jmc

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Aug
22
 awarded  Supporter
May
9
 comment Recovering a number from a remainder list
@anon Concerning your second comment, the residue of such an $x$ consists of the images under the canonical projections. In some sense they form the defining data of $x$, so yes, it is equivalent to knowing $x$.
May
9
 comment Recovering a number from a remainder list
@anon Concerning the first comment: I think it is still correct to say that the residues need not determine an integer (exactly because the $x \in \hat{\mathbb{Z}}$ that they determine is not in the image of $\mathbb{Z}$). I might indeed have added the word 'profinite', but I did not want to make it to difficult at first. I added extra theory in the next paragraphs on $p$-adics, so readers should be able to find their way to the deeper explanation. [Finally, yes, I might have written this answer a bit quickly, and it might have been better if I spent more time on it.]
May
8
 comment Recovering a number from a remainder list
Indeed the map $\mathbb{Z} \to \hat{\mathbb{Z}}$ is injective. But my point is, that a list of residues need not come from an integer (but if it does, then it is uniquely determined).
May
8
 answered Recovering a number from a remainder list
May
8
 comment Principal divisors on a compact Riemann surface
Shouldn't the sum in the definition of f(z) be a product?
May
8
 comment What are some examples of a mathematical result being counterintuitive?
@MichaelHardy Ok, that makes more sense (-;
May
8
 comment What are some examples of a mathematical result being counterintuitive?
So the one-point space with your favourite topology is counterintuitive? Interesting...
May
8
 comment How much is a topological space $X$ determined by the category of sheaves of abelian groups on $X$?
To write down something pretty obvious: If you don't put conditions on $X$ and $Y$ you can certainly not conclude that $X$ and $Y$ are homeomorphic if $\mathsf{Sch}_{\mathsf{Ab}}(X)$ and $\mathsf{Sch}_{\mathsf{Ab}}(Y)$ are equivalent. Take two finite spaces with the trivial topology, that have different cardinality. [I think my point is that if you can say something, you will say it about $\mathsf{Open}(X)$ and $\mathsf{Open}(Y)$, since sheaves do not see points, but only opens.]