@Tobias finally finished that cool exercise 15. Took me a while to come up with some steps, really nice. Wish we had that interesting group theory exercises in my abstract algebra class
It's amazing how many things you can deduce from a simple condition like every non-trivial subgroup is a p-group for some p
@ÍgjøgnumMeg If you wish to do the other exercises, there are two errors in them. Unfortunately, it does not seem like it is possible to update the document on pure in any easy way
I would actually love a seperate course on finite groups where you do more advanced stuff like Burnside transfer, Frattini groups or stuff like that, but there's no prof at my uni working on finite groups, so I guess it's unliekly
@ÍgjøgnumMeg It is in exercise $3$, where I originally had the second coordinate be with values in $\{0,1\}$ instead of $\pm 1$, and I forgot to change this in the last part of the exercise
@MatheiBoulomenos Yeah, the university decided that group theory wasn't worth teaching so they offloaded it into some stupid 4 week course that nobody really cared about
@MatheiBoulomenos All of my algebra/number theory knowledge has been self-taught because the university doesn't value pure subjects at all, unfortunately :(
I am having an algebra crisis in the sense that I want to learn Galois theory but I can't figure out how or where to start. One of the problems being that I know a lot of the things here and there and superficially, but also that it's hard for me to read algerba
@BalarkaSen It will please you to know that my motivational example for introducing rings will have a slight geometric leaning (integer points on a circle with the square root of an integer as radius)
@TobiasKildetoft That's fair, I'm supposed to start a "first" course on Algebra (in my final undergrad. year!), which will be a free module (ha) as far as I'm concerned
@Tobias our course went like this: monoids (though we only proved trivial facts), basic stuff on groups, rings (though we really did mostly Gauss's lemma), fields / Galois theory, then groups again including Sylow and solvablity, and then applications of Galois theory
@LeakyNun I feel like my brain just can't cope with algebra. I felt the same shitty way when learning algebraic geometry two years back from an actual reading course.
@PyRulez It looks as though we only have weak lower bounds on Tree(3), not the actual value or upper bound expressions. What math can we use to determine if some number is larger than it? — benzeneAug 16 at 19:12
@benzene the third note has some upper bounds. We don't have tight upper bounds, but we do have upper bounds. I listed some, in fact. — PyRulezAug 16 at 19:13
@ÍgjøgnumMeg so you mean like in high school? That's really cool, we only did boring stuff like calculus or linear algebra in $\Bbb R ^3$ in my country's equivalent of high school
@MatheiBoulomenos Yeah, basically, Abitur-level, there was one tiny bit of group theory in one of the modules and I got really interested in it and then I watched a documentary on Fermat's Last Theorem and found out about Algebraic Number Theory
@MatheiBoulomenos And now the majority of my time spent at university has been self-studying algebra and number theory alongside the boring crap that my uni teaches
@ÍgjøgnumMeg I actually considered the original faulty proof of FLT as another possible motivation for the introduction of rings, but it seemed to get a bit too technical to explain
@MatheiBoulomenos The definition of "regular prime" seems weirdly unmotivated when you first read the statement of the proof until you hear about class groups
@TobiasKildetoft No what I mean is, it seems like a weird condition to put on a class of primes until you realise why you need the condition, if you know what I mean
@ÍgjøgnumMeg But if you have a group associated to a prime, then it seems very natural to me to ask whether the prime divides the order of its own group
@Tobias does group cohomology also have applications in pure finite group theory? I'm taking a seminar right now and we mostly focus on number theory stuff apart from the homological algebra machinery
@TobiasKildetoft I know about the interpretation of $H^1$ and $H^2$ in terms of extensions. I was just wondering if you can prove some interesting lemmas using that
@MatheiBoulomenos As I said, I mostly come across it in the more general setting of Ext groups, and where the representations themselves are the objects of interest
Well, I'm used to define the Picard group of a ring using invertible modules, then you don't need to invoke Serre-Swan. But I guess that doesn't work for non-affine schemes
I guess that definition is non-standard, I just know it from some exercises I did
@Balarka I think it's something like vector bundles are exactly sheaves where the global sections of the sheaf are projective modules over the the projective sections of the scheme
Whenever I want to do mathematics, I feel like just looking at it and understand as much as a visitor of an art gallery understands by looking at paintings.. More like someone who enjoys looking at stuff, but is unable to do something by himself..
I'm thinking the argument should be something like this. One can pick three appropriately chosen independent holomorphic vector fields on $\Bbb P^1$, and send $(p, \xi)$ to $(\xi(v_1(p)), \xi(v_2(p)), \xi(v_3(p))$. I think with appropriately chosen $v_i$ this will satisfy eg $w_3^2 = w_1 w_2$.
There's a proof that every finite division ring is commutative, but that involves a little bit more than the class equation (namely cyclotomic polynomials)
@MikeMiller If you blowup $X$ at $p$ that's a subspace of $X \times \Bbb P^2$ with equations like $x_i u_j = x_j u_i$ where $u_i$ are homogeneous coordinates for $\Bbb P^2$ and $x_i$ are coordinates for $X$. Working that out explicitly should tell you something, I guess. I am trying to see the picture.
If you blowup, the exceptional divisor gets a natural $O(-1)$ over it as a normal bundle neighborhood. Somehow the local structure of the cone makes it $O(-1) \otimes O(-1)$.
Strange
(I think your idea is better)
So you get a cylinder $\Bbb P^1 \times \Bbb C$ (as intuition tells you after you desingularize) but with a twist on $\Bbb P^1 \times \{0\}$
Quote: "If $X$ is a space having a countable basis, then any discrete subspace $A$ of $X$ must be countable." What exactly does this mean? Does it mean that if $A$ is a subset such that its subspace topology is discrete, then $A$ must be a countable set?
To resolve that singularity blow up the origin of A^2. You get O(-1) with the involution given fiberwise by -1. This carries a map to O(-2) invariant under the action.
This should be equivalent IMO to the blowup of A^2/+-1 = cone at the singular point
In fact I think there should be some easy statement like "A map of varieties that induces a map of tangent cones induces a map on blowups"
Hello!! Suppose that the plane P contains the points A, B, C. Then we have that $\vec{OA}$ and $\vec{BC}$ are on the plane, or not? Then is $\vec{OA}\times \vec{BC}$ perpendicular to the plane P?
I want to check which of the following is perpendicular to the plane P: 1) $\vec{OA}\times \vec{BC}$ (which is not) 2) $\vec{AB}-\vec{AC}$ 3) $\vec{OA}\times\vec{OB}-\vec{OA}\times\vec{OC}$ 4) $\vec{OA}\times\vec{BC}-\vec{OB}\times\vec{BC}$
I think 2 is not, because it is parallel to the plane, isn't it? About the other two I don't really know. Could you give me a hint? @LeakyNun
Ah we have the following: 3) $\vec{OA}\times\vec{OB}-\vec{OA}\times\vec{OC}=\vec{OA}\times (\vec{OB}-\vec{OC})=\vec{OA}\times (-\vec{BO}-\vec{OC})=-\vec{OA}\times \vec{BC}$ 4) $\vec{OA}\times\vec{BC}-\vec{OB}\times\vec{BC}=(\vec{OA}-\vec{OB})\times\vec{BC}=-(\vec{AO}-\vec{OB})\times\vec{BC}=-\vec{AB}\times\vec{BC}$
@TedShifrin for real inner-product spaces we have the result $\langle x,y \rangle = \dfrac 12 (\|x+y\|^2 - \|x\|^2 - \|y\|^2)$, can we generalize this result for complex inner-product spaces?
