136
votes
Short exact sequences every mathematician should know
There is one obvious sequence that underlies all vector analysis and a lot that builds up on it, no matter if its applied analysis, PDE, physics or the original foundations of algebraic topology. Yet ...
76
votes
Short exact sequences every mathematician should know
This might be very basic, but the short exact sequence
$$
0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0
$$
is both an injective resolution of $\mathbb{Z}$, and a flat resolution of $\...
75
votes
Short exact sequences every mathematician should know
The exponential sheaf sequence:
$$0\to 2\pi i\,\mathbb Z \to \mathcal O_M {\buildrel\exp\over\to}\mathcal O_M^*\to 0$$
where $\mathcal O_M$ is the sheaf of holomorphic functions on the complex ...
67
votes
Short exact sequences every mathematician should know
I find it hard to believe that three days have gone by and no one has explicitly mentioned
$$
0 \to \Bbb Z \to \Bbb R \to \Bbb S^1 \to 0
$$
61
votes
Short exact sequences every mathematician should know
I think a short exact sequence that every teacher should know is
$$ 0 \to \mathbb R^d \to \mathrm{Isom}(\mathbb R^d) \to \mathrm{O}(\mathbb R^d) \to 0, $$
maybe for $d=2$ or $d=3$. Better still, ...
47
votes
Short exact sequences every mathematician should know
The short exact sequence
$$ 0 \to \mathrm{rad}({\mathfrak g}) \to {\mathfrak g} \to {\mathfrak g}/\mathrm{rad}({\mathfrak g}) \to 0$$
separates a Lie algebra ${\mathfrak g}$ into its solvable radical $...
46
votes
Understanding a quip from Gian-Carlo Rota
I think Abdelmalek Abdesselam and William Stagner are completely correct in their interpretation of the words "Behind" and "one immutable source" as describing one theory, the theory of symmetric ...
46
votes
Short exact sequences every mathematician should know
"Every mathematician should know" is too much to ask, but I do think the following is a great short exact sequence that captures a vital phenomenon:
$$0 \to K(H) \to B(H) \to Q(H) \to 0.$$
$...
45
votes
Short exact sequences every mathematician should know
I strongly doubt there is any short exact sequence that every mathematician should know, but I certainly wish that those of them who know that for a (co)chain complex $(C,d)$
$$
0\to\operatorname{Im}(...
41
votes
Short exact sequences every mathematician should know
An example of a short exact sequence satisfying your first desiderata, but one which you probably won't fully understand till you are further along in homological algebra, is the Universal Coefficient ...
40
votes
Accepted
Morava K-theories for dummies?
This is a result in algebraic topology, where we study the structure of topological spaces $X$. One early way to do this is to calculate a thing called $H_*(X)$, the ordinary homology of $X$. Later ...
- 55k
39
votes
Vladimir Voevodsky's works
Perhaps one of the biggest ideas that VV was pursuing is the Initiality Conjecture for Martin-Löf type theory (with universes). A rough idea is that from the rules for a type theory, one can define a ...
35
votes
Short exact sequences every mathematician should know
Within the category of Banach spaces and bounded linear maps,
$$0\to c_0 \to \ell_\infty \to \ell_\infty / c_0 \to 0$$
is a paradigm example of a short exact sequence that does not split, contrary to ...
29
votes
Vladimir Voevodsky's works
Aside from his work on the foundations of mathematics, which others have already elaborated on, earlier in his career Voevodsky also proved the Milnor conjecture in algebraic geometry. The Milnor ...
29
votes
Short exact sequences every mathematician should know
How about the short exact sequence that expresses that every group can be expressed in terms of generators and relators? For any group $G$, there is a short exact sequence (in fact many) of the form $$...
28
votes
Short exact sequences every mathematician should know
For any abelian group $A$, there is a short-exact sequence $$0 \to T(A) \to A \to A/T(A) \to 0,$$
where $T(A)$ is the torsion subgroup of $A$, and $A/T(A)$ is torsion-free.
27
votes
Accepted
Torsion in the Atiyah–Hirzebruch spectral sequence of a classifying space
Of course, in any spectral sequence $E_{r+1}$ is a subquotient of $E_r$ (the kernel of $d_r$ divided by the image of $d_r$). And in general new torsion can appear in the sense of torsion elements in $...
- 54.4k
27
votes
Short exact sequences every mathematician should know
The kernel-cokernel exact sequence: in an abelian category, given $A \xrightarrow{f} B \xrightarrow{g} C$, the following sequence is exact
$$ 0 \to \ker f \to \ker gf \to \ker g \to \text{coker } f \...
26
votes
Short exact sequences every mathematician should know
Given a finitely generated module $M$ over a commutative Noetherian ring $R$, there is a short exact sequence $$0\to M_1 \to R^n \to M\to 0$$
where you map $1$ in each $R$ to a generator of $M$ and $...
24
votes
Accepted
H-space structures on non-sphere suspensions?
If $Y$ is a connected CW-complex of finite type which is both an H-space and a co-H-space, then $Y$ has the homotopy type of $S^1$, $S^3$, $S^7$ or a point. This is a result of Robert West:
Robert W. ...
- 35k
24
votes
Understanding a quip from Gian-Carlo Rota
I think you are misinterpreting the quote. In the last sentence, the word "source" does not mean "source of these theories (K-theory, categories, group representations", but "source of the theory of ...
23
votes
Short exact sequences every mathematician should know
I suppose many algebraic topologists would agree that the short exact sequence
$$0\longrightarrow \mathbb Z/p \longrightarrow \mathbb Z/p^2 \longrightarrow \mathbb Z/p\longrightarrow 0$$
giving rise ...
23
votes
Accepted
Does Waldhausen K-theory detect homotopy type?
The answer to the question
Does the homotopy type of $𝐴(𝑋)$ determine the homotopy type of $𝑋$?
is No in general. As you say, $A(X)$ is determined by the homotopy type of $\Sigma^\infty \Omega X_+...
- 10.7k
21
votes
The $K$-theory homology of the Eilenberg-MacLane spectrum
We have $KU_*(H\mathbb Z) = \pi_*(KU\wedge H\mathbb Z)$; this ring is concentrated in even dimensions and carries an isomorphism between the additive and multiplicative formal group law, hence is ...
- 4,659
21
votes
Accepted
Why does K-theory need schemes to be Noetherian?
You don't need the Noetherianness hypothesis to talk about K-theory. But the definition you propose in your question is not suited for the most general case. From a notion of K-theory we want at least ...
- 16.2k
21
votes
Short exact sequences every mathematician should know
The Tate extension. Let $k$ be a field, and let $V$ be the space $k((t))$ be the space of Laurent series with coefficients in $k$, considered as a topological vector space. If we write $\operatorname{...
21
votes
Magic behind idempotent-complete categories a.k.a. why (sometimes) be Karoubian is sexier than be Abelian
I don't know if this is going to answer your question but here's some relevant background. Splitting idempotents has a very special property from a categorical point of view: it is an absolute colimit ...
- 115k
21
votes
Accepted
Problems concerning subspaces of $M_{n}(\mathbb{Q}) $
Let's call this maximal dimension function $\rho_{\mathbb{Q}}:\mathbb{N}\to\mathbb{N}$, i.e., $\rho_{\mathbb{Q}}(n)$ is the largest possible dimension of a subspace $N\subset M_n(\mathbb{Q})$ such ...
- 106k
20
votes
Accepted
Entering to the K-theory realm
I think that doing algebraic K-theory properly certainly requires a good background on stable homotopy theory, that is to say the homotopy theory of spectra. Unfortunately there are not many textbooks ...
- 16.2k
20
votes
The $K$-theory homology of the Eilenberg-MacLane spectrum
Since for any two spectra $E,F$ we have
$$
E_n(F)=[\mathbb{S}^n,E\wedge F]\cong [\mathbb{S}^n,F\wedge E] = F_n(E),
$$
you may as well ask about $H_n(KU;\mathbb{Z})$, the integral homology of the ...
- 35k
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