Tagged Questions
3
votes
0answers
52 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 ...
10
votes
1answer
155 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
vote
1answer
60 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:= ...
6
votes
1answer
57 views
Showing that $\mathbb{C}[x,y]^{\mu_n}$ and $\mathbb{C}[x,y,z]/(xy-z^n)$ are isomorphic as rings
The problem: Let $\mu_n$ act on $\mathbb{C}[u,v]$ with weights $(1,-1)$. I would like to show that the rings $\mathbb{C}[u,v]^{\mu_n}$ and $\mathbb{C}[x,y,z]/(xy-z^n)$ are isomorphic.
Explanation of ...
2
votes
1answer
52 views
Bijection between hom sets of $k$ - algebras
Let $R:= k[x_1,\ldots,x_r]$, $S:= k[x_{r+1},\ldots,x_{r+s}]$ and $Q:= k[x_1,\ldots,x_{r+s}]$. Let $I \subseteq R$ and $J \subseteq S$ be ideals. I have in texts in algebraic geometry that for any $k$ ...
4
votes
2answers
91 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 ...
1
vote
0answers
41 views
Morphisms from the group variety of $n$-th roots
In Milne - Lectures on étale cohomology, example 6.10 i came across the following.
We fix a variety $X$ and work in the category $Var/X$ of varieties over $X$ (so with fixed morphisms to $X$!) and ...
6
votes
1answer
129 views
A valuation ring
In Qing Liu, Algebraic Geometry and Arithmetic Curves, page 116, exemple 4.1.8, one has $\mathcal{O}_K$ a discrete valuation ring with uniformizing parameter $t$, $P\in\mathcal{O}_K[S]$ an Eisenstein ...
4
votes
0answers
77 views
Some elementary facts
What is the simplest and the most conceptual proof of some basic facts on algebraic geometry?
1) Hilbert's Nullstellensatz
2) Regular functions on projective variety - only constants
3) elemination ...
6
votes
1answer
139 views
Questions about subalgebras of finitely generated $k$-algebras
Let $k$ be a field (if necessary assume $k$ to be algebraically closed). Let $A$ be a finitely generated $k$-algebra and let $B$ be a subalgebra of $A$. Remark that $B$ doesn't have to be noetherian, ...
1
vote
1answer
119 views
What is Proj $\mathbb{C}[x,y][z]/\langle xz-yz\rangle$?
Assuming that $x,y$ have weight $0$ and $z$ has weight $1$,
$$
R= \mathbb{C}[x,y][z]/\langle xz-yz\rangle = \mathbb{C}[x,y]\oplus
( \oplus_{i\geq 1}\mathbb{C}[x]z^i),
$$
what closed subvariety is ...
0
votes
1answer
57 views
Graded rings and their localizations
Let $A$ be a $\mathbb{Z}_{\geq 0}$-graded ring, $f \in A$ - homogenious, and $I \subset A$ - homogenious ideal. Let $A_f$ be its localization, and $A_{(f)}$ - subring of elements of degree 0. How to ...
3
votes
1answer
146 views
Discrete Valuation Rings
Let $V = \mathbb A^1(k)$ ($k$ is an algebraically closed field), $\Gamma(V) = k[X]$ and let $K = k(V) = k(X)$.
Prove that for each $a \in k = V$, $\mathcal{O}_a(V) := \{f\in K(V): f$ is defined at ...
3
votes
0answers
50 views
Arithmetic progressions of units in a domain
Let $R$ be a domain with unity, and suppose that $R^\times$ has finite rank as an abelian group.
Can $R^\times$ contain infinitely long arithmetic progressions?
Can $R^\times$ contain arithmetic ...
2
votes
1answer
57 views
Limits of subrings and surjectivity
Let $A$ be a ring and let $\mathcal{F}$ be the inductive system of subrings of $A$ which are of finite type over $\mathbb{Z}$:
$$
\mathcal{F} = \{ \mathbb{Z}[a_1,\dots,a_n] \subseteq A \mid n \geq 0, ...
4
votes
1answer
102 views
Ring homomorphism and affine scheme
How to describe all ring homomorphisms $f: A \rightarrow B$, such that
corresponding affine scheme morphism $f: Spec \, B \rightarrow Spec \, A$ is open immersion?
5
votes
2answers
140 views
Injectivity of Homomorphism in Localization
Let $\alpha:A\to B$ be a ring homomorphism, $Q\subset B$ a prime ideal, $P=\alpha^{-1}Q\subset A$ a prime ideal. Consider the natural map $\alpha_Q:A_P\to B_Q$ defined by ...
1
vote
1answer
143 views
Krull dimension and transcendence degree
What is the simpliest proof of the fact,
that integral algebra $R$ over a field $k$
has the same Krull dimension as transcendence
degree $deg.tr_k R$?
Is it possibple to use only Noether normalization ...
6
votes
4answers
197 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
170 views
Irreducible Components of the Prime Spectrum of a Quotient Ring and Primary Decomposition
Recently I encountered a problem (the first exercise from chapter four of Atiyah & McDonald's Introduction to Commutative Algebra) stating that if $\mathfrak{a}$ is a decomposable ideal of $A$ (a ...
4
votes
2answers
99 views
Finite presentation of algebra of invariants
(1) Let $R$ be a ring, let $A$ be a finitely presented $R$-algebra, and let $G$ be a finite group of $R$-automorphisms of $A$. Is the algebra of invariant $A^G$ finitely presented over $R$?
I can ...
4
votes
1answer
215 views
Invertible elements in the ring $K[x,y,z,t]/(xy+zt-1)$
I would like to know how big is the set of invertible elements in the ring $$R=K[x,y,z,t]/(xy+zt-1),$$ where $K$ is any field. In particular whether any invertible element is a (edit: scalar) multiple ...
3
votes
1answer
74 views
Are minimal prime ideals in a graded ring graded?
Let $A=\oplus A_i$ be a graded ring. Let $\mathfrak p$ be a minial prime in $A$, is $\mathfrak p$ an graded ideal?
Intuitively, this means the irreducible components of a projective variety are also ...
1
vote
2answers
64 views
Understanding the image under the map $\mathbb{C}[t]\stackrel{f^*}{\rightarrow} \dfrac{\mathbb{C}[x,y]}{\langle xy\rangle}$ given by $t\mapsto x+y$
Consider the map of affine schemes
$$
\operatorname{Spec}\left( \dfrac{\mathbb{C}[x,y]}{\langle xy\rangle }\right)\stackrel{f}{\rightarrow} \operatorname{Spec}\mathbb{C}[t]
$$
whose corresponding ...
1
vote
3answers
105 views
The closure of $\overline{\{x\}}$ being irreducible and relating the generic point to its associated irreducible scheme
If $x$ is a point in $X$ where $X$ is a scheme, we write $\overline{\{ x\}}$ for the closure of $x$ in $X$.
$\mathbf{Question \;1}$: I am a bit confused why $\overline{\{ x\}}$ is irreducible. ...
8
votes
0answers
108 views
Tensoring is thought as both restricting and extending?
I hope these questions are not too trivial.
Let $I$ be an ideal in $R$. Write $I'\subseteq R[t]$. Then the notion of tensoring
$$
(R[t]/I')\otimes_{\,\mathbb{C}[t]} \mathbb{C}[t]/\langle t-c ...
0
votes
0answers
52 views
Complete intersection but not a domain
This may be a rather trivial question but here it goes.
I am looking for a scheme defined by (more than one irreducible and reduced) homogeneous equation in a polynomial ring that is a complete ...
3
votes
1answer
110 views
Difference between $\left< x\right> \cap \left< x,y\right>^2$ and $\left< x,y\right>^3$
Consider the ideals $I = \left< x\right> \cap \left< x,y\right>^2 = \left<x^3,x^2y, xy^2\right>$ and $J=\left< x,y\right>^3=\left< x^3, x^2y, xy^2, y^3
\right>$ in ...
1
vote
0answers
29 views
Understanding $Bl_{\mathcal{I}}(k^4)/S_2$ where $\mathcal{I}$ is defined by $(x_1-x_2,x_3-x_4)$
Let $k_4=Spec(k[x_1, x_2, x_3,x_4])$ and $\mathcal{I}$ is the ideal sheaf defined by $(x_1-x_2,x_3-x_4)$. Then
$$
Bl_{\mathcal{I}}(k^4) = Proj (\oplus_{i\geq 0} I^i t^i)
$$
where ...
2
votes
0answers
70 views
Degree 1 elements in a graded ring from a blow-up perspective
This may be an elementary question but I hope this question will benefit others as much as myself.
Let $k^4 = Spec \; k[x_1, x_2, x_3, x_4]$.
Writing $Bl_{\mathcal{I}}(k^4)$ as $R = ...
0
votes
0answers
58 views
understanding a graded ring in geometric terms
Consider a ring $k[x_1,x_2,z]$ where the variables $x_1$ and $ x_2$ have degree $0$ and $z$ has degree 1 and $k$ is an algebraically closed field.
It is clear that $Proj(k[x_1,x_2,z])=k^2 \times ...
2
votes
2answers
150 views
What if $\operatorname{char}\mathbb{K}$ is not $0$ or if $\mathbb{K}$ is not algebraically closed? (Nullstellensatz)
Given a field $\mathbb{K}$ which is algebraically closed and of characteristic 0, we can say exactly what the maximal ideals of $\mathbb{K}[x_1,\dots,x_n]$ are and they correspond to points in ...
3
votes
1answer
111 views
On limits, schemes and Spec functor
I have several related questions:
Do there exist colimits in the category of schemes? If not, do there exist just direct limits? Do there exist limits? If not, do there exist just inverse limits? ...
5
votes
1answer
129 views
Integral closure in the total ring of fractions
My question is linked with normalization of reduced algebraic curves that are not necessarily irreducible.
Let $(A,\mathfrak{m})$ be a local reduced noetherian ring with Krull dimension $1$, let ...
1
vote
2answers
117 views
can singular points become nonsingular after a base change
Let $X$ be a normal surface over a field $k$. Assume that $X$ is singular.
Does there exist a field extension $L/k$ (finite or infinite) such that $X_L$ is nonsingular?
The answer is no in general. ...
5
votes
3answers
186 views
When is a local algebra reduced?
Let $k$ be a field and let $A$ be a local $k$-algebra which has finite dimension over $k$. Let $\mathfrak{m}$ be the maximal ideal of $A$ and let $k' = A / \mathfrak{m}$ be the residue field.
For ...
1
vote
1answer
124 views
Is the functor $\mbox{Rings}\rightarrow \mbox{Sets}$ given by $R \mapsto \{\pm 1 \in R\}$ corepresentable?
Is the function $\mbox{Rings}\rightarrow\mbox{Sets}$ given by $R\mapsto \{\pm 1\in R\}$ corepresentable?
Of course this might be problematic in characteristic 2 since this set is then a singleton, ...
4
votes
2answers
126 views
Subrings of formal series rings
Let $k$ be a field and $A = k[[x_1, \dots, x_n ]]$ be the ring of formal series in $n$ variables. Consider $g_1, \dots, g_m \in A$ such that $g_1(0) = \cdots = g_m(0) = 0$. For every $f \in k[[t_1, ...
3
votes
1answer
245 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 ...
4
votes
1answer
317 views
What is the connection between the definition of complete intersection variety and complete intersection ring?
An algebraic variety is called a complete intersection if its defining ideal is generated
by codimension many polynomials.
A Noetherian local ring $R$ is called a complete intersection if its ...
7
votes
0answers
116 views
Minimal systems of generators for commutative rings
Let $S$ be some base ring (a commutative ring or even just a field), and $R$ a commutative ring containing $S$ which is finitely generated (as an algebra) over $S$. What conditions guarantee that any ...
