ℍᵃⁿⁿᵃʰ 🍀 @ZevenKorian@mathstodon.xyz

still pissed codimension of a smooth map is only upper semicontinuous

i found a dissertation whose topic was precisely giving a combinatorial proof of that theorem and i basically had to leaf through it to understand the statement

that being said i guess the risk was calculated because i can usually find a statement for each one and udnerstand it more or less immediately

except the Alexander duality whew that one

and where to draw the line like i'm following a series of lectures on Milnor's bouquet theorem and whew the author won't shy away from throwing around homology theorems

idk if you ask me the hardest problem in math is not P=NP, the Riemann hypotesis or the Collatz conjecture

it's determining how much detail you need to give to your target audience lol

update the notion of basis as a linearly independent generating set is well defined for modules over unit rings BUT two given bases may not be bijective

example: if \(R\) is a unit ring, \(R \cong R\oplus R\cong R\times R\)

i'd even consider selling the pdf but i don't know how much i should charge; it's kinda lengthy but it's not like i am an algebra student so they're definitely not going to be the best

maybe i could speedrun getting my 4th year commutative algebra notes up to speed

it s a v good ring, really; it is noetherian, semi-local, etc

it's not an arbitrary module as such, well; the only thing that the parameter ring lacks to be a division ring (which pretty much grants you everything a vector space should be) is that the maximal ideal is non-trivial (i.e. we have non-zero elements without inverse)

the other day i was thinking whether i can say a linearly independent free generating set (in an arbitrary module) could be called a basis

i remember the notion of basis is somewhat flawed in general, and the example may have been related to taking the direct sum of a ring with itself \(n\) times, but i cannot really remember the finer details

anyway just to be sure let's do things formal

#Introduction
I am a Singularity theory PhD student interested in complex hypersurface germs \(f: (\mathbb{C}^n,S) \to (\mathbb{C}^{n+1},0)\). The nature of my work makes it so I find myself going back and forth between differential and algebraic geometry. I also have some passing interest in the noble art of diagram-chasing.

I speak about math in a very laid-back and offhand way, though.

sub category as in

how functor category theory is 2-category

this would be 0-category ig

left me wondering if you can make some weird and overly engineered sub-category that turns arrows into functors

anyway today in our hatcher study group we were discussing on how Eilenberg-Steenrod holds for singular cohomology (still baffled it's not Exactly the same as dualising singular homology tfw) and i just realised left-right equivalence of maps is kind of a """""naturality""""" condition

also i decided to stop specifying "differential" or "algebraic" geometry because i really do live in a coboundary of both lol