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### Finding a good homogeneous coordinate ring for a smooth projective variety

TL;DR Given a smooth projective variety $X\subseteq \mathbb{P}^m$ (I guess this means that the homogeneous coordinate ring $S(X)$ is regular whenever localizing at a non-maximal graded prime ideal). ...
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### When is $\text{gr}(V)\cong V$?

Let $V$ be a vector space with an increasing filtration $V_j, j\in \mathbb Z$, we assume the filtration is Hausdorff $\bigcap_j V_j=0$ and exhaustive $\bigcup_j V_j=V$. Consider the associated graded ...
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### Non trivial massey triple product [closed]

Suppose that $(A, d)$ is graded commutative algebra with three generators $a, b, c$, each of degree $1$ and $da = db =0$, $dc = ab$. Then, how can I construct a non-trivial Massey product? What ...
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### Algebra is noetherian if associated graded algebra is noetherian

$\newcommand{\cn}{\colon} \newcommand{\<}{\leqslant} \newcommand{\>}{\geqslant} \newcommand{\ss}{\subset} \newcommand{\gr}{\mathrm{gr}}$ Let $A$ be associative commutative algebra with unity ...
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### Let $S$ be a $\mathbb Z^{\ge 0}$-graded ring over $A$. If $S_+$ is finitely generated, then $S$ is a finitely generated graded $A$-algebra?

Let $S$ be a $\mathbb Z^{\ge 0}$-graded ring over $A$. Suppose $S_+= (a_1, \dots, a_n)$ where $a_i$ are homogeneous of positive degree. I want to show that $S$ is a finitely generated graded $A$-...
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### About the structure of a Hopf algebra on universal enveloping algebras of Lie algebras

We know that the universal enveloping algebra construction provides a functor from Lie algebras to cocommutative Hopf algebras which is left adjoint to the primitive functor. Furthermore, if we ...
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### Ideals in localisation of graded ring

Let $k$ be a field and $I$ a homogeneous ideal of $k[x_1,...,x_n]$. Consider the graded ring $R=k[x_1,...,x_n]/I$ with unique homogeneous maximal ideal $\mathfrak m=(x_1,...,x_n)R.$ Let $J$ be a ...
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### Is the Weyl algebra $A_1(\Bbb{C})$ graded?

For this question, the Weyl algebra is the algebra of differential operators on $\Bbb C$: $$A_1(\Bbb{C})=\Bbb{C}\langle x,\partial\rangle/(\partial x- x\partial -1)$$ Although for a lot of purposes it ...
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1 vote
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### $\Omega^*(M)$ as a graded algebra over the ring of smooth functions. Is there a notion of basis?

In Introduction to Smooth Manifolds by John M. Lee p.360, he defines $\Omega^*(M) =\displaystyle\bigoplus_{k=0}^n\Omega^k(M)$, the differential forms on a manifold $M$, as being an associative, ...
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