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3
votes
1answer
93 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 The result is supposely similar to the well-known structure theorem in the non-graded case. So let $M$ be a ...
3
votes
1answer
84 views

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 ...
2
votes
1answer
87 views

Endomorphisms preserving bilinear form

Let $V=V_0 \oplus V_1$ be a $\mathbb{Z_2}$ finite dimensional graded vector space of dimension $2n$. Let $x_1,..,x_n,y_1,...,y_n$ be a basis of $V$ such that the $x_i$ form a basis for $V_0$ and the ...
1
vote
1answer
58 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 ...
0
votes
1answer
28 views

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 ...
9
votes
0answers
212 views

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 ...
5
votes
0answers
77 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 ...
4
votes
0answers
85 views

Flatness of homomorphisms of graded-commutative rings

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 ...
4
votes
0answers
206 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 ...
2
votes
0answers
297 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 ...
2
votes
0answers
71 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 $\mathfrak a$ ...
2
votes
0answers
128 views

$\mathrm{Hom}$ and $^*\mathrm{Hom}$ for graded modules: Exercise 1.5.19(f) of Bruns-Herzog

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 ...
2
votes
0answers
123 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 ...
2
votes
0answers
79 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 = ...
1
vote
0answers
80 views

Projective covers of graded modules.

I want to prove that there exist projective covers in the category of graded modules over an algebra. I am fairly new to "this" kind of mathematic and don't really know where to start: I found the ...
1
vote
0answers
68 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 ...
1
vote
0answers
35 views

Why do we have $\operatorname{Proj}(\oplus_{i=0}^\infty t^i \left< x\right>^i)\cong \operatorname{Proj}(\oplus_{i=0}^\infty t^i \left< x^2\right>^i)$?

I'm just curious but why is it that $$ \operatorname{Proj}\left(\oplus_{i=0}^\infty t^i \left< x\right>^i\right) $$ isomorphic to $$ \operatorname{Proj} \left(\oplus_{i=0}^\infty t^i \left< ...
1
vote
0answers
30 views

Understanding $Bl_{\mathcal{I}}(k^4)/S_2$ where $\mathcal{I}$ is defined by $(x_1-x_2,x_3-x_4)$

Let $k_4=Spec(k[x_1, x_2, x_3,x_4])$ and $\mathcal{I}$ is the ideal sheaf defined by $(x_1-x_2,x_3-x_4)$. Then $$ Bl_{\mathcal{I}}(k^4) = Proj (\oplus_{i\geq 0} I^i t^i) $$ where ...
0
votes
0answers
24 views

Confused with a terminology used in a book Group Representation Vol 1 Part A by G Karpilovsky

I am confused with a terminology used in a book Group Representation Vol 1 Part A by G Karpilovsky (Chapter 10, page 441). Which is as follows: Let $A=\oplus_{g \in G} A_g$ be a $G$-graded ...
0
votes
0answers
69 views

Extending a lemma about Castelnouvo-Mumford regularity

I'm learning Castelnouvo-Mumford regularity of associated graded ring. There is a lemma: It's from "Castelnuovo-Mumford regularity postulation number and relation types" by Markus Brodmann and ...
0
votes
0answers
45 views

Suggest a good book or reference on graded modules over polynomial rings

I am looking for reference books or papers on graded modules over the polynomial ring $k[x_0, \ldots, x_n]$. Any good commutative algebra text like Eisenbud's Commutative Algebra already contains a ...
0
votes
0answers
99 views

Graded version of Grothendieck's Non-vanishing Theorem

Is there a graded version of the Grothendieck Non-vanishing Theorem? (Theorem 6.1.4 of the book Local Cohomology. An Algebraic Introduction with Geometric Applications written by M. P. Brodmann and ...
0
votes
0answers
26 views

Splitting of Abelian Groups in Graded Modules

Suppose I have a graded module $M=\oplus M_i$ over a graded ring $R=\oplus R_i$. Assume $M_i\cong A \oplus B$, where A and B are abelian groups. Since $M$ is a graded ring, we know that $R_k(A\oplus ...
0
votes
0answers
119 views

When is $\operatorname{gr}_I (M)$ finite?

When is $\operatorname{gr}_I (R)$ (I mean associated graded ring of $I$) finite? When is $\operatorname{gr}_I (M)$ finite? ($R$ is Noetherian ring and $M$ finite $R$-module.)
0
votes
0answers
66 views

Extension of multiplication to the tensor algebra.

In this wikipedia article http://en.wikipedia.org/wiki/Tensor_algebra#Construction We construct $T(V)$ as the direct sum of vector spaces $T^kV$ for $k=0,1,2,…$ $$ T(V)= ...
0
votes
0answers
78 views

Question about initial forms

I am working through Eisenbud's Commutative Algebra, and in Chapter $5$ he defines the following map. Say we have a filtration of modules ${\cal F}:M=M_0\supset M_1\supset\cdots$. Then for $f\in M$, ...
0
votes
0answers
63 views

geometrical interpretation of $\mathbb{Z}/2\mathbb{Z}$ graded space

According to wikipedia, a $\mathbb{Z}/2\mathbb{Z}$ graded space (super vector space) $V$ is a a vector space which can be decomposed in a direct sum $V=V_0 \oplus V_1$ where elements of $V_0$ are ...
-1
votes
0answers
30 views

Some problems with Castelnuovo-Mumford regularity

I have some problems to complete my thesis. In this paper: "Upper bound for the Castelnuovo-Mumford regularity of associated graded modules" - Cao Huy Linh Lemma 4.1 Please explain for me about ...