/sci/ - Science & Math

>>12616552
OK, so I can give you an example that will not be a toy example. It's known as the Universal Coefficient Theorem. I don't know how much you know about homology, so I'll give you the basics first. If it sparks your interest, I recommend Rotman's book on algebraic topology.

First of all, a chain complex of [math]R[/math]-modules would be a sequence of modules and their homomorphisms [math]C_* = \cdots \to C_{n+1} \xrightarrow{\partial_{n+1}} C_n \xrightarrow{\partial_n} C_{n-1} \to \cdots[/math], one for each integer, such that [math]\partial_n \circ \partial_{n+1} = 0[/math] for all indices. This is the same thing as [math]\text{im} (\partial_{n+1}) \le \ker(\partial_n)[/math], so we may loo at the homology of this complex, and it is given by [math]H_n(C_*) = \ker(\partial_n)/ \text{im}(\partial_{n+1})[/math], for any integer n. Notice that vanishing homology for some n means exactness at [math]C_n[/math] in the complex. Let's then assume we are interested in the homology of some space [math]X[/math], and that [math]C_* = C_*(X; \mathbb{Z})[/math] is for example the singular chain complex (and now the ring is the integers). The groups in the complex vanish for negative indices, and for non-negatives we obtain [math]H_n(X; \mathbb{Z}) = H_n(C_*)[/math], the nth (integral) singular homology group of [math]X[/math]. So far so good, no need for exact sequences.

>>12586266
I actually checked it. There's a new formula that takes the height(/length? Is a human high or long?) into account and the calculator gave 17.6 classically and 16.7 in the new way. Winters are a bit chilly, to be desu.

>>12586277
>I pictured an outgoing not-so-old, but maybe your supervisor just hangs out with peers from time to time
60 this year. I hope I get to see him live around that, so that I may give him a present.
>When are you meeting him? It sounded like you sort of promised stuff until a date.
I think the semester starts next week or the one after that, so probably then. I promised to try get some examples done, but no luck with that so far.
>Did you enjoy that algebraic number theory course?
If I remember correctly, somewhat yes. Not the best thing ever, but it was nice to get some more motivation for ring theory (many parts of which are essentially created to solve number theoretical problems).