In mathematics, in particular abstract algebra, a graded ring is a ring that is a direct sum of abelian groups $R_i$ such that $R_i R_j \subset R_{i+j}$. (Def: http://en.m.wikipedia.org/wiki/Graded_ring)

351 questions
Filter by
Sorted by
Tagged with
48 views

58 views

### Reference for: a nontrivially graded integral domain is never quasi-local

Let $\Gamma$ be a torsionless grading monoid and $R=\bigoplus_{\alpha\in\Gamma}R_\alpha$ be a $\Gamma$-graded integral domain. I'm interested in the following result: if $R$ is nontrivially graded,...
34 views

7 views

### How to extend to a unique derivation on the graded tensor algebra $TV$?

If $V$ is a finitely generated graded module and $TV$ its graded tensor algebra then, Any degree $k$ linear map $V \to TV$ extends to a unique derivation of $TV$. I found this in page 45 of the book "...
17 views

### Reason to apply the Koszul sign rule everywhere in graded contexts

I'm copy-pasting this question I asked in MO that received no answer. The Koszul sign rule is a sign rule that arises from graded commutative algebras. For instance, let $\bigwedge(x_1,\dots, x_n)$ ...
25 views

45 views

### Error in Hatcher on how multiplication is defined in $H^*(X;R)$?

Should the multiplication $(\sum_i \alpha_i) (\sum_j \beta_j) = \sum_{i,j} \alpha_i\beta_j$ actually be $(\sum_i \alpha_i) \smile (\sum_j \beta_j) = \sum_{i,j} (\alpha_i \smile \beta_j)$?
10 views

### Why is this sequence of graded algebras wrong?

I'm studying Hilbert sequences, and when trying to understand the proof of $h_A(t)= \frac{1}{(1-t)^m}$ when $A= k[x_1,\dots,x_m]$ ($m \geq 2)$, I'm kinda stuck. I can prove it using the standard ...
102 views

27 views

42 views

### generators of associated graded ring

Let $k$ a field, $A = k[x,y]/(y^2-x^3)$ and $\mathfrak{a} = (x,y)$ and let $G_{\mathfrak{a}}(A) := \bigoplus_{n \geq 0} \mathfrak{a}^n/\mathfrak{a}^{n+1}$ be the associated graded ring of $A$ with ...
121 views

### On the $k$-vector space dimension of graded pieces of an Artinian $k$-algebra $k[x,y]/J$

Let $R=k[x,y]$ be a polynomial ring in two variables over an infinite field $k$. Let $\mathfrak m=(x,y)$. Let $J$ be a homogeneous ideal whose radical is $\mathfrak m$. Consider the standard grading ...
58 views

Assume $A$ is a $\mathbb{Z}$-graded ring, $M$ is a graded $A$ module. If for any finitely generated homogeneous ideal $I \subset A$, the canonical map $I\otimes_{A} M \to IM$ is an isomorphism, how to ...
37 views

17 views

206 views

### Bihomogeneous Nullstellensatz

I'm reading 'Arithmetically Cohen-Macaulay Sets of Points in $\mathbb P^1\times\mathbb P^1$' by Elena Guardo and Adam Van Tuyl. (One can read it partially on Google books.) I doubt whether the '...
### The associated graded ring of the localization $k[x_1,\dots,x_n]_{(x_1,\dots,x_n)}$
I was reading the Atiyah-Macdonald p. 121: Example. Let $A$ be polynomial ring $k[x_1,\dots,x_n]$ localized at the maximal ideal $\mathfrak{m}=(x_1,\dots,x_n)$. Then $G_{\mathfrak{m}}(A)$ is a ...