2
votes
1answer
44 views
Is there an open mapping theorem for affine morphisms?
Let $A$ and $B$ be rings. If $\varphi : A \longrightarrow B$ is such that $^a\varphi : Spec(B) \longrightarrow Spec(A)$ is bijective, then in what conditions $^a\varphi$ is a homeomorphism? Or, more ...
4
votes
1answer
76 views
+50
a case where contraction of a principal ideal is principal
Let $K$ be a field and $R_1,\cdots,R_n$ DVRs of $K$ with $m_i$ the maximal ideal of $R_i$. Define $A=\cap R_i$. Then $A$ is semilocal with maximal ideals $p_i=m_i \cap A$. Also, $A_{p_i} = R_i$.
...
5
votes
0answers
99 views
Question about the nullstellensatz for projective schemes
Assume that $ G $ is a graded ring. Assume that $A$ is a relevant homogeneous ideal (that is, it does not contain the irrelevant ideal $ \oplus_{n > 0}G_n$). I am having trouble proving the ...
11
votes
1answer
74 views
Uniformly solvable families of polynomials
It is a famous theorem that there is no "quintic formula", i.e. there is no formula which expresses the roots of a quintic polynomial $x^5+a_4x^4+\cdots+a_0$ in terms of $a_4,\ldots,a_0$ and rational ...
2
votes
0answers
26 views
Support of a direct sum of local cohomology modules
Let $R$ be a Noetherian ring with unit, $I$ be an ideal of $R$. Let $M$ be a finitely generated $R$ module. How can we show the following:
$$\operatorname{Supp}(\bigoplus_{j\ge ...
3
votes
1answer
42 views
All local cohomology modules being zero
Let $R$ be a Noetherian ring with unit, $I$ be an ideal of $R$ and let $M$ be a finitely generated $R$-module. Suppose $H_{I}^j(M)=0$ for all $j$, then how can one show that $M=IM$?
The converse of ...
2
votes
1answer
98 views
+50
proof of the Krull-Akizuki theorem (Matsumura)
This set of questions refers to the proof of the Krull-Akizuki theorem given in Matsumura's Commutative Ring Theory, pages 84-85. For those who don't have the text, i will provide the details.
The ...
4
votes
1answer
49 views
studying the topology of a real algebraic set
Let $f_1,\ldots,f_n \in \mathbb{R}[x_1,\ldots,x_m]$ be polynomials with real coefficients and let $I$ be the ideal that they generate. Denote by $V_{\mathbb{R}}(I)$ the corresponding real variety, ...
3
votes
0answers
41 views
why is an open faithfully-flat morphism fpqc?
Why is an open faithfully-flat morphism fpqc?
In other words, why must an open faithfully flat morphism $X\rightarrow Y$ have the property that around every $x\in X$, there is an open nbhd $U$ of ...
3
votes
0answers
57 views
Integral homomorphism induces a closed map on spectra
I'm trying to prove the following:
Let $f:A\rightarrow B$ is a integral homomorphism (e.g. $B/f(A)$ is a integral extension). Consider $f^{*}: \operatorname{Spec}B \rightarrow ...
6
votes
2answers
79 views
Coordinate ring in projective space. What are they?
When $X$ is an algebraic variety of affine $n$-space, then the coordinate ring of $X$ are polynomials restricted to $X$.
But when $X$ is a variety of projective $n$ space, what are the elements ...
4
votes
1answer
46 views
rational functions on projective n space
How to prove that the field of rational functions on whole of projective n space is constant functions. By rational function I mean quotients of homogeneous polynomials of same degree ...
2
votes
1answer
91 views
What does “Hauptidealsatz” mean in “Krull's Hauptidealsatz”?
What does "Hauptidealsatz" mean in "Krull's Hauptidealsatz"? Thank you very much.
4
votes
1answer
60 views
Primary decomposition of power of a prime.
Let $R$ be a commutative Noetherian ring with unit. Suppose $P$ is a prime ideal that is not maximal. How can we go about finding a normal (reduced) primary decomposition of the power of $P$, say a ...
10
votes
1answer
158 views
Classification of local Artin (commutative) rings which are finite over an algebraically closed field.
A result in deformation theory states that if every morphism $Y=\operatorname{Spec}(A)\rightarrow X$ where $A$ is a local Artin ring finite over $k$ can be extended to every $Y'\supset Y$ where $Y'$ ...
-1
votes
1answer
68 views
Irreducible polynomials and affine variety
Let $k$ be any field, and let $f,g\in k[x,y]$ be two irreducible polynomials such
that $g$ is not divisible by $f$. Prove that $V(f,g)\subseteq A_k^2$ is finite.
3
votes
1answer
34 views
square system of polynomial equations having infinite number of solutions
Suppose we have a system of $n$ polynomial equations in $n$ unknowns over $\mathbb{C}$ and suppose that the corresponding ideal generated by these equations is not the unit ideal $(1)$. Under what ...
2
votes
0answers
137 views
Finite morphisms of schemes are closed
I want to prove that finite morphisms of schemes are closed, but I cannot prove the affine case, namely:
Given a finite morphism of rings $\varphi :B \to A$ prove that the induced morphism of ...
-1
votes
1answer
72 views
Question about local ring (from a sentence in Hartshorne)
$A$ is a noetherian domain with the property that $A_{\mathfrak q}$ is a DVR for any prime $\mathfrak q \subset A$ of height $1$. $K$ is the fraction field of $A$, and $f \in K(t)$ is a nonconstant ...
3
votes
1answer
62 views
All the Associated Primes are minimal.
Let $R$ be a commutative Noetherian ring with unit and let $I$ be a fixed ideal. I am sorry if the following turns out to be a very silly question.
1) Suppose $\operatorname{Ass}(R/I)$ are all ...
1
vote
1answer
62 views
Localization of $K[x,y|x^2-y^3]$ and $K[x,y|xy]$ at $\langle x,y\rangle$ and $\{\text{non-zero-divisors}\}$ (exercise in SICA)
In Greuel & Pfister's A Singular Introduction to Commutative Algebra, p. 38, there is written:
So we have rings
$$\begin{array}{l l}
R_1:= K[x,y|x^2\!-\!y^3], & R_4:= K[x,y|xy],\\
R_2:= ...
4
votes
1answer
90 views
Vakil 14.2.E: $L\approx O_X(div(s))$ for s a rational section.
I am working through Vakil's Ch 14 (march2313 version) on invertible sheaves and am having trouble on 14.2.E.
The question (in notation to be defined) is this: how do I show that each point in the ...
1
vote
1answer
41 views
Finding a presentation of $A$-algebra $B$
Find a presentation of the $A$-algebra $B$, where $B=\mathbb{Z}[1/2]\subseteq \mathbb{Q}$ and $A= \mathbb{Z}$.
I want to prove it but I can't understand what want to me! Please describe to me.
5
votes
1answer
59 views
Function field question from Silverman's AEC
Just before Proposition 1.7 on page 5 of AEC (2nd ed), Silverman defines $M_P$ as an ideal in the affine coordinate ring.
Then he states Proposition 1.7 (the intrinsic characterization of ...
3
votes
0answers
33 views
Computing a rational function at a point in terms of a uniformising parameter
I am not quite sure how to ask this precisely, but vaguely I would like to know how difficult it is to write a function on an algebraic curve at a point $P$ as a power series of a uniformising ...
7
votes
0answers
94 views
Class group of $k[x,y,z,w]/(xy-zw)$
I had a homework problem (II.6.5 in Hartshorne) to compute the (Weil divisor) class group of $X=\operatorname{Spec} k[x,y,z,w]/(xy-zw)$. I have accomplished this; however, I used some results I don't ...
2
votes
0answers
48 views
Uniqueness of minimal resolution
Let $R$ be a domain, and $a_1,\dots,a_r$ be a regular sequence of $R$. Let $b_1,\dots,b_r$ be another regular sequence, such that two regular sequences generate the same ideal, i.e. ...
11
votes
1answer
113 views
Intersection of powers of maximal ideals
Let $A=\mathbb K[X_1,\ldots,X_n]$ be a polynomial ring over some field $\mathbb K$. Let $\mathfrak p\subseteq A$ be a prime ideal. Let $Z(\mathfrak p)=\{ \mathfrak m\subset A\text{ maximal}\mid ...
13
votes
1answer
256 views
$\mathbb A^n(k)$ and $\mathbb A^n(k)\setminus \{0\}$ are not homeomorphic
Let $k$ be an algebraic closed field. Why $\mathbb A^n(k)$ and $\mathbb A^n(k)\setminus\{0\}$ (for $n>1$) are not homeomorphic with respect to the Zariski topology?
2
votes
1answer
48 views
Proposition 3 in Chapter I.7 (Dimension) of Mumford's Red Book
In Mumford's Red book, chapter I.7 (Dimension), the proof of Proposition 3 (1.) has the step:
If $B=f^{\star -1}(A)$, apply the going-up theorem to $S/B\subset R/A$.
What does the inclusion ...
3
votes
0answers
70 views
Finite type ring extension + condition = finite extension?
Is the following true ?
If $A \subset B$ is finite type extension (i.e. $B$ is a finitely generated $A$-algebra) of integral domains such that the set $\{\mathfrak ...
5
votes
1answer
199 views
Intuition behind Hilbert's Nullstellensatz
maybe that's a pointless question, however I'm having problems in "understanding" (accepting) the Hilbert's Nullstellensatz. I understand the proof, however I cannot understand the concept in a more ...
5
votes
1answer
108 views
What points of affine space can be mapped to zero by an étale morphism?
Let $K$ be a field and $n$ a positive integer.
For what points $x\in\mathbb{A}^n_K$ can I find an étale morphism $f_x:\mathbb{A}^n_K\to \mathbb{A}^n_K$ mapping $x$ to zero and how does such a ...
4
votes
1answer
75 views
simple application of Bezout's Theorem
Let $f(x),g(x) \in \mathbb{C}[x_1,\cdots,x_n]$ be two irreducible homogeneous polynomials of degree $n,m$ respectively. Does Bezout's Theorem say that the system of equations $f(x)=0, g(x)=0$ has ...
3
votes
1answer
36 views
Algebraic description of a stalk in the fppf topology
Let $X$ be a scheme and $x\in X$ a point. The stalk of $X$ at $x$ in the Zariski topology is the local ring $\mathcal{O}_{X,x}$. The stalk of $X$ at $x$ in the étale topology is the strict ...
0
votes
0answers
67 views
Rational functions on the punctured affine plane. [closed]
How to show that the set of rational functions on $\mathbb A^2$ defined on $\mathbb A^2-(0,0)$ is $K[x,y]$. Can this be generalized?
0
votes
2answers
75 views
Valuation but not Noetherian Rings
For valuation rings I know examples which are Noetherian.
I know there are good standard non Noetherian Valuation Rings. Can anybody please give some examples of rings of this kind?
I am very ...
1
vote
1answer
56 views
Looking for a “prime-ish” family of subsets
Is there a nontrivial (what I mean is below) example of a compact Hausdorff space $X$ and a family $\mathscr{F}$ of subsets of $X$ with the following pair of properties?
$\mathscr{F}$ is ...
3
votes
1answer
42 views
Question about algebraically independence.
Let $R=k[Y_1, \ldots, Y_m]/P$, where $k$ is a field and $P$ is a prime ideal of $R$. Suppose that $Y_1, \ldots, Y_m$ are algebraically independent over $k$. Let $y_1=Y_1+P, \ldots, y_m=Y_m+P$. Can we ...
4
votes
2answers
66 views
Identifying the ideal generated by the variety $V(y^2-x^3)$
I am having trouble showing the following result:
Suppose that $k$ is an infinite field and consider the affine variety $V(y^2-x^3)$. If $I(V)$ denotes the ideal of all polynomials vanishing on ...
3
votes
1answer
84 views
Koszul complex of locally free sheaves
Let $X$ be a complex variety; one can also assume it is smooth if this helps. $\mathcal{E}$ is a locally free sheaf of rank $r$ on $X$, and $s \in H^0(X, \mathcal{E})$. Then one has a Koszul complex ...
13
votes
2answers
146 views
Is a linear combination of minors irreducible?
Let $X=(X_{ij})_{1\le i,j\le n}$ be a matrix of indeterminates over $\mathbb C$. For choices $I,J\subseteq\{1,\ldots,n\}$ with $|I|=|J|=k$ denote by $X_{I\times J}$ the matrix $(X_{ij})_{i\in I,j\in ...
2
votes
1answer
61 views
Homogeneous forms of degree $d$ in quotient ring
We have a nice description for the space of all homogeneous elements of degree $d$ in $R = k[x_1,\ldots,x_{n+1}]$, namely it is isomorphic to $$(x_1,\ldots,x_{n+1})^d/(x_1,\ldots,x_{n+1})^{d+1}.$$
...
10
votes
1answer
146 views
Geometric meaning of completion and localization
Let $R$ be a commutative ring with unit, $I$ an ideal of $R$ and consider the following three constructions.
The localization $R_I$ of $R$ at $I$ (i.e. the localization of $R$ at the multiplicative ...
1
vote
0answers
29 views
When is the completion of an A-algebra at a height-1 prime just A[[X]]?
Let $A$ be a ring (commutative with unity), and let $B$ be a regular finite-type $A$-algebra of relative dimension 1 over $A$. (ie, Spec $B$ is a regular curve over $A$).
Let $\mathfrak{p}$ be a ...
6
votes
2answers
114 views
Irreducible components of the variety $V(X^2+Y^2-1,X^2-Z^2-1)\subset \mathbb{C}^3.$
I want to find the irreducible components of the variety $V(X^2+Y^2-1, \ X^2-Z^2-1)\subset \mathbb{C}^3$ but I am completely stuck on how to do this. I have some useful results that can help me ...
9
votes
2answers
179 views
Usefulness of completion in commutative algebra
After studying about the completion of a module $M$ over a ring $A$ (e.g. $I$-adic completion), I am left with the following questions:
(i) What is the usefulness of the concept of completion in ...
7
votes
1answer
202 views
constructing a projection onto a variety
Consider the vector space $\mathbb{C}^n$. Given any linear subspace $S$ we can choose a complement of $T$ in $V$, i.e. $\mathbb{C}^n=S \oplus T$ and we can subsequently define a projection ...
3
votes
2answers
88 views
Regular in codimension 1
Apologies if this is an obvious question. I've really gotten my head tangled up in knots trying to approach it from the right angle, and I'm not getting anywhere - so I thought I'd ask.
A scheme is ...
6
votes
4answers
110 views
dimension of a coordinate ring
Let $I$ be an ideal of $\mathbb{C}[x,y]$ such that its zero set in $\mathbb{C}^2$ has cardinality $n$. Is it true that $\mathbb{C}[x,y]/I$ is an $n$-dimensional $\mathbb{C}$-vector space (and why)?
