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23 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 ...
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30 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 ...
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82 views

Problem with Eisenbud's Lemma “Symmetry of Diagonalization”?

In his proof of Lemma A2.5 in his book Commutative Algebra with a View towards Algebraic Geometry, Prof. Eisenbud writes something like this: Let R be a commutative ring, $M$ an $R$-module, $S(M)$ ...
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79 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 ...
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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 ...
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118 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.)
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64 views

Grading on the graded direct product

This question is related to this one. Probably it's obvious but could you tell me what is the grading on the graded direct product? I was thinking about $^*\Pi M^i=\oplus_j(\Pi_i M^i_j)$ where ...
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$\mathrm{Hom}$ 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 ...
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38 views

Graded comodule isomorphism

Let $R$ be a semisimple ring. Let $C$ be a graded, connected $R$-coring (i.e. coalgebra in the monoidal category of $R$-bimodules), $C=\oplus_{n\geq 0} C_n$, such that $C_0 \simeq R$. Denote by $C_+$ ...
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54 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)= ...
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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 $f\in M$, ...
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61 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 ...