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Apr
11
comment Definition of split of exact sequence
Why would you think that $ℤ \cong nℤ \oplus ℤ/nℤ$?
Apr
11
comment When is $G\cong\operatorname{End}(G)$?
@Gaussler Yeah, but it doesn’t work for rings without one (like $2ℤ$) either.
Apr
11
comment When is $G\cong\operatorname{End}(G)$?
@Gaussier For $R = ℤ$, $f = \mathrm{id}_ℤ + \mathrm{id}_ℤ$ is not multiplicative (nor does it preserve $1$), because $f(1·1) = 2 ≠ 4 = f(1)·f(1)$ (that’s right – $\mathrm{Ring}$ is not an additive category).
Apr
9
comment Find an automorphism
Find instead two surjective homomorphisms $ℤ → G$ and use the first isomorphism theorem. Alternatively, directly find two different isomorphisms $ℤ/nℤ → G$ which send $1$ to …?
Apr
8
comment Why are functors exact if they preserve all short exact sequences?
@lenticcatachresis I neither speak nor read Spanish but I might just be able to guess my way through this with a dictionary. With David’s answer, though, this will be unnecessary. Many thanks eitherway!
Apr
8
accepted Why are functors exact if they preserve all short exact sequences?
Apr
8
comment Why are functors exact if they preserve all short exact sequences?
@Berci Thanks, so generally $\operatorname{ker}~if = \operatorname{ker}~f$ whenever $i$ is a monomorphism and $\operatorname{img}~fp = \operatorname{img}~f$ whenever $p$ is an epimorphism – which can be proven by checking the universal properties, right?
Apr
8
comment Why are functors exact if they preserve all short exact sequences?
@Jim I know that, but that is hardly enlightening and not very satisfying unless you prove/understand Mitchell’s embedding theorem first which I haven’t and which seems to me too much of an effort for this problem.
Apr
8
comment Why are functors exact if they preserve all short exact sequences?
Thanks! This seems to work for the category of modules, say, where there is a notion of injectivity and kernels/cokernels can be defined as submodules rather than some limits/colimits. Does this approach generalise to arbitrary abelian categories? (Sorry for being so lazy not to think about it in detail.)
Apr
8
accepted The pushout of an open/closed injective map is open/closed
Apr
8
asked Why are functors exact if they preserve all short exact sequences?
Apr
6
comment An upper bound of a complex number
For real numbers $a$ and $t$, $e^{iat}$ is on the unit circle, hence $|e^{iat}| = 1$?
Apr
6
answered Relative homotopy and composition of maps
Apr
6
comment If $F$ has characteristic $p$ and $f(x)=x^p-a\in F[x]$, then $f(x)$ is either irreducible over $F$ or $f(x)$ splits in $F$.
The Frobenius homomorphism is used in (or rather mentioned to justify) the step $X^p-α^p = (X-α)^p$. Divisors of polynomials don’t change by expanding the field of coefficients – if $g ∈ F[X]$ is monic and $g | f$ in $F[X]$, then trivially $g | f$ in $L[X]$, hence $g = (X-α)^n$ in $L[X]$ for some $n ∈ ℕ_0$. Then you can expand $g$ in $L$ and conclude like I’ve done it.
Apr
6
comment Why $dz\wedge d\bar{z} = d|z|\wedge d\phi$ for $z \in \mathbb{C}$?
But what is the exterior derivative of a complex number?
Apr
6
comment Why $dz\wedge d\bar{z} = d|z|\wedge d\phi$ for $z \in \mathbb{C}$?
What does “$dz$” mean if $z$ is an actual complex number?
Apr
6
revised the number of inversions in the permutation “reverse”
added 6 characters in body
Apr
6
revised the number of inversions in the permutation “reverse”
added 6 characters in body
Apr
6
answered the number of inversions in the permutation “reverse”
Apr
6
comment Intuition for theorem about compact subsets of topological groups
So you do know a proof of the statement, but you still want some intuition for it, right?