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Aug
20
comment Cover a sphere by two closed subsets not containing a closed self-antipodal connected subset?
@Timkinsella Good. Make it an answer and I'll accept it. Thanks!
Aug
20
asked Cover a sphere by two closed subsets not containing a closed self-antipodal connected subset?
Jul
30
revised Generalizing a statement about direct limits in the category of $A$-modules to other categories
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Jul
30
asked Generalizing a statement about direct limits in the category of $A$-modules to other categories
Jul
24
comment Surjectivity implies injectivity of finitely generated modules, localization?
@user26857 Right, but I only asked for a geometric meaning, not claimed that there should be. I asked since I feel that I don't really understand the determinant trick and it's still a very tricky stuff to me. I don't understand the underlying reason why viewing $M$ as an $A[X]$-module is that efficient, say, to obtain Frobenius normal forms for matrices, or in proof of Nakayama's lemma.
Jul
23
revised Surjectivity implies injectivity of finitely generated modules, localization?
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Jul
23
comment Surjectivity implies injectivity of finitely generated modules, localization?
@MartinBrandenburg Yeah, it might be frustrating. Incidentally, is there any geometric viewpoint of viewing $M$ as an $A[X]$ module by action $X.m=f(m)$?
Jul
23
comment Surjectivity implies injectivity of finitely generated modules, localization?
@MartinBrandenburg I didn't understand your point, but by Nakayama, the surjectivity of $M\otimes_k k\xrightarrow{f\otimes1}M\otimes_k k$ implies the surjectivity of $M\xrightarrow fM$. See, for example, Atiyah & Macdonald, Ex 2.10.
Jul
22
revised Surjectivity implies injectivity of finitely generated modules, localization?
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Jul
22
revised Equation for the length of a chord parallel to either the minor or major axis in an ellipse
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Jul
22
answered Without Lebesgue
Jul
22
revised An Orthogonal Projection with Weighted Norm
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Jul
22
comment Surjectivity implies injectivity of finitely generated modules, localization?
@user26857 Maybe $N$ isn't essential.
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
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Jul
14
revised The geometric interpretation for extension of ideals?
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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 $k(P)=\operatorname{Frac}(B/P)$?