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Jul
22
asked Surjectivity implies injectivity of finitely generated modules, localization?
Jul
20
revised Counterexamples for lcm-gcd identity and modular law for rings
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Jul
20
revised Counterexamples for lcm-gcd identity and modular law for rings
added 149 characters in body
Jul
14
revised The geometric interpretation for extension of ideals?
added 2 characters in body
Jul
14
accepted The geometric interpretation for extension of ideals?
Jul
14
comment The geometric interpretation for extension of ideals?
In fact I'm not sure about these. I didn't go into these deep theories, but I looked up Atiyah & Macdonald pp 47, ex 21 iv) to find the definition of the fiber. I haven't gone into sheaves, schemes, etc, but only trying to obtain some rough idea on this. Sorry for my ignorance. Thanks!
Jul
14
comment The geometric interpretation for extension of ideals?
But the fiber of ${}^af$ over a prime ideal $P$ is $\operatorname{Spec}(k(P)\otimes_BA)$, not $\operatorname{Spec}((B/P)\otimes_BA)$, where $k(P)=\operatorname{Frac}(B/P)$?
Jul
14
asked The geometric interpretation for extension of ideals?
Jul
6
asked Counterexamples for lcm-gcd identity and modular law for rings
Jul
5
accepted Topological/homotopical classification for 1-dim CW-complexes?
Jul
5
comment Topological/homotopical classification for 1-dim CW-complexes?
Very good. Thanks!
Jul
5
comment Topological/homotopical classification for 1-dim CW-complexes?
Yes, and in fact, the if part is tautological, so we only need to tackle with the only if part.
Jul
3
comment Why not $SL_n (\mathbb R)$ in this exercise
There's another way to show that $A\in SL_n(\mathbb Z)\implies A^{-1}\in SL_n(\mathbb Z)$. Note that $A^{-1}=(\det A)^{-1}\operatorname{adj}(A)$ where $\operatorname{adj}(A)$ is the adjugate matrix.
Jul
3
comment Topological/homotopical classification for 1-dim CW-complexes?
Yes, but when they are removed, I cannot see any reason that two non-isomorphic graphs are not homeomorphic.
Jul
3
comment $f:\mathbb{R}\to \mathbb{R}$ continuous and $\lim_{h \to 0^{+}} \frac{f(x+2h)-f(x+h)}{h}=0$ $\implies f=$ constant.
There's only a minor gap: It only shows that $\lim_{h\to0^+}(f(x+h)-f(x))/h=0$, not $f'(x)=0$. But in this weaker condition $f$ is also a constant (It's known that if the Dini derivative $D^+f\ge0$ everywhere, then $f$ is nondecreasing).
Jul
3
comment Topological/homotopical classification for 1-dim CW-complexes?
But then are non-graph-isomorphic thrown graphs non-homeomorphic? I guess so but I cannot conceive a proof.
Jul
2
awarded  Curious
Jul
2
awarded  Inquisitive
Jul
2
comment Topological/homotopical classification for 1-dim CW-complexes?
The difference between graph-iso and homeo is that intuitively we can merge two adjacent edges and delete the joint vertex if it's only connected with these edges. It's inequivalent, but to my intuition, it could be shown that homeo problem is somewhat no easier than graph-iso problem.
Jul
2
comment Topological/homotopical classification for 1-dim CW-complexes?
Thanks. I do have Hatcher but I didn't read it. Any idea on topological classification?