Questions tagged [graded-rings]
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)
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Hartshorne I.2.6 - questions about Boercherds' solution
Hartshorne Exercise I.2.6: $Y$ is a projective variety with homogeneous coordinate ring $S(Y)$, then $\dim (Y) = \dim S(Y) + 1$.
I think I'm not understanding a simple thing. It relates to the part of ...
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Filtration on field of fractions of graded ring
Let $R = \bigoplus_{i \in \mathbb{Z}} R_i$ be a commutative graded ring, that is, $R_i \cdot R_j \subseteq R_{i+j}$ for every $i,j \in \mathbb{Z}$. Assume that $R$ is a domain, so that it has a field ...
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If a graded ring $S$ is a UFD, then $S_0$ is a UFD
Let $S$ be a commutative graded ring ($S = \oplus_{n \in Z} S_n$). If $S$ is a UFD, can we deduce $S_0$ is a UFD?
I think $S_0$ is a UFD. Because we can gather all the finite product of the ...
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Homogenization of a prime ideal is prime
I'm studying from Qing Liu's book and I'm a bit stuck in his construction of $\operatorname{Proj}(B)$.
In particular Lemma 2.3.35 says that if $I$ is a prime ideal then $I^h= \bigoplus _{d\geq 0} I\...
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There can't only be non-homogenous prime ideals between two homogenous primes. (Vakil 12.2.G, part c)
I'm stuck and looking for advice on part c of Question 12.2.G of Vakil's FOAG. The question states:
(a) Suppose $X \subset \mathbb{P}^n$ is an irreducible projective $k$-variety. Show that the affine ...
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Blowup of a simplicial affine toric variety at the fixed point of the torus action
In this question, all cones are strongly convex, rational, polyhedral cones. We shall adopt the convention that, if a lowercase Greek letter $\sigma$ denotes a simplicial cone, then the uppercase ...
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What is $\operatorname{Proj}A[x]$ when $A[x]$ has the trivial grading?
$\newcommand{\Proj}{\operatorname{Proj}}
\newcommand{\p}{\mathfrak{p}}
\newcommand{\Spec}{\operatorname{Spec}}$
Let $A$ be a commutative ring, and $A[x]$ has the trivial grading where every element ...
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Proj construction and Nilpotent Homogenous Elements in Graded Ring [duplicate]
Let $A= \oplus_{n \ge 0} A_n$ a Noetherian graded ring and $f \in A_d$ a nonzero homogeneous element of degree $d>0$. The natural ring map $q:A \to A/(f)$ induces a well defined map $q^*: \...
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Quotient of Graded Rings $S \to S/f$ inducing Homeomorphism on Proj
Let $A$ be a Noetherian ring, and let $X$ be a closed subscheme of of $\mathbb{P}^r_A$. We define the homogeneous
coordinate ring $S:=S(X)$ of $X$ for the given embedding to be $A[x_0, ..., x_r]/I$ (...
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Graded ring generated by finitely many homogeneous elements of positive degree has Veronese subring finitely generated in degree one
Let $S=\bigoplus_{k\ge 0}S_n$ be a graded ring which is generated over $S_0$ by some homogeneous elements $f_1,\dotsc, f_r$ of degrees $d_1,\dotsc, d_r\ge 1$, respectively. I want to show that there ...
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Correspondence between $G$-graded $(A,A)$-bimodules and $G$-graded (left) $A^e$-modules
Let $G$ be a group (note $G$ is not necessarily abelian), and let $A$ be a $G$-graded algebra over a field $k$. We know that a $k$-central $(A,A)$-bimodule corresponds to a (left) $A^e$-module via $a \...
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On the homeomorphism $U_f\subset \operatorname{Proj}A\leftrightarrow \operatorname{Spec}(A_f)_0$
Let $A$ be a $\mathbb Z_{\geq 0}$ graded ring. Then we have that $\operatorname{Proj}A$ is the set of homogenous prime ideals which do not contain the irrelevant ideal $A_+$. We put a topology on this ...
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Name for algebra which is commutative up a group action
I am wondering if there is a name for an algebra which is commutative up to some group action.
To be more concrete, assume $A= \bigoplus A_n$ is a graded algebra, so $A_n \cdot A_m \subset A_{n+m}$, ...
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Projective and free modules
In the graded context, if $R = K[x_1,...,x_n]$ where $K$ is a field, is a projective R-module a free R-module?
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On the bijection between homogenous prime ideals of $A_f$ and prime ideals of $(A_f)_0$
Let $A$ be a $\mathbb{Z}^{\geq0}$ graded ring, and $f$ an element of positive degree. It is well known that the homogenous prime ideals of $A_f$ are in bijection with prime ideals of $(A_f)_0$, that ...
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