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Sep
11
comment Is every local ring a valuation ring?
But, $F$ isn't the field of fractions of $\{0,1\}\subseteq F$?
Sep
11
comment Difference between two concepts of homotopy for simplicial maps?
@ZhenLin It seems that you are right, if the definition of the product is essentially same as the one in Goerss & Jardine. It might be a mistranlation from Russian.
Sep
11
comment Difference between two concepts of homotopy for simplicial maps?
@QiaochuYuan Simply homotopy of $f,g\colon X\to Y$ here, informally speaking, is just a simplicial map $F\colon\Delta[1]\times X\to Y$ from a cylinder $\Delta[1]\times X$ (triangulated canonically, where $\Delta[1]$ is just the simplicial set associated with $I=[0,1]$, i.e., $1$-simplex) to $Y$, both of which are simplicial sets. Sorry for my ignorance. I don't know what you're referring to. Thanks, anyway.
Aug
27
comment 6.17 Theorem : Show that $f \ \ \in \mathfrak R(\alpha)$ if and only if $ f\alpha' \ \ \in \mathfrak R$ ( walter rudin)
It follows from Rudin's argument that the upper and lower (Darboux) integrals of $fd\alpha$ and $f\alpha'dx$ are the same.
Aug
27
comment Differential identity and wedge products
$f\omega$ is just $f\wedge\omega$ where $f$ is considered as a $0$-form, and $d(\omega_1\wedge\omega_2)=d\omega_1\wedge\omega_2+(-1)^p\omega_1\wedge d\omega_2$ where $\omega_1$ is a $p$-form.
Aug
27
comment $f\in C(\mathbb{R})$. What does it mean?
But it's usually denoted as $C^n(\mathbb R)$ or $\mathcal C^n(\mathbb R)$ as well as $\mathcal C^0(\mathbb R)$ without parentheses.
Aug
27
comment A question on short exact sequences.
I found learning diagram chasing isn't that easy for a beginner from books, but there're two materials which seems more accessible: Hatcher's Algebraic Topology, pp115, subsection Relative Homology Group, and Eisenbud's Commmutative Algebra, pp637, subsection A3.7.
Aug
26
comment Differentiability of non-analytic complex functions
Well, and by Looman-Menchoff theorem, a continuous complex function is holomorphic in some region if and only if it satisfies Cauchy-Riemann equations everywhere.
Aug
24
comment Homology of a finite graph follows from Mayer-Vietoris sequence?
@LeeMosher Thanks. I've edited to include these.
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!
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
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
comment Surjectivity implies injectivity of finitely generated modules, localization?
@user26857 Maybe $N$ isn't essential.
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
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.