# Tagged Questions

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 ...
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 $Y'$ ...
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:= ...
1answer
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### 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 ...
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 $k$ ...
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 ...
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 ...
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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 ...
0answers
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### 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 ...
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, ...
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 ...
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 ...
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 ...
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 ...
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, ...
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?
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
2answers
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### 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 ...
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. ...
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 ...
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 ...
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 ...
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 ...
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 = ...
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 ...
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 ...
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? ...
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 ...
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. ...
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 ...
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, ...
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, ...
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 ...
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 ...
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 ...
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