Questions about commutative rings, their ideals, and their modules.
23
votes
0answers
381 views
The Ring Game on $K[x,y,z]$
I recently read about the Ring Game on Mathoverflow, and have been trying to determine winning strategies for each player on various rings. The game has two players and begins with a commutative ...
21
votes
0answers
345 views
A short proof for $\dim(R[T])=\dim(R)+1$?
If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and nontrivial ...
18
votes
0answers
261 views
What is the Picard group of $z^3=y(y^2-x^2)(x-1)$?
I'm actually doing much more with this affine surface than just looking for the Picard group. I have already proved many things about this surface, and have many more things to look at it, but the ...
14
votes
0answers
110 views
Hilbert's original proof of basis theorem
Does anyone know Hilbert's original proof of his basis theorem--the non-constructive version that caused all the controversy? I know this was circa 1890, and he would have proved it for ...
12
votes
0answers
93 views
A ring isomorphic to its finite polynomial rings but not to its infinite one.
I was messing with the ring $k[x_1,\dots,x_n,\dots]$ of polynomials in numerable many variables in order to solve an exercise of Atiyah, and the following question came to me and made me curious:
...
9
votes
0answers
135 views
Maximal ideal space of $c_{\mathcal{U}}$
Let $\mathcal{U}$ be an filter over $\mathbb{N}$. Define
$$c_{\mathcal{U}} = \{{(x_n)\in \ell_\infty\colon \lim_{\mathcal{U}, n}x_n =0\}},$$
which is a C*-algebra. Is there an accessible topological ...
9
votes
0answers
89 views
checking that an element of a module is zero, point-wise
Let $M$ be a module over a commutative ring $R$. Let $s \in M$ be an element such that for any $x \in \mathrm{Spec}\,R$, the image of $s$ in $M \otimes \kappa(x)$ is 0 (where $\kappa(x)$ is the ...
9
votes
0answers
182 views
Minimal spectrum of a commutative ring
Can anyone explain to me why the minimal prime ideals of a commutative ring (with the subspace topology inherited from the Zariski topology) form a totally disconnected space, or give a reference? I ...
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 ...
7
votes
0answers
114 views
Find all maximal subrings of $\mathbb{C}[x]$
Definition: A maximal subring $S$ of $R$ is a subring such that if $S \subseteq T \subseteq R$ then $T=S$ or $T=R$.
Find all maximal subrings of $\mathbb{C}[x]$.
Clearly $\mathbb{C}[x^2,x^3]$ ...
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 ...
7
votes
0answers
149 views
Trivial intersection of algebraic sets?
The question came up while reading a bit more into the Hilbert-Zariski theorem I asked about the other week.
Suppose $V$ is an algebraic variety over arbitrary field $k$. (For this situation, I'll ...
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 ...
6
votes
0answers
94 views
An example of a commutative ring in which every primary ideal is prime
It is clear that every prime ideal in a commutative ring is primary. The converse is false; for example, in the ring $\mathbb{Z}$ the ideal $(p^2)$ is an example of a primary ideal that is not prime ...
6
votes
0answers
69 views
What is the image of the map $\hom(V,V) \to \hom(\wedge^k V,\wedge^k V)$?
The title says it all. For the uninitiated: Any map $f:V \to W$ induces a map $\wedge^k V \to \wedge^k W$ by $v_1 \wedge \cdots \wedge v_k \mapsto f(v_1)\wedge \cdots \wedge f(v_k)$, so $\wedge^k(-)$ ...
6
votes
0answers
85 views
Divisor class group of an affine surface
In this topic the OP considers the following surface: $X=\mathcal{Z}(z^3-y(y^2-x^2)(x-1))$. (The field it's not explicitely mentioned, but for geometric reasons this can be algebraically closed.) He ...
6
votes
0answers
137 views
Punctured plane algebraic over a finite field?
I've been asked to prove, by my algebraic geometry teacher, that the punctured affine plane $\mathbb{A}_k^2 \backslash \{0,0\}$ is not an algebraic set, i.e. is not the zero set of any set of ...
6
votes
0answers
104 views
Question about definition of $\mathrm{Ext}$
One can define $\mathrm{Ext}^n(M,N)$ (where $M,N$ are $R$-modules) in two ways, either by taking an injective resolution of $N$ and applying $\mathrm{Hom}(M,-)$or by taking a projective resolution and ...
6
votes
0answers
99 views
On the order of $\mathbb{Z}[X]/(f,g)$ and the resultant $R(f,g)$.
I suspect that $\#\mathbb{Z}[X]/(f,g)=|R(f,g)|$ holds for any two non-constant polynomials $f,g\in\mathbb{Z}[X]$, where $R(f,g)$ is the resultant of $f$ and $g$. I am however unable to prove it. I'd ...
6
votes
0answers
120 views
When does base change preserves Homs
Let $A \to B$ be a ring homomorphism and $M,N$ be two $A$-modules. Consider the natural map
$\alpha_{M,N} : \mathrm{Hom}_A(M,N) \otimes_A B \to \mathrm{Hom}_B(M \otimes_A B,N \otimes_A B)$
Consider ...
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 ...
5
votes
0answers
64 views
Description of $\mathrm{Ext}^1(R/I,R/J)$
Let $R$ be a commutative ring with unit and $I$ and $J$ are nonzero ideals of $R$. Do we have a nice description for $\mathrm{Ext}^1_R(R/I,R/J)$?
What do I mean by a nice description? For example ...
5
votes
0answers
58 views
Injective Hull and and some Hom set.
Let $R$ be a commutative ring with unit. Suppose $P\in Spec(R)$ and let $E=E(R/P)$ be the injective hull of $R/P$. What can we say about $Hom_R(R/P, E)$. We know that $R/m\cong Hom_R(R/m, E)$, where ...
5
votes
0answers
75 views
When is the pushforward / direct image of a reflexive sheaf locally free?
I have seen a number of theorems that guarantee the direct image of a reflexive sheaf to be reflexive again, or for the direct image of a locally-free sheaf to be locally free again.
This makes me ...
5
votes
0answers
82 views
Hilbert symbol over a ring
Normally the Hilbert symbol over a field $\mathbb{F}$ is defined for $a,b\in\mathbb{F}^*$ as follows:
$$ (a,b)=\begin{cases}1,&\text{ if }z^2=ax^2+by^2\text{ has a non-zero solution }(x,y,z)\in ...
5
votes
0answers
67 views
The geometric interpretation of Gorenstein local ring
Many local rings have geometric interpretations. Cohen–Macaulay rings correspond to equi-dimensionality, and regular local rings correspond to non-singularity, but what is a geometric interpretation ...
5
votes
0answers
94 views
Non-reflexive module isomorphic to its double dual
Could you give me an example of a non-reflexive module isomorphic to its double dual?
I found an example here but I cannot understand it, do you have any simpler examples?
By this question we should ...
4
votes
0answers
36 views
Lattices as invertible module
Let $E$ be an etale algebra over $\mathbb{Q}$. In other words, $E$ is a finite sum of number fields. Let $L$ be a lattice in $E$, and $R$ the order associated to $L$. More explicitly, $$R=\{ e\in ...
4
votes
0answers
55 views
Artinian rings are perfect
Is there a simple way to prove that an Artinian ring is perfect? (in the commutative case)
4
votes
0answers
64 views
Maximal ideals in the algebra of continuously differentiable functions on [0,1]
This is an exercise in Rudin's Functional Analysis, in the chapter on commutative Banach algebras. My (uneducated) guess was that every homomorphism on $C^{1}[0,1]$ is an evaluation at some point of ...
4
votes
0answers
52 views
A question on an answer on Math Overflow about Artin approximation
I have a question on an answer of this Math Overflow question.
Let $(A,I)$ be a commutative excellent normal local domain. The completion
$$
\hat A=\underset{\longleftarrow}{\operatorname{lim}} ...
4
votes
0answers
138 views
Defining multiplication on a Koszul complex
Let $R$ be a Nothearian commutative ring and $x$ and $y$ two elements in $R$. We construct the Koszul complex on $x$ and $y$. We start by the following two chain complexes:
$$
C_2=0\to ...
4
votes
0answers
95 views
Definition of analytically unramified rings
A noetherian local ring A is said to be analytically unramified if the complete local ring $\hat{A}$ is reduced.
I don't see why it makes sense to call such a ring analytically unramified. The ...
4
votes
0answers
78 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 ...
4
votes
0answers
142 views
Construction of graded rings and modules
In Algebraic Geometry and Homological Algebra - as far as I know - we often consider graded rings and modules so as to encode more information, say, some sort of (computational) complexity. For ...
4
votes
0answers
76 views
Is reducedness an open condition?
If $X$ is a (general) scheme and $X$ is reduced at $p$, i.e. $\mathscr{O}_{X,p}$ is reduced, does there necessarily exist an open neighborhood of $p$ on which $X$ is reduced, i.e. $\mathscr{O}_X(U)$ ...
4
votes
0answers
87 views
Addition and multiplication are continuous in the $I$-adic topology
Can you tell me if this is correct?
Let $R$ be a ring and let it have the $I$-adic topology for some ideal $I$ in $R$. I want to show that $+: R \times R \to R$ is continuous at $(x_0, y_0)$.
Proof: ...
4
votes
0answers
163 views
radical of an ideal
Let $R$ be a commutative ring with identity and $I$ a proper ideal of $R$. We define $L$-radical of $I$, denoted by $\sqrt[L]{I}$, the intersection of all primary ideals of $R$ containing $I$. It is ...
4
votes
0answers
88 views
What is the injective hull of $\mathbb{C(x,y)}/\mathbb{C[x,y]}$?
What is the injective hull of $\mathbb{C(x,y)}/\mathbb{C[x,y]}$ as a $\mathbb{C}[x,y]$-module? Is it isomorphic to any familiar module?
4
votes
0answers
51 views
How to extract roots in a complete local ring using binomial series
Let $A$ be a local ring with maximal ideal $m$ that is $m$-adically complete, and assume $1/2 \in A^\times$. I've read in several places that for any $x \in m$, a square root of $1 + x$ in $A$ is ...
4
votes
0answers
76 views
Mod-$R$, Mod-$S$ and Mod-$R \otimes S$
Let $R,S,T$ be commutative rings and assume that $R,S$ are $T$-algebras.
In an answer to this question, Pierre-Yves Gaillard gives an example of an $R \otimes_T S$-module that cannot be written as ...
4
votes
0answers
64 views
Injective map on power series ring
Suppose $k$ is a field and let $n > m$. Does there exist injective homomorphisms of the form:
$$ k [[x_1, x_2, \ldots, x_n]] \rightarrow k[[x_1, x_2, \ldots, x_m]]$$
4
votes
0answers
130 views
Is there an integral domain with a lot of residue fields of the same characteristic?
Is there a commutative integral domain $R$ in which:
every nonzero prime ideal $Q$ is maximal, and
for every prime power $q\equiv 3 \bmod 8$, there is a maximal ideal $Q$ of $R$ such ...
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 ...
3
votes
0answers
48 views
Homogeneous ideals are contained in homogeneous prime ideals
Let $I$ be a homogeneous ideal of a graded ring $S$. I want to show that there exists a homogeneous prime ideal which contains $I$.
I proved the following:
Let $T$ be the set of all homogeneous ...
3
votes
0answers
38 views
Injective dimension is locally finite but not globally
Let $R$ be a commutative ring. Could someone provide me an example where $\operatorname{id}_{A_{\mathfrak p}}(M_{\mathfrak p})$ is finite for all $\mathfrak p\in \operatorname{Spec}(R)$, but ...
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 ...
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 ...
3
votes
0answers
55 views
Hilbert’s zeros theorem, an application. (The algebraic variation)
Theorem: (Hilbert) If $k$ is a field, $A$ is a finitely generated $k$-algebra, and $M$ is a maximal ideal in $A$, then the factor $A/M$ is a finite extension of $k$. In particular if $k$ is ...
