Questions tagged [examples]
For questions requesting examples of a certain structure or phenomenon
555 questions
1
vote
2
answers
738
views
Weakly initial sets - examples and nonexamples
A weakly initial set in a category C is a set of objects I of C such that every object a of C has at least one arrow from an object contained in I.
The question is then, does Fields have a weakly ...
- 53.3k
7
votes
9
answers
3k
views
What category without initial object do you care about?
Recently I have been listening to some constructions that have been designed to accommodate categories without an initial object. The speaker has given some idea of a category or two that he cares ...
- 6,870
29
votes
5
answers
9k
views
Examples where Kolmogorov's zero-one law gives probability 0 or 1 but hard to determine which?
Inspired by this question, I was curious about a comment in this article:
In many situations, it can be easy to
apply Kolmogorov's zero-one law to
show that some event has probability 0
or 1, ...
CommunityBot
- 1
0
votes
1
answer
237
views
Geometric explanation of an orbit space: Integer action on the affine line
Let $k$ be a field of char $0$ and let $\mathbb{Z}$ act on $\mathbb{A}^1_k$ by the action induced by $G\to\mathrm{Aut}_k(k[X]), n\mapsto X+n$. It is rather easy to show that the orbit space $\mathbb{A}...
- 1,439
4
votes
1
answer
3k
views
Ringed and locally ringed spaces
A pair $(X,O_X)$ is a ringed space if $X$ is a topological space and $O_X$ is a sheaf of rings. If every stalk $O_{X,x}$ is a local ring, then we say that $(X,O_X)$ is a locally ringed space.
In the ...
- 45.7k
18
votes
17
answers
6k
views
What is your favorite isomorphism? [closed]
The other day I was trying to figure out how to explain why isomorphisms are important. I pulled Boyer's A History of Mathematics off the bookshelf and was surprised to find that isomorphism isn't ...
CommunityBot
- 1
14
votes
4
answers
2k
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 ...
- 123
15
votes
1
answer
1k
views
Comodule exercises desired
This Question is inspired by a Quote of Moore's
"There are two ‘evil’ influences at work here:
1. we are toilet trained with algebras not coalgebras
2. some of us are addicted to manifolds and so ...
- 3,726
5
votes
1
answer
329
views
Example of a quasitopological group with discontinuous power map
A quasitopological group is a group $G$ with topology such that multiplication $G\times G\rightarrow G$ is continuous in each variable (i.e. all translations are continuous) and inversion $G\...
- 36
19
votes
4
answers
4k
views
What are your favorite finite non-commutative rings?
When you are checking a conjecture or working through a proof, it is nice to have a collection of examples on hand.
There are many convenient examples of commutative rings, both finite and infinite, ...
- 1,793
7
votes
4
answers
9k
views
Good example of a non-continuous function all of whose partial derivatives exist
What's a good example to illustrate the fact that a function all of whose partial derivatives exist may not be continuous?
- 25.7k
5
votes
2
answers
920
views
Operator Valued Weights
One of the basic tools in subfactors is the conditional expectation. If $N\subset M$ is a $II_1$-subfactor (or an inclusion of finite factors), then there is a unique trace-preserving conditional ...
- 4,624
0
votes
1
answer
487
views
Must finite groups with isomorphic commutators and quotients be isomorphic?
Let G and H be finite groups. Let G' = [G,G] and H' = [H,H] be the corresponding derived groups (commutator subgroups) of G and H. I am looking for an example where G' is isomorphic to H' and G/G' is ...
- 3,176
9
votes
1
answer
3k
views
Sequence that converge if they have an accumulation point
I am looking for classes of sequence, that converge iff they contain a converging sub-sequence.
The basic example of such sequences are monotone sequences of real numbers.
A more interesting examples ...
- 18.7k
1
vote
1
answer
716
views
An example of a space which is locally relatively contractible but not contractible?
A space $X$ is called locally contractible it it has a basis of neighbourhoods which are themselves contractible spaces. CW complexes and manifolds are locally contractible. On the other hand, the ...
- 52.6k
6
votes
2
answers
1k
views
Gaining intuition for how submodules behave
I'm studying elementary commutative algebra this semester, largely following Atiyah-MacDonald. I often find myself in a situation where I'm interested in whether some property of an R-module M is ...
CommunityBot
- 1
5
votes
2
answers
878
views
What is an example of a non-regular, totally path-disconnected Hausdorff space?
I need this for a counterexample: the multiplication in the fundamental group $\pi_1(\Sigma X_+)$, when it is equipped with the topology inherited from $\Omega \Sigma X_+$, fails to be continuous for ...
- 7,193
1
vote
1
answer
3k
views
Example for deterministic function with unbounded total variation and bounded quadratic variation
It is well known that e.g. $sin(1/x)$ is of unbounded total variation (in the interval [0,1] assuming $f(0)=0$). (Preliminary numerical tests suggest that) it is also of unbounded quadratic variation. ...
- 41.1k
19
votes
4
answers
2k
views
Example of a smooth morphism where you can't lift a map from a nilpotent thickening?
Definition. A locally finitely presented morphism of schemes $f\colon X\to Y$ is smooth (resp. unramified, resp. étale) if for any affine scheme $T$, any closed subscheme $T_0$ defined by a square ...
- 23.8k
31
votes
6
answers
3k
views
What can you do with a compact moduli space?
So sometime ago in my math education I discovered that many mathematicians were interested in moduli problems. Not long after I got the sense that when mathematicians ran across a non compact moduli ...
- 6,247
9
votes
0
answers
281
views
Krull rings and determinantal invariants
During another attempt to come to grips with Hillman's excellent book Algebraic Invariants of Links, I am having difficulty figuring out why Krull rings are the setting for Chapter 3- the natural ...
- 22.1k
12
votes
1
answer
494
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 \...
- 2,083
48
votes
5
answers
15k
views
Algebraically closed fields of positive characteristic
I'm taking introductory algebraic geometry this term, so a lot of the theorems we see in class start with "Let k be an algebraically closed field." One of the things that's annoyed me is that as far ...
CommunityBot
- 1
2
votes
1
answer
1k
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 $X\...
CommunityBot
- 1
1
vote
0
answers
1k
views
Again about Bing's house with two rooms [duplicate]
Possible Duplicate:
How to show that the “bing’s house with two rooms” is contractible?
I don't know why my question is closed? here, I make my question clearly, when "hollowing ...
CommunityBot
- 1
1
vote
1
answer
1k
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!
...
CommunityBot
- 1
3
votes
1
answer
4k
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?
- 19.6k
0
votes
1
answer
216
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 ...
- 5,232
22
votes
2
answers
4k
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 ...
- 7,108
20
votes
8
answers
3k
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 $R$ ...
CommunityBot
- 1
13
votes
2
answers
4k
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 ...
- 7,108
2
votes
0
answers
254
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 ...
CommunityBot
- 1
3
votes
4
answers
1k
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 ...
- 6,546
6
votes
4
answers
1k
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.
CommunityBot
- 1
0
votes
1
answer
1k
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. ...
- 318
15
votes
27
answers
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 ...
CommunityBot
- 1
3
votes
5
answers
2k
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 ...
CommunityBot
- 1
7
votes
3
answers
2k
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 ...
2
votes
1
answer
265
views
Hausdorff Derived Series
There is a short section in the book Locally Compact Groups by Markus Stroppel (Chapter B7) on the notion of a "Hausdorff Solvable Group", which he defines as a topological group with a descending ...
- 56.6k
5
votes
2
answers
765
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 ...
CommunityBot
- 1
3
votes
4
answers
1k
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 ...
- 2,132
4
votes
2
answers
320
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 ...
- 6,292
8
votes
3
answers
1k
views
Gerbes for a cyclic group. (or maybe G_m too)
Let μn be the group scheme of n-th roots of unity. If X is a scheme and L is a line bundle on X, then I can construct a μn-gerbe Y over X by letting the S-points of Y be a S-point of X, a line ...
CommunityBot
- 1
7
votes
3
answers
585
views
Weil divisors on non Noetherian schemes
Let X be an integral scheme that is separated (say over an affine scheme). Define a Weil divisor as a finite integral combination of height 1 points of X, where the height of a point of X is the ...
- 281
3
votes
1
answer
928
views
Simple applications of Atiyah-Bott localization
I am looking for some simple and concrete -- but still non-trivial and illustrative -- applications of Atiyah-Bott localization in the context of equivariant cohomology.
Do you know any good ones?
- 156k
0
votes
3
answers
234
views
Where to find nice diagrams of trees and other graphs? [closed]
Are there some publicly available, vector format diagrams of trees and other graphs? They aren't hard to make, but they sure do take a lot of time (for me).
- 5,425
18
votes
9
answers
2k
views
What representative examples of modules should I keep in mind?
So here's my problem: I have no intuition for how a "generic" module over a commutative ring should behave. (I think I should never have been told "modules are like vector spaces.") The only ...
- 2,992
12
votes
2
answers
2k
views
Non-quasi separated morphisms
What are some examples of morphisms of schemes which are not quasi separated?
- 11.3k
13
votes
5
answers
5k
views
Examples and intuition for arithmetic schemes
How should a beginner learn about arithmetic schemes (interpret this as you wish, or as a regular scheme, proper and flat over Spec(Z))? What are the most important examples of such schemes? Good ...
- 15.1k
13
votes
3
answers
815
views
Constructing a degeneration (as a group scheme) of G_m to G_a
SGA 3, expose 12, remark 1.6 says that one can easily construct a group scheme over a discrete valuation ring with generic fiber Gm and special fiber Ga.
What is such an example?
- 24.1k