2
votes
1answer
97 views

The Zariski topology on $\operatorname{Spec} A$ as an intial topology

Given any commutative ring $A$ let $\operatorname{Spec} A$ be the space of prime ideals of $A$. Can we interpret the Zariski topology as an initial (or final) topology with respect to some canonical ...
-1
votes
0answers
140 views

If $A$ is complete for $I$-adic and $J$-adic topologies, then $A$ is also complete for the $(I+J)$-adic topology

If $A$ is complete for both $I$-adic and $J$-adic topologies, then $A$ is also complete for the $(I+J)$-adic topology. (Matsumura, CRT, Exercise 8.1) How can I solve this problem? A is a ring ...
2
votes
2answers
72 views

Irreducible components of $Spec(A) $

A topological space $X$ is called irreducible if given $A_{1}, A_{2} $ open sets $ \neq \emptyset $ then $A_{1} \cap A_{2} \neq \emptyset$. The maximal irreducible topological subspaces of $X$ are ...
3
votes
1answer
31 views

Is there a topological characterisation of non-Archimedean local fields?

A local field is a locally compact field with a non-discrete topology. They classify as: Archimedean, Char=0 : The Real line, or the Complex plane Non-Archimedean, Char=0: Finite extensions of the ...
1
vote
0answers
31 views

Is an ideal generated by a compact subset finitely generated?

Let $R$ be a commutative topological ring and let $K$ be a compact subset of $R$. Denote by $I$ the ideal generated by $R$. Then is it true (or under what assumptions on $R$ (besides Noethernity)) is ...
1
vote
1answer
63 views

A property of $I$-adic topologies

Let $R$ be a commutative ring with multiplicative identity and $I$ a proper ideal of $R$ such that the intersection of its powers is the zero ideal. It can be shown that if the $I$-adic topology is ...
0
votes
0answers
32 views

Noetherianty of ring of continuous functions [duplicate]

Let $X$ be a topological space. Is there any topological property on $X$ that be equivalent to $C(X,\mathbb R)$ be noetherian ring?
0
votes
1answer
134 views

When is the ring of continuous functions Noetherian?

Let $X$ be a topological space. Is there any topological property on $X$ that be equivalent to $C(X,\mathbb R)$ being a noetherian ring?
4
votes
1answer
59 views

Is there a simple way to state continuity for $I$-adic topology?

Let $R$ be a commutative ring with the $I$-adic topology defined by an ideal $I$, and let $S$ be a commutative ring with the $J$-adic topology for an ideal $J$. How would you translate saying that a ...
1
vote
1answer
160 views

Relation between spectrum of a ring and its quotient ring and localization.

Let $A$ be a commutative ring. $I$ be an ideal of $A$, $S$ be a multiplicative closed subset. We know that : there is 1-1 correspondence between the prime ideals $\mathfrak{p}\in Spec A$ containing ...
3
votes
1answer
86 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
73 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, ...
13
votes
1answer
292 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?
1
vote
1answer
67 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 ...
2
votes
2answers
50 views

limits of sequences of topological rings

Let $A$ be a ring and $I$ an ideal of $A$ such that $A$ is complete in the $I$-adic topology. Let $a \in I$. Then the sequence $y_n=1-a+a^2-a^3+\cdots+(-1)^n a^n$ converges in $A$. By definition of ...
2
votes
1answer
34 views

Showing that the natural map into the completion is continuous

Let $M$ be an $A$-module and $M=M_0 \supset M_1 \supset \cdots$ a sequence of submodules, which we define to be a fundamental system of neighborhoods of $0$. Thus we make $M$ into a topological group. ...
6
votes
1answer
79 views

algebraic distance of an element of a ring from an ideal

Let $A$ be a commutative ring and $I$ an ideal. Does there exist a notion of "distance" of an element $x \in A$ from the ideal $I$? This "distance", need not be of the form $A\rightarrow \mathbb{R}$; ...
3
votes
0answers
131 views

Topology of maximal ideal space

We know that strictly positive integers with multiplication form a semigroup $S$. Let $\mathscr A=\ell ^1(S)$ with convolution. What is $\mathscr A$'s maximal ideal space? It seems enough to find ...
13
votes
1answer
364 views

Equivalent definitions of Noetherian topological space

It is well known that we have many different definitions of noetherianity for rings. Namely, given a ring $R$, the following are equivalent: 1) every ideal of $R$ is finitely generated. 2) $R$ ...
10
votes
1answer
246 views

Completion as a functor between topological rings

In the following all rings are assumed to be commutative and unitary. Preliminaries: For any topological ring $R$ we can form its completion $\widehat{R}$ by taking all Cauchy sequences modulo null ...
17
votes
3answers
536 views

What does the topology on $\operatorname{Spec}(R)$ tells us about $R$?

Let $R$ be a commutative ring with a unit. $\newcommand{\spec}{\operatorname{Spec}}\spec(R)$ denotes the set of all prime ideals in $R$, and it can be topologized using the Zariski topology. Last ...
2
votes
2answers
122 views

Irreducibility and Subspace Topology

Is the following statement true? "Let $X$ be a topological space, $Y$ a subspace and $S$ a closed and irreducible subset of $X$. Then $Y \cap S$ is not necessarily irreducible in $Y$." Counterexamples ...
4
votes
1answer
140 views

Combinatorial Dimension of a Topological Space - Ascending versus Descending Chains

The (combinatorial) dimension of a topological space is defined as the supremum of the lengths over all strictly ascending chains of closed irreducible subsets (e.g. Hartshorne). Can it also be ...
4
votes
2answers
114 views

$\operatorname{Spec}A$ and $T_1$ separation axiom

Let $A$ be a commutative ring. Why $\operatorname{Spec}A$ almost never satisfies the $T1$-separation axiom (Matsumura, Commutative Ring Theory, p.25)?
7
votes
3answers
303 views

A theorem due to Gelfand and Kolmogorov

For any topological space $X$, we can define $C(X)$ to be the commutative ring of continuous functions $f\,:\,X\rightarrow \mathbb{R}$ under pointwise addition and multiplication. Then $C(-)$ becomes ...
0
votes
2answers
89 views

What do I need to know to understand the completion of the field of rational functions of a non-singular projective curve?

So the title gives the jist of my question. Specifically, let $X$ be a non-singular projective curve, $P$ a point on $X$, $v_P$ the discrete valuation associated to the ring $\mathcal{O}_P$. Then I ...
2
votes
3answers
289 views

Zariski topology on prime $\mathrm{Spec}$ of a ring $R$

Let $R$ be a commutative unital ring. Let $\mathrm{Spec}(R) = \{ \mathfrak p \subset R \mid \mathfrak p \text{ a prime ideal of } R \}$. We define a set $C$ to be closed in this space if and only if ...
5
votes
2answers
485 views

$p$-adic completion of integers

I'm trying to do the following exercise: Let $p$ be a prime and for $n\geq 1$ let $\alpha_n :\mathbb Z/p \mathbb Z \to \mathbb Z/p^n \mathbb Z$ be the injection of abelian groups given by $1 \mapsto ...
21
votes
2answers
593 views

The prime spectrum of a Dedekind Domain

Let $A$ be a Dedekind Domain, let $X = \operatorname{Spec}(A)$. Are all open sets in $X$ basic open sets? Thinking about the Zariski topology (in the classical sense) of a non-singular affine curve, ...
7
votes
1answer
177 views

The product of two spectral spaces

Notice: the following statements about the product topologies are all Cartesian product topology, we are in the category of topology not the category of schemes. In this page of sober space, it said ...
9
votes
3answers
236 views

Zariski topology in the complex plane: an example

I want to find the closure under the zariski topology, of this set $ \left\{ {\left( {x,y} \right) \in {\Bbb C}^2 ;\left| x \right| + \left| y \right| = 1} \right\} $ I have no idea what I can do
1
vote
0answers
104 views

Do modules have any topology?

Is there any kind of topology, natural or unnatural, that modules do have? Is there any geometric interpretation for flat modules? Is "exactness" of a sequence, any kind of geometric condition? ...
7
votes
2answers
161 views

Why is $\text{Supp}(M)$ connected in the Zariski topology?

Suppose $M$ is a indecomposable module, so that it cannot be written as $M_1\oplus M_2$ for $M_1\neq M$ and $M_2\neq M$, which is finitely generated over a commutative ring $R$. Why is ...
4
votes
1answer
108 views

$C$ is irreducible iff $C=\mathscr{Z}(\mathfrak{p})$ for some prime ideal $\mathfrak{p}$?

Let $A$ be a commutative Noetherian ring, and $C$ a closed subset of $\operatorname{Spec}(A)$. In some reading, it is an unproven proposition that $C$ is irreducible iff ...
7
votes
3answers
207 views

$\operatorname{Spec} (A)$ as a topological space satisfying the $T_0$ axiom

I have been spending a few days now proving the last bit of the following problem of Atiyah Macdonald: Prove that $X = \operatorname{Spec}(A)$ as a topological space with the Zariski Topology ...
16
votes
2answers
714 views

Compactness of $\operatorname{Spec}(A)$

In an exercise in Atiyah-Macdonald it asks to prove that the prime spectrum $\operatorname{Spec}(A)$ of a commutative ring $A$ as a topological space $X$ (with the Zariski Topology) is compact. Now ...
2
votes
1answer
93 views

$\mathfrak{a}$-adic topology on a submodule equivalent to the induced topology

Suppose $A$ is a Noetherian ring, $M$ a finite $A$-module and $N \subset M$ a submodule. Furthermore, $\mathfrak{a} \subset A$ is an ideal. Consider the $\mathfrak{a}$-adic topology on M, i.e. the ...
5
votes
1answer
186 views

When is the ring of continuous functions absolutely flat?

This question was created in a discussion. Let $X$ be a topological space. Denote by $C(X; \mathbb{R})$ the ring of real-valued continuous functions defined on $X.$ Characterize those compact ...
3
votes
1answer
116 views

What is $A^-$ in topology?

Hartshorne Algebraic Geometry Proposition 1.5 (page5) In a notherian topological space $X$, every nomempty closed subset $Y$ can be expressed as a finite union $Y=Y_1 \cup \cdots \cup Y_r$ of ...
6
votes
2answers
352 views

Atiyah-Macdonald Ex 7.20 Constructible sets

Let $X$ be a topological space and let $\mathscr{F}$ be the smallest collection of subsets of $X$ which contains all open subsets of $X$ and is closed with respect to the formation of finite ...
2
votes
2answers
148 views

For a ring R with an ideal I, the I-adic topology makes R into a topological ring

Let $R$ be a commutative ring with identity. Let $I$ be an ideal of $R$. Suppose, we give a topology on $R$ where a set is open if and only if it is a union of cosets of powers of $I$. Then, is $R$ a ...
3
votes
3answers
212 views

Hausdorffness of a topological abelian group

In the chapter on completions in Atiyah Macdonald, they define a topological abelian group. Let $G$ be such a group. Denote $H$ to be the set of intersection of neighbourhoods of $0$. Then it is shown ...