K}[x_{1},\ldots,x_{n}]$ is considered graded with standard grading (i.e. $\... 0answers 87 views ### Cones of max-Spec Let$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$is a finitely ... 0answers 98 views ### Structure theorem of modules in the graded case [duplicate] I ran into an exercise in D. Passman's book A Course in Ring Theory (page 130) asking me to prove the structure theorem of modules over PIDs in the graded case. Find all finitely generated graded ... 0answers 166 views ###$\mathrm{Hom}$and$^*\mathrm{Hom}$for graded modules: Exercise 1.5.19(f) of Bruns-Herzog [duplicate] Assume$R$is a graded ring and$M$and$N$graded modules. Denote by$^*\mathrm{Hom}_R(M,N)$the set of all homogeneous$R$-linear maps from$M$to$N$. How can I prove that if$M$is finitely ... 3answers 159 views ###$0$-th exterior power, empty product of modules and their tensor product$\def\finiteprod#1#2{#1_{1}\times#1_{2}\times\dots\times#1_{#2}}$In Lang's algebra he defines the tensor product as a universal object in the category of multilinear maps from$\finiteprod En$where ... 3answers 141 views ### What does$p+q=k$mean in the index of summation? I need help solving something I don't understand. OK so the problem is this: $$H^k(X,C)=\bigoplus_{p+q=k} H^{p,q}(X),$$ What does the$\;p+q=k\;$mean? Thank you anybody that helps! :) 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 ... 1answer 285 views ### Compute Hilbert function of a monomial ideal I'd like to know whether there exist easy methods that compute the Hilbert function of a graded$k$-algebra, without computer programs. My homework asks to me to compute the Hilbert function of$R/I$... 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 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 PoincarĂ© series. Suppose now that$P_A(x)=P_B(z)$. Question. Is it implies that$A \cong B?$2answers 176 views ### Decomposition of finitely generated graded modules over PID I found this decomposition theorem used in a paper I'm reading, but it isn't referenced and I can't seem to find it in any of the books I have: Every graded module$M$over a graded PID decomposes ... 1answer 119 views ### Poincaré series of quotient module I am trying to calculate the PoincarĂ© series$P(M,t)$with respect to the standard degree grading of the graded$\mathbb{C} [x,y,z,w]$-module$ M=\mathbb{C}[x,y,z,w]/I$, where$I = (x,w) \cap (z,w) $.... 1answer 68 views ### on exactness of the functors$M \mapsto \hat{M}$and$M \mapsto \hat{A}\otimes_{A}M$if$A$is a Noetherian ring,$M$a finitely generated module,$I$is an ideal of$A$, and$\hat{A}$is the$I-adic$completion of$A$, then we know$\hat{A}\otimes_{A}M\cong\hat{M}$. Also on Atiyah&... 1answer 106 views ### Does the category of graded rings have limits? Let$\mathfrak{C}$be the category of ($\mathbb{Z}$)-graded-commutative rings. Does this category have limits in it? I am particulary interested in power series rings over a field. Is there a ... 1answer 62 views ### Noetherian assumptions in basic properties of coherent sheaves of modules Using Hartshorne's definition of 'coherent sheaf': Proposition 5.11c Let$S$be a graded ring,$M$a graded$S$-module,$X=\operatorname{Proj} S$. Then$\tilde M$is a quasi-coherent$\mathscr O_X$... 1answer 64 views ### Shift of a simple graded module I am trying to understand the simple graded modules over a graded ring$R$(all the gradings over$\mathbb{Z}$). I know that there exists a bijection between the simple graded$R$-modules and simple ... 1answer 28 views ### Sign convention when commuting shifts and tensor product In Markl, Schnider, and Stasheff's Operads in algebra, topology, and physics, they give the observation$\mathfrak{s}^{-1} \mathcal{E}nd_V \cong \mathcal{E}nd_{V[-1]}$(lemma 3.16) as motivation for ... 1answer 274 views ### Definition of multiplicity Q.1. Bruns_Herzog define multiplicity (in the case of graded rings and modules) as My question is that: why multiplicity for$d=0$it is defined as$\ell(M)$? Is there a kind of ... 1answer 152 views ### About Betti Numbers I'm studying the book 'The Geometry of Syzygies' of David Eisenbud, but I'm having problem with the following step, in page 7 he says the we have a free resolution to the set of ten points in$\mathbb{...
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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 in ...
Let $S$ be a $\mathbb{Z}$-graded ring and $F$ a $\mathbb{Z}$-graded module that is free of finite rank $n$. Then we can write $F = \oplus_{i=1}^n S(\nu_i)$, where $S(\nu_i)$ is a graded ring ...
Let $A$ be a (skew-) commutative DG-algebra, and let $M,N$ be two DG $A$-modules. I am looking for a reference which describes the functor $-\otimes_A -$ and its basic properties (associtivity, ...