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

@ZevenKorian I knew that if I followed some folks on Mathstodon sooner or later I'd see a toot like this, that makes absolutely no sense to me, but I was willing to take the risk.

Correct on all counts.

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, $$R \cong R\oplus R\cong R\times R$$

Does anyone have any idea what this character is for: ⋳
"element of with vertical bar at end of horizontal stroke (U+22F3)"

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

@christianp My library is full of this one

anyway just to be sure let's do things formal

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

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