User loch | chat.stackexchange.com

loch

Apr 11, 2021 14:04

the answer wrote deg_nF there and I suspect the intended meaning of that is 'lowest power of x_n which divides all terms in the expression'

or in other words it's just factoring the largest power of x_n you possibly can

loch

Mar 30, 2021 04:25

i can imagine students used to being spoonfed algorithms might ask such a question
but i never taught linear algebra so lol

loch

Mar 30, 2021 01:47

@XinYuanLi i think if you just wanted to convicne yourself why $f^{-1}(\emptyset) = \emptyset$ you can just write out the definition -- the LHS is $\{ w \in \Omega: f(\omega) \in \emptyset \}$, but clearly this has to be empty lol

loch

Mar 30, 2021 01:40

yes but even if it's not surjective you still have $f^{-1}(S) = \Omega$

loch

Mar 30, 2021 01:37

@XinYuanLi why do you need surjectivity?

loch

Mar 27, 2021 06:55

in that case i think S^{-1}A would be k

loch

Mar 27, 2021 06:51

no

loch

Mar 27, 2021 06:46

@love_sodam that seems to agree with what im saying! the total ring of fractions is what you get when you invert all non-zero divisors, and in that case your original ring injects to the total ring of fractions

loch

Mar 27, 2021 06:42

@leslietownes functional analysis is more weird :^)

loch

Mar 27, 2021 06:41

if you wanted to show that $dim(S^{-1}(A) < dim(A)$ can be arbitrarily loose, then i think you could consider e.g. $k[x_1,...,x_n] \times k$ and invert the multiplicative set $S = \{ (0,1) \}$

geometrically you're looking at the disjoint union of an affine n-space and a point and taking the open set consisting of the point

loch

Mar 27, 2021 06:37

@love_sodam if by ring of fraction you mean en.wikipedia.org/wiki/Total_ring_of_fractions, then it seems like ring of fraction is never zero since R injects into it

loch

Mar 21, 2021 04:38

@AminIdelhaj nice! i think it's less painful than learning hard math :^)

loch

Mar 21, 2021 03:44

*not sure about cool

loch

Mar 21, 2021 03:43

@AminIdelhaj im not sure --- im learning some logarithmic geometry on the math side, and learning some stats in my own time

loch

Mar 21, 2021 03:42

@AminIdelhaj hey! nothing much -- what about you?

loch

Feb 28, 2021 00:55

i think leakys question is why it would make sense to give that a structure of a variety (scheme)

loch

Feb 26, 2021 18:40

@BalarkaSen i don't think i've ever seen people calling a map 'spec k[x]/(x^2) -> X' a geometric point?

loch

Feb 25, 2021 21:58

then you might want to review that first

but essentially the answer consists of two parts:

1) computing the first derivative of the function w.r.t. all the w_i's, and set them all the zero, and the answer argues that the zero is unique

2) check that this really does define a minimum, which is the part about E(w+h) (which amounts to checking that the hessian matrix is positive definite)

this should remind you of single variable calculus, where to find the minimum of a function you first 1) compute the first derivative and set it to zero, 2) check that it genuinely defines a (local) minim…

loch

Feb 25, 2021 21:38

@fido9dido are you familiar with multivariable calculus?

loch

Feb 23, 2021 21:32

@TedShifrin glad that you got vaccinated!

loch

Feb 23, 2021 19:32

not too bad
what about you?

loch

Feb 23, 2021 19:13

hello @TedShifrin
long time indeed!

loch

Oct 30, 2020 17:10

@AminIdelhaj ive been slacking off to figure out industry options so i have to catch up on math now lol

loch

Oct 30, 2020 17:09

@MikeMiller nope 2022

loch

Oct 30, 2020 16:58

@AminIdelhaj long time no see!

loch

Jul 8, 2020 09:48

(as far as im aware) - but i think the papers are pretty readable (definitely more readable than the levine paper lol)

loch

Jul 8, 2020 09:48

if any

loch

Jul 8, 2020 09:48

no there arent a lot of notes available

loch

Jul 8, 2020 09:48

yes

loch

Jul 8, 2020 09:46

i just know that to see the stuff 'in action', you might want to look at the stuff that wickelgren did on counting things over non-algebraically closed fields

loch

Jul 8, 2020 09:41

@Alex the answer is not much esp. with the technical stuff :p

loch

Jul 7, 2020 02:45

@Alex i remember trying to read this a long time ago

loch

Jun 21, 2020 04:50

@BalarkaSen ncatlab.org/johnbaez/show/Entropy+as+a+functor :0)

loch

Jun 7, 2020 10:38

@LeakyNun unless things changed u need both

loch

May 29, 2020 03:44

@StanShunpike you'd see polytopes showing up in toric geometry

loch

May 25, 2020 14:32

that sounds right
do you not need any assumptions on f ? (idk)

loch

May 25, 2020 14:25

so i guess step 1 of the story is - convince yourself that it's ok to do topology with sheaves (e.g. singular cohomology = sheaf cohomology of constant sheaf etc.)

and that sheaves come with a bunch of abstract nonsense so you can apply them to topology

(i dont know what the further steps are)

loch

May 25, 2020 14:21

it allows people to do topology without seeing the topology
for example your various long exact sequences in topology actually come from morphisms you get from adjunctions of functors...

but more seriously somehow people figured out that sheaves are very powerful in topology - eg like balarka mentioned its good for singular spaces

and singular spaces really do show up quite often in geometry, esp. 'relatively' - eg fibers of a morphism between two smooth spaces can be singular. in topology there's leray spectral sequence where when you have a fibration you can relate the topology of the b…

loch

May 25, 2020 14:06

@BalarkaSen self-harm is bad

loch

May 25, 2020 14:05

unfortunately my topology is too abysmal for me to decide if what you're saying makes sense :(

but anyway -- i think of GRR as just a relative version of Riemann-Roch (taking Y to be a point recovers Hirzebruch-Riemann-Roch). this 'makes sense', except one might wonder what does a 'relative version of Riemann-Roch' mean ---

so maybe it's worth staring at Hirzebruch Riemann-Roch for a second. tl;dr - it says that the holomorphic euler characteristic of a vector bundle can essentially be computed by integrating some chern classes of the vector bundle (and the todd class, which is data from …

loch

May 24, 2020 10:10

sad

loch

May 21, 2020 04:08

Torsion-free abelian groups of rank 1

Infinitely generated abelian groups have very complex structure and are far less well understood than finitely generated abelian groups. Even torsion-free abelian groups are vastly more varied in their characteristics than vector spaces. Torsion-free abelian groups of rank 1 are far more amenable than those of higher rank, and a satisfactory classification exists, even though there are an uncountable number of isomorphism classes. == Definition == A torsion-free abelian group of rank 1 is an abelian group such that every element except the identity has infinite order, and for any two non-identity...

loch

May 14, 2020 04:51

@TedShifrin hi @TedShifrin

loch

May 13, 2020 06:34

@feynhat sounds right

loch

May 11, 2020 19:25

meant to say integrals of polynomials oops

loch

May 11, 2020 10:42

@user736948 xy, xyx, xyxy,...

loch

May 8, 2020 17:53

what if you use rational trigonometry

loch

May 6, 2020 01:27

@WilliamSun i think it’s very hard to learn from stacks project and would recommend you learn commutative algebra through other sources instead. The typical recommendation is atiyah macdonald which i is good but i personally prefer ones with a geometric flavour

loch

May 5, 2020 17:43

i did

loch

May 5, 2020 17:31

well, you should think of prime ideals as corresponding to irreducible (reduced) (closed) varieties.

in this case, let's take k to be alg. closed. Then your maximal ideals are closed points (0-diml varieties). They look like (x-a,y-b)

Your prime ideals which are not maximal or 0 are necessarily going to have height 1 (if i rmb my commutative algebra correctly), which just means geometrically that they correspond to 1-dimensional varieties in \C^2.

But of course that means that they are hypersurfaces - you can show that in particular they're given by the zero locus of an irred polynomial.…

loch

Mathematics

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