# Questions tagged [hilbert-polynomial]

For question about hilbert polynomial in commutative algebra.

49 questions
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### Two definitions of Hilbert series/Hilbert function in algebraic geometry

In classical algebraic geometry, suppose $I$ is a reduced homogeneous ideal in $k[x_0,\cdots,x_n]$, where $k$ is algebraically closed field, then $I$ cuts out a projective variety $X$, whose Hilbert ...
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### Turing Reduction, Hilbert's 10th Problem

I have the following problem : Using the fact the following language is undecidable H: The set of all multivariate polynomial with integer coefficients [p] such ...
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### Question on flat morphisms and Hilbert polynomials

I have been recently reading Dr. Kaledin's notes on algebraic geometry. There is a statement in lecture 16 about which I feel confused. Оказывается, что для пучков на проективном пространстве, ...
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### Binomial coefficients undefined in in the Hilbert polynomial for projective space

Let $k$ be a field and let $X= \mathbb{P}_{k}^{r}$ be the projective space (as a scheme) of dimension $r$ over $k$. Let $\mathcal{O}(d)$ denote the degree $d$ twisted structure sheaf. Then we define ...
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### Hilbert polynomial of the pullback of a line bundle, a particular case

I am working on a particular case of the following problem. Let $X$ be a projective algebraic surface, $L$ a base point free invertible sheaf on $X$ and $\varphi:X\rightarrow \mathbb{P}^n$ the ...
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### Hilbert Series of $k[x_1, … x_n]$

For the algebra $A = k[x_1, \dots, x_n]$ graded by degree. How does one find the Hilbert series. For a single variable, the hilbert series is simply $1+t+t^2+\dots = 1/(1-t)$.
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### Question about Hilbert polynomials and Reduction of ideals

Let $(R, m)$ be a C-M local ring with infinite residue field and $I$ an $m$-primary ideal. Does there always exist a minimal reduction $J$ of $I$ such that $r_J(I) = n(I) + d$?
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### Push-forward of coherent sheaves and Hilbert polynomials

Let $k$ be an algebraically closed field of characteristic zero, $X, Y$ be projective $k$-schemes. Fix closed immersions $i:X \hookrightarrow \mathbb{P}^n$ and $j:Y \hookrightarrow \mathbb{P}^m$ for ...
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### Multiplicity is always positive

Let $R$ be a homogeneous noetherian ring with $R_0$ artinian (e.g. $R = k[x_1,\dots, x_n]$), and $M$ a finitely generated graded $R$-module. I want to show that $e(M)$, the multiplicity of $M$, is ...
### In zero dimensional graded module the degree $1$ component is non zero.
I am reading the commutative algebra book by W. Bruns and H. Herzog. I am stuck at the Corollary 4.1.8 which comes from lemma 4.1.7. Actually in the corollary $d=0$ case does not come from the ...