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2
votes
1answer
597 views

Example of restriction of a finite morphism which is not finite

Every closed immersion is a finite morphism. Therefore, restriction of a finite morphism to a closed subset is always a finite morphism itself. Can you give an example of quasi-projective varieties ...
1
vote
1answer
436 views

Example of inclusion which is not a finite morphism [closed]

Every closed immersion is a finite morphism. Can you give an example of quasi-projective varieties $X\subset Y$ such that inclusion $X\hookrightarrow Y$ is not finite? Same with Y projective? Thanks! ...
2
votes
1answer
2k views

How to show that the “bing's house with two rooms” is contractible? [closed]

I can't image this, Someone can give a clear illustration?
10
votes
4answers
1k views

Regular spaces that are not completely regular

In the undergraduate toplogy course we were given examples of spaces that are $T_i$ but not $T_{i+1}$ for $i=0,\ldots,4$. However, no example of a space which is $T_3$ but not $T_{3.5}$ was given. ...
11
votes
1answer
452 views

Tensor products and two-sided faithful flatness

Let $f: R \to S$ be a morphism of Noetherian rings (or more generally $S$ can just be an $R-R$ bimodule with a bimodule morphism $R \to S$). Suppose $f$ is faithfully flat on both sides, so $M \to M ...
10
votes
2answers
2k views

non principally polarized complex abelian varieties

I've read in (abstracts of) papers that there are abelian varieties over fields of positive characteristic that admit no prinicipal polarization. Apparently its not the easiest thing to find an ...
4
votes
3answers
780 views

An example where GCD depends on the domain

First some notation. Given a domain $R$ and $x,a,b \in R$, I write $x=gcd(a,b)_R$ to mean that $x$ is one gcd of $a$ and $b$ in $R$. I want to find an example of an GCD-domain $R$, a subdomain $S ...
89
votes
53answers
19k views

What are your favorite instructional counterexamples?

Related: question #879, Most interesting mathematics mistake. But the intent of this question is more pedagogical. In many branches of mathematics, it seems to me that a good counterexample can be ...
6
votes
2answers
1k views

Example of connected-etale sequence for group schemes over a Henselian field?

Can someone give a really concrete example of such a sequence? I am looking at several notes related with such things, but haven't seen any well-calculated example. And I'm really confused at this ...
18
votes
8answers
1k views

Cryptomorphisms

I am curious to collect examples of equivalent axiomatizations of mathematical structures. The two examples that I have in mind are Topological Spaces. These can be defined in terms of open sets, ...
2
votes
0answers
242 views

Forgetting extra structure inducing Symmetries

This is a major edit of the original post after receiving helpful comments. It is often the case when one adds additional structure to make a problem more tractable. When one attempts to forget this ...
4
votes
1answer
788 views

On using field extensions to prove the impossiblity of a straightedge and compass construction

Let $z \in \mathbb{C}$. Consider the following statements: The point $z$ can be constructed with straightedge and compass starting from the points $\{ 0,1\}$. There is a field extension $K / ...
1
vote
5answers
648 views

Is there a finitely complete category with terminal object but NO subobject classifier?

This came up today while thinking about topoi in seminar, as the title suggests my question is; Is there a finitely complete category with terminal object but NO subobject classifier? Hopefully ...
6
votes
1answer
1k views

Connection between bi-Hamiltonian systems and complete integrability

As I understand, the lack of indication on how to obtain first integrals in Arnol'd-Liouville theory is a reason why we are interested in bi-Hamiltonian systems. Two Poisson brackets $\{ \cdot,\cdot ...
17
votes
2answers
3k views

A finitely generated, locally free module over a domain which is not projective?

This is a followup to a previous question What is the right definition of the Picard group of a commutative ring? where I was worried about the distinction between invertible modules and rank one ...
2
votes
4answers
478 views

Ternary relations that are not binary functions

By far the most prominent elementary relations that are not functions are binary and the most prominent elementary ternary relations are in fact binary functions. "Elementary" shall mean "part of the ...
4
votes
4answers
541 views

Kähler manifold which is not algebraic

Can someone provide examples of Kähler manifolds which are not algebraic? This question came to my mind seeing the post of Andrea Ferretti.
0
votes
1answer
778 views

Maximal subgroups of abelian groups and Q-algebras

Let $G$ be an abelian group which does not have a maximal subgroup. Does it follow that $G$ is a $\mathbb{Q}$-algebra? It is easy to see that $\mathbb{Q}$-algebras do not admit any maximal subgroups. ...
5
votes
1answer
678 views

Example for an integral, rectifiable varifold with unbounded first variation

I'm just looking for an example of an integral, rectifiable varifold, which has no locally bounded first variation. Recapitulation for every $m$-rectifiable varifold $\mu$ exists a $m$-rectifiable ...
35
votes
5answers
3k views

What is Yoneda's Lemma a generalization of?

What is Yoneda's Lemma a generalization of? I am looking for examples that were known before category theory entered the stage resp. can be known by students before they start with category theory. ...
6
votes
2answers
493 views

Examples of Completions and Algebraic Closures

It is widely known that the algebaric closure of the $p$-adic completion $\mathbb{Q}_p$ of $\mathbb{Q}$ isn't complete anymore. It's completion is complete and known as $\mathbb{C}_p$. I have read ...
0
votes
1answer
192 views

Motivation for Cosuspended Category Axioms

Today I was wondering about the axioms given by Bernhard Keller for Cosuspended Categories. The axioms of a triangle feel very much like exactness, but not quite. The last axiom about the large ...
10
votes
8answers
1k views

What are examples illustrating the usefulness of Krull (i.e., rank > 1) valuations?

In modern valuation theory, one studies not just absolute values on a field, but also Krull valuations. The motivation is easy enough: If $k$ is a field, a valuation ring of $k$ is a subring $R$ ...
54
votes
33answers
8k views

Experimental Mathematics

I would like to ask about examples where experimentation by computers have led to major mathematical advances. Motivation I am aware about a few such cases and I think it will be useful to gather ...
34
votes
15answers
6k views

Strong induction without a base case

Strong induction proves a sequence of statements $P(0)$, $P(1)$, $\ldots$ by proving the implication "If $P(m)$ is true for all nonnegative integers $m$ less than $n$, then $P(n)$ is true." for ...
24
votes
20answers
6k views

Nontrivial question about fibonacci numbers?

I'm looking for a nontrivial, but not super difficult question concerning Fibonacci numbers. It should be at a level suitable for an undergraduate course. Here is a (not so good) example of the sort ...
11
votes
7answers
2k views

Simple show cases for the Yoneda lemma

I've been given a very simple motivating and instructive show case for the Yoneda lemma: Given the category of graphs and a graph object $G$, seen as a quadruple $(V_G,\ E_G,\ S_G:E\rightarrow V,\ ...
45
votes
2answers
4k views

Galoisian sets of prime numbers

The question is about characterising the sets $S(K)$ of primes which split completely in a given galoisian extension $K|\mathbb{Q}$. Do recent results such as Serre's modularity conjecture (as proved ...
7
votes
4answers
794 views

Examples of Poisson Schemes

A Poisson Manifold is a real manifold $M$ along with a Lie bracket $[\cdot,\cdot]$ on $C^\infty(M)$ which is a derivation in each variable. Poisson manifolds are interesting for a few reasons, among ...
80
votes
42answers
16k views

What are the most attractive Turing undecidable problems in mathematics?

What are the most attractive Turing undecidable problems in mathematics? There are thousands of examples, so please post here only the most attractive, best examples. Some examples already appear on ...
6
votes
5answers
2k views

An example of two elements without a greatest common divisor

Is there an easy example of an integral domain and two elements on it which do not have a greatest common divisor? It will have to be a non-UFD, obviously. "Easy" means that I can explain it to my ...
2
votes
5answers
1k views

non-abelian groups of prescribed order

Is there a construction that will give a non-abelian group of order $p^mr$ where $p$ is a prime, $r$ and $p$ are relatively prime and $m$ is an arbitrary non-negative integer? I suspect in this ...
5
votes
1answer
446 views

obstruction to smooth lifting of smooth schemes

According to general theory, for a square zero thickening defined by an ideal I: SpecA -> SpecA', there is an obstruction of lifting a smooth scheme X over A to a smooth scheme over A' living in ...
1
vote
1answer
2k views

What are examples of theorems get extensions based on simple observation?

Here are some examples illustrate what I meant: Bonnet-Myers:Bonnet in 1855 proved n=2 case, Myers in 1941 extended to any dimension using the same idea. Bishop-Gromov Volume comparison: Bishop knew ...
7
votes
3answers
941 views

What are some interesting sequences of functions for thinking about types of convergence?

I'm thinking about the basic types of convergence for sequences of functions: convergence in measure, almost uniform convergence, convergence in Lp and point wise almost everywhere convergence. I'm ...
41
votes
28answers
20k views

Applications of the Chinese remainder theorem

As the title suggests I am interested in CRT applications. Wikipedia article on CRT lists some of the well known applications (e.g. used in the RSA algorithm, used to construct an elegant Gödel ...
19
votes
11answers
3k views

When does 'positive' imply 'sum of squares'?

Does anyone have examples of when an object is positive, then it has (or does not have) a square root? Or more generally, can be written as a sum of squares? Example. A positive integer does not ...
177
votes
63answers
89k views

Proofs without words

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results? (One could ask if this is of interest to mathematicians, and I ...
21
votes
3answers
1k views

What manifolds are bounded by RP^odd?

Real projective spaces ℝPn have ℤ/2 cohomology rings ℤ/2[x]/(xn+1) and total Stiefel-Whitney class (1+x)n+1 which is 1 when n is odd, so it follows that odd dimensional ones are boundaries of compact ...
13
votes
27answers
3k views

Justifying a theory by a seemingly unrelated example

Here is a topic in the vein of Describe a topic in one sentence and Fundamental examples : imagine that you are trying to explain and justify a mathematical theory T to a skeptical mathematician ...
6
votes
2answers
854 views

Example of non-closed convex hull in a CAT(0) space

this is related to this question but is simpler, and hopefully is well-known. There are a number of references that say that the convex hull of a collection of points in a CAT(0) space need not be ...
4
votes
2answers
410 views

Can we distinguish the algebraic and continuous duals of a Banach space without choice (or HBT)?

The algebraic dual of a normed vector space is the space of all linear functionals to the ground field (either $\mathbb{R}$ or $\mathbb{C}$ for this question). The continuous dual is the subspace of ...
2
votes
4answers
723 views

Examples of divisors on an analytical manifold

I am trying to understand divisors reading through Griffith and Harris but it is difficult to come up with any particular interesting example. I have browsed through Hartshone's book but everything is ...
45
votes
27answers
4k views

books well-motivated with explicit examples

It is ultimately a matter of personal taste, but I prefer to see a long explicit example, before jumping into the usual definition-theorem path (hopefully I am not the only one here). My problem is ...
4
votes
2answers
247 views

Is there a poset with 0 with countable automorphism group?

Is there a poset P with a unique least element, such that every element is covered by finitely many other elements of P (and P is locally finite -- actually, per David Speyer's example, let's say that ...
20
votes
6answers
2k views

Failure of smoothing theory for topological 4-manifolds

Smoothing theory fails for topological 4-manifolds, in that a smooth structure on a topological 4-manifold $M$ is not equivalent to a vector bundle structure on the tangent microbundle of $M$. Is ...
9
votes
5answers
998 views

Is a functor which has a left adjoint which is also its right adjoint an equivalence ?

I am looking for a counter-example of two functors F : C -> D and G : D->C such that 1) F is left adjoint to G 2) F is right adjoint to G 3) F is not an equivalence (ie F is not a quasi-inverse of ...
9
votes
4answers
1k views

Negative Gromov-Witten invariants

I understand the heuristic reason why Gromov-Witten invariants can be rational; roughly it's because we're doing curve counts in some stacky sense, so each curve $C$ contributes $1/|\text{Aut}(C)|$ to ...
60
votes
23answers
22k views

What are the most misleading alternate definitions in taught mathematics?

I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in ...
15
votes
3answers
1k views

Algebraic varieties which are topological manifolds

Inspired by this thread, which concludes that a non-singular variety over the complex numbers is naturally a smooth manifold, does anyone know conditions that imply that the topological space ...