1
vote
1answer
62 views

The geometric interpretation for extension of ideals?

Suppose $f\colon B\to A$ is a ring homomorphism, and $I\subseteq B$ is an ideal. What's the geometric interpretation for the extension $f(I)A$ of the ideal $I$? Especially, I'm interested in the case ...
3
votes
0answers
109 views

Counterexamples for lcm-gcd identity and modular law for rings

In Miles Reid's Undergraduate Commutative Algebra, Exercise 1.3, we need to find counterexamples of lcm-gcd identity and modular law in the ring $A=k[X,Y]/(XY)$: $(I+J)(I\cap J)=IJ$; ...
1
vote
1answer
40 views

A condition for a homogeneous ideal to be prime

The following is the problem 11 of Chaper 8 Section 4 of Ideals, Varieties, and Algorithms by Cox, Little and O'Shea. A homogeneous ideal is said to be prime if it is prime as an ideal in ...
2
votes
1answer
40 views

Computing the Zariski cotangent space

I'm an extreme beginner with algebraic geometry and am trying to get used to things. Say I have some (algebraically closed) field $k$, in $k^2$ I want to compute the Zariski cotangent space, let's say ...
0
votes
0answers
61 views

What are the open and closed sets in $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)$?

What are the open and closed sets in $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)$ ? $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)=\{ (0),\ (\tilde{x}-a,\tilde{y}-b),\ b^2=a^3\}$.
0
votes
0answers
41 views

How to decide if an ideal in $\mathbf Q[X,Y]/(P)$ is principal?

Let $P(X,Y)$ be an irreducible polynomial in $\mathbf Q[X,Y]$. Given an ideal $I$ of the quotient ring $\mathbf Q[X,Y]/(P)$ (say given by a set of generators) how can I decide if $I$ is principal or ...
4
votes
1answer
60 views

$I(V_1)+I(V_2) \neq I(V_1 \cap V_2)$?

Let $V_1,V_2 \subset \mathbb{A}^n(k)$ affine varieties ($k$ field). I've proved $I(V_1)+I(V_2) \subset I(V_1 \cap V_2)$, but I don't know how to prove $\supset$. I think that's maybe because that ...
3
votes
1answer
86 views

Minimal primary decomposition

Let $m$ be an integer ${\geq}3$ and $f(x,y,z)=y^m(x+y^3)-z^3$ in $k[x,y,z]$. Find the singular points of $f$ and find a minimal primary decomposition of the jacobian of $f$. I find the set of ...
0
votes
1answer
60 views

generators of an ideal

I've been thinking about this exercise but I can't get the solution. In $\mathbb{R}^3$ , I consider the usual axis: $l_1=\{ x_1=x_2=0 \}$, $l_2=\{x_1=x_3=0\}$ and $l_3=\{ x_2=x_3=0 \}$. Calculate ...
2
votes
0answers
42 views

Ideal of a Vanishing set $I(V(F[X,Y]))$ and how to repeat the computation.

The video I am getting this from is found here: https://www.youtube.com/watch?v=spHxUPvrkXw, it is around 5 minutes in. The first part of the question is: for $F[X,Y] = Y^2 - X^3 = 0$ find ...
7
votes
2answers
111 views

Techniques for showing an ideal in $k[x_1,\ldots,x_n]$ is prime

An affine variety $X$ over a field $k$ is irreducible if and only if its defining ideal $I(X)$ is prime (in this post we use the convention that varieties are not necessarily irreducible). Hence, it ...
3
votes
1answer
40 views

Complexifications of degree 3 subschemes in $\mathbb A^2_{\mathbb R}$

I am trying unsuccessfully to solve exercise II-20 (page 65) from the book "The geometry of schemes" by Eisenbud and Harris. In this exercise it is stated that there are two non-isomorphic subschemes ...
4
votes
1answer
44 views

If $I\leq K[X_0,\dots,X_n]$ for $K$ a field is an ideal whose radical is homogeneous, is it always the case that $I$ is homogeneous?

If $I\leq K[X_0,\dots,X_n]$ (for $K$ a field, let's say algebraically closed) is an ideal whose radical is homogeneous, is it always the case that $I$ is homogeneous? I'm trying to understand ...
2
votes
0answers
42 views

Find the projective closure of the ideal $I=\langle y-x^2,z-x^3\rangle$

When I looked at this example, my first instinct was to homogenize only the generators of $I=\langle f_1 := y-x^2,f_2:=z-x^3\rangle$ in a new variable $w$. But then, I realized that I would miss some ...
2
votes
2answers
49 views

Non-radical ideal giving the empty set.

Let $R$ be the polynomial ring of $n$ variables over $\mathbb C$. It is known that a radical ideal $I (\ne R)$ defines a non-empty set $\mathbf V(I) \subset \mathbb C^n$. I am looking for a ...
2
votes
1answer
65 views

Algebraic Geometry and Maximal ideals

I am solving the following problem but couldn't figure out a strategy to solve: Does $(x^3-17, y^2)$ generate maximal ideals in the quotient ring $R=\mathbb{C}[x,y]/I$ where $I$ is the principal ...
3
votes
1answer
99 views

What does $J_1\cap J_2=\emptyset$ mean algebraically for two varieties in $\Bbb{C}^n$?

Let $J_1, J_2$ be two varieties in ${\Bbb C}^n$. Then $$ J_i=V(I_i)\quad i=1,2. $$ for some $I_i\subset\Bbb{C}[x_1,\cdots, x_n]$ and $$ J_1\cap J_2=V(I_1\cup I_2) $$ and $$ J_1\cup J_2=V(I_1I_2). $$ ...
15
votes
1answer
136 views

Is there a geometric meaning of a prime power not being primary?

I guess that the standard example of a prime power that is not a primary ideal is $$\mathfrak p^2 :=(x,z)^2\subset k[x,y,z]/(xy-z^2):=A.$$ Because $\mathfrak p^2 = (x^2,xz,xy)$, we see that $x\not ...
7
votes
1answer
107 views

Ideal with large Grobner basis with respect to one monomial order

What is an example of a set of at most four polynomials $f_1,\ldots,f_n$ (in any number of variables) such that $\{f_i\}$ is a Grobner basis of $I=\langle f_i\rangle$ with respect to one monomial ...
5
votes
2answers
69 views

Ideal for two polynomials in three variables

Consider the set $B=\{(t^2,t^3,t^4)\mid t\in \mathbb{C}\}$. It is a subvariety of $\mathbb{C}^3$, because it is equal to $V(y^2-x^3,z-x^2)$. How can we find the ideal $I(B)$? I think it is $I(\langle ...
2
votes
1answer
51 views

Ideal of variety in polynomial ring

Let $k$ be a field, let $A$ be an ideal of $k[x_1,\ldots,x_n]$, and let $B$ be an ideal of elements $f\in k[x_1,\ldots,x_n]$ such that $f^n\in A$ for some $n$. I want to show that $B\subseteq ...
3
votes
1answer
107 views

Exercise 1.2.4 and Example 4.3.6 in Liu

I want to prove that if $X$ is a noetherian scheme then any flat closed immersion into $X$ is open, that is, if $A$ is noetherian then $\varphi:\operatorname{Spec}(A/I)\to\operatorname{Spec}(A)$ is ...
3
votes
1answer
48 views

If singular set is finite then the ideal is radical

Let $F\in K[X,Y]$ and if the zero set $V(F,\frac{\partial F} {\partial x},\frac{\partial F} {\partial y})$ is finite then $\sqrt {(F)} = (F)$. I don't see the relation between $\frac{\partial F} ...
5
votes
1answer
151 views

What ideal is this?

Let $k$ be a field and $R = k[X]$ all polys over $k$ in $X$. Choose $p \in R$ and define $I_p = \{ f \in R : f\circ p(X) \in I \}$, where $I$ is some ideal in $R$. Then $I_p$ is an additive ...
2
votes
1answer
48 views

Irreducible algebraic sets with intersecting parts

Let $V = V(F)$ be an irreducible hypersurface in $A^n(k)$. To show: If $W$ is an irreducible algebraic set in $A^n(k)$ with $V \subset W$, then $V = W$. The ideas I got so far: Since $V, W$ are ...
3
votes
2answers
127 views

Is $(x^2+y^2-1,z-iy)$ a prime ideal in $\mathbb C[x,y,z]$?

Is $(x^2+y^2-1,z-iy)$ a prime ideal in $\mathbb C[x,y,z]$? How can I prove it? I need this to decompose the algebraic set $V(x^2+y^2-1,x^2-z^2-1)$ into irreductible components.
3
votes
4answers
256 views

Proof that the ideal $(xy, xz)$ in $\mathbb{A}^3$ is radical but not prime

The proof that $(xy, xz)$ is not prime seems easy. In particular, $xy \in (xy, xz)$, but neither $x$ nor $y$ is in $(xy, xz)$. On the other hand, I don't know how to prove that $(xy, xz)$ is ...
1
vote
1answer
75 views

Reading multiplicity of cusps , singularity etc from initial polynomial.

Here I have an example which I found. Can someone help me to understand what's happening here? The following are my concerns: 1) What do we need to do co-ordinate transformation? 2) How does the ...
0
votes
1answer
81 views

Prove fact about polynomial in uncountable fields

$F$-uncountable field. $I_{i}$-ideal in $F[x_{1},...,x_{n}]$ $F^{n}=\cup_{i=1}^{\infty}V(I_{i})$   $V(I_{i})\subseteq V(I_{i+1})$ Prove that $\exists k, V(I_{k})=F^{n}$ All that I've find is that ...
2
votes
3answers
98 views

Affine variety over a field

Suppose we have an algebraically closed field $K$. An affine variety is the common zero locus of a collection of polynomials $f_{\alpha} \in K[z_1, \dots, z_n]$. So basically it is the set of points ...
4
votes
2answers
97 views

Where do I use the fact that $F$ is algebraically closed in this proof?

I have to do the following. Let $F$ be an algebraically closed field. $I\in F[X_1,...,X_n]$ an ideal. Denote by $S(I)$ the subset in $F^n$ consisting of all $n$-tuples $(a_1,...,a_n)\in F^n$ such that ...
2
votes
2answers
209 views

Product of ideals corresponding to vanishing of points is equal to their intersection

Let $k$ be some field, and let $v,v',v''$ be three distinct points in $k^3$. Let $\mathfrak{m}_v = (X_1 - v_1,X_2 - v_2,X_3 - v_3)$ be the ideal in $k[X_1,X_2,X_3]$ corresponding to the polynomials ...
4
votes
3answers
299 views

Points and maximal ideals in polynomial rings

Let $k$ be a field, then I want to prove the following statement: for every $P=(b_1,\ldots,b_n)\in K^n$, the ideal $\mathfrak{m}_P=(x_1-b_1,\ldots,x_n-b_n)$ is maximal in the polynomial ring ...
1
vote
0answers
70 views

Looking for a binomial system solver

I am interested in solving binomial systems of the form $$ \begin{cases} a_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} + b_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} &= 0 \\ ...
1
vote
1answer
76 views

Ideals / Direct sum decomposition

Let $u = (u_1 , \ldots , u_n ) \in \mathbb{A}^n$. Let $I$ be the ideal of $A = \mathbb{C}[x_1 , \ldots x_n ]$ generated by the elements $x_1 - u_1 , \ldots , x_n - u_n$. (i) Show that as a ...
0
votes
0answers
80 views

places of function field and closed point of a scheme

Given an integral scheme $X$, let $K(X)=\mathrm{Frac}(R)$ be its function field, where $\mathrm{Spec}(R)$ is some non-empty open affine subscheme of $X$. Take the maximal ideal $P$ of some DVR of ...
0
votes
1answer
70 views

Graded rings and their localizations

Let $A$ be a $\mathbb{Z}_{\geq 0}$-graded ring, $f \in A$ - homogenious, and $I \subset A$ - homogenious ideal. Let $A_f$ be its localization, and $A_{(f)}$ - subring of elements of degree 0. How to ...
7
votes
4answers
423 views

Spectrum of $\mathbb{Z}[x]$

Can someone point me towards a resource that proves that the spectrum of $\mathbb{Z}[x]$ consists of ideals $(p,f)$ where $p$ prime or zero and $f$ irred mod $p$? In particular I remember this can be ...
1
vote
1answer
101 views

Comparing two different conditions for an ideal to correspond to a closed subscheme

Let $I$ be an ideal in the graded ring $S = A[x_0, \ldots, x_r]$. In Exercise II.5.10(a), Hartshorne defines the saturation $\bar I$ of $I$ to be the set $$\bar I = \{s \in S \mid \text{for all } i = ...
4
votes
3answers
920 views

Ideal of the twisted cubic

The twisted cubic is the image of the morphism $\phi : \mathbb{P}^1 \to \mathbb{P}^3 , (x:y) \mapsto (x^3:x^2 y:x y^2:y^3)$, it is given by $X = V(ad-bc,b^2-ac,c^2-bd)$. Now I would like to compute ...
1
vote
1answer
898 views

Saturated ideal

Let $k$ be a field, let $I \triangleleft k[X_1,\dots,X_n]=S$ be an ideal and fix $f \in S$. The saturated ideal of $I$ is $I^{sat}=I:f^\infty=\{g \in S \mid \exists m \in \mathbb{N} \ s.t. \ f^mg \in ...
2
votes
1answer
73 views

Hilbert function and isom of varieties

I'm just curious about a certain concept: If two ideals $I$ and $J$ in a polynomial ring $R$ have the same Hilbert function (note: I'm not talking about the Hilb polynomial), then are their supports ...
6
votes
2answers
255 views

(Minimal?) Polynomials using the Nullstellensatz

I'm struggling with an exercise that was asked in class: Let $\alpha = \sqrt[3]{3} + \sqrt{7}\sqrt[4]{2}.$ Show that there is a polynomial $p$ in the ideal $I=\left<a^3 - 3, b^2 - 7, c^4-2, ...
3
votes
1answer
200 views

Comparing algebraic varieties over a shared subset of variables

I'm currently experimenting with polynomial ideals and Gröbner bases, and I seem to be lacking some terminology/understanding. I have two systems of polynomial equations $P$ and $Q$ over a field ...
-1
votes
1answer
421 views

Doubt on class group

I started reading Class group after some one's advice ,so I got the following doubts,I would be happy if someone clarify the doubts, I understood that the class group measures the failure of the ...
4
votes
1answer
357 views

Is the number of prime ideals of a zero-dimensional ring stable under base change?

Let $A$ be a zero-dimensional ring of finite type over a field $k$ and let $X= \textrm{Spec} \ A$ be its spectrum. Note that $X$ is a finite set. Suppose that $k\subset K$ is a finite field extension ...