Search Results

This class of rings is studied by Arnold and Gilmer in the paper Jimmy T. Arnold and Robert Gilmer 'The Dimension Sequence of a Commutative Ring' American Journal of Mathematics , Vol. 96, No. 3 (19 …

For 1), Nagata's counterexample goes as follows: Let $\mathbb{N}=A_1\cup A_2 \cup \cdots$ be a partition such that $|A_i|<|A_{i+1}|$. Take $S=k[x_1,x_2,\ldots]$, prime ideals $I_i=\langle x_i | i\in …

I think this problem can in fact be handled by Gröbner basis theory in the case $A$ is a polynomial ring. Since $I\cdot J \subseteq I\cap J$ for any two ideals, one can simply compute a Gröbner basis …

It is easy to make examples of such subrings. For example, take $A=k[x,y]$ and consider the subring $$ B=k[x^a y^b : 0\le \frac{b}{a}<\sqrt{2}]. $$Geometrically, $B$ is spanned by monomials whose e …

I guess what you are after is the Hilbert function of ideal. More precisely, if you have a multigraded polynomial ring $k[x_1,\ldots,x_n]$ and the ideal is given by $I=(f_1,\ldots,f_r)$, the Hilbert f …

Let $X\subset \mathbb{P}^n$ be a smooth projective variety with ideal sheaf $I_X$. The conormal sequence is given by $$ 0\to I_X/I_X^2\to \Omega_{\mathbb{P}^n}|_X\to \Omega_{X}\to 0. $$ For which var …

Let $R$ be a Noetherian ring and let $I$ be an ideal in $R[x]$. Then the following facts hold: $I$ is prime in $R[x]$ $\Longleftrightarrow$ $I\cap R$ is prime in $R$ and $\overline{I}$ is prime in $ …

The geometric meaning of Corollary 18.17 in Eisenbud is that given an equidimensional scheme $X$ of dimension $n$ and a surjective morphism $\pi:X\to V$ where $V$ is a regular scheme of dimension $n …

The most basic motivation for studying resolutions is computing the Hilbert function/polynomial of $Y=V(I)$. The geometric information in this polynomial is essentially the dimension, the embedded deg …

This follows from the Hilbert syzygy theorem. If $R=\mathbb{C}[x_0,\ldots,x_n]^G$ and $R^G$ is generated by elements $f_1,\ldots,f_r$ of degrees $a_1,\ldots,a_r$ respectively, $R^G$ has a free resolut …

Let $X$ be a smooth projective variety and $Z$ a closed subscheme. Let $X'$ be the blow-up of $X$ with center $Z$. Under what conditions on $Z$ is $X'$ Cohen-Macaulay? In the case $Z$ is non- …