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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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### 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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### 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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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 ...
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### 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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### Sign convention for derivatives in a $\mathbb{Z}_2$ graded space

Suppose $V=V_0\oplus\theta V_1$ is a $\mathbb{Z}_2$ graded super vector space. Note: Since $\theta^2=0$, this implies $\theta\mathrm{d}\theta=-\mathrm{d}\theta\cdot\theta$. However, I wish to know ...
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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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### 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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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: $R \to ... 0answers 337 views ### Multiplicity and regular sequences We define multiplicity of a module$M$of dimension$d>0$as $$e(M) := \operatorname{lc} (P_M) (d-1)!,$$ where$P_M$denotes the Hilbert polynomial of$M$and$\operatorname{lc}(P_M)$its leading ... 0answers 217 views ### Construction of graded rings and modules In Algebraic Geometry and Homological Algebra - as far as I know - we often consider graded rings and modules so as to encode more information, say, some sort of (computational) complexity. For ... 0answers 84 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 68 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 ... 0answers 31 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 ... 0answers 114 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$W_\bullet$... 0answers 106 views ### Is the length of the composition series of a free module identical to the number of its bases? Let$A_0$be an Artinian ring,$M$a free$A_0$-module. Then, is the length of the composition series of$M$identical to the number of its bases? It seems to me that it is not. If$\mathfrak a$is ... 0answers 110 views ### 'Finitely generated modules' versus 'Finitely generated algebra' I'm reading Commutative Algebra by Bourbaki, and I'm on chapter III of the book. I really think that the author made a typo there, but I'm not so sure. It's Proposition 1, and its Corollary on page ... 0answers 175 views ### Computing ext over graded rings This question came up as I was reading Beilinson, Ginzburg, Soergel paper Koszul Duality Patterns in Representation Theory. Suppose that$A$is a Koszul ring (for the definition of Koszul ring ... 0answers 83 views ### Degree 1 elements in a graded ring from a blow-up perspective This may be an elementary question but I hope this question will benefit others as much as myself. Let$k^4 = Spec \; k[x_1, x_2, x_3, x_4]$. Writing$Bl_{\mathcal{I}}(k^4)$as$R = Proj(\...
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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 ...
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 ...