today she had a tantrum about not being able to wear the shirt she wanted to day care. it did wonders for my mental health.
because the day care is a phony baloney anticapitalist utopia, they have a strict policy of no animated logos or other figures on clothing. if we hadn't taken it off they would have swapped for something else she wouldn't have liked.
@TedShifrin I expected some statement only on $q = n$ like $H_q(X;G) = G$ for $q = n$ and no condition on other degrees. For example, torus and sphere make sense. May be some statement holds for compact orientable manifolds?
hi... i had a question about tangent vectors. as i understand it one can define a tangent vector at a point $p$ on a real smooth manifold $M$ as a $p$-derivation $D:C^\infty_p(M)\rightarrow\mathbb{R}$. i am wondering if there is a way to generalize this to commutative algebras, i.e. a purely algebraic way to understand $C^\infty_p(M)$ with reference only to $C^\infty(M)$.
Hello, can someone please verify that this is true: for all $i\in N$ if $f_i:A_i\to B_i$ are continuous, then $\prod f:A_1\times A_2\times\cdots\to B_1\times B_2\times\cdots$ is continuous.
@leslietownes it seems to me the proper definition here would be a derivation $C^\infty_p(M)$, but the trouble i'm having is reconstructing $C^\infty_p(M)$ from $C^\infty(M)$. sorry, i think i'm being unclear
hmm, i suppose in the case of real smooth manifolds one can simply use $C^\infty(M)$ to define tangent vectors, though. is there any advantage to using $C^\infty_p$?
hmm, maybe it's overly optimistic of me then to wish for a generalization encompassing those other categories... i imagine then we can't use localizations of the global sections either? but i'm very inexperienced with these other areas
well, i guess i was curious if there was a way of defining tangent spaces based on how it's done in the real case which extends a bit further than the real case
i think i'm fumbling around in territory which i'm not suited to study yet though
It can be written in terms of derivations too, it makes no difference. Let $(A, \mathfrak{m})$ is a commutative local $k$-algebra, where $k = A/\mathfrak{m}$ is the residue field.
A derivation is a $k$-linear map $D : A \to k$ such that $D(ab) = [a]D(b) + [b]D(a)$.
$[a]$ is like the "value" of $a$ at $\mathfrak{m}$, which is $0$ in $k = A/\mathfrak{m}$.
The $k$-vector space of derivations is $(\mathfrak{m}/\mathfrak{m}^2)^*$
This observation tells you that one can do much of calculus in an abstract setup with rings and so on (by which I really mean schemes, but never mind).
Of course, whether such perverse generalizations of calculus deserve to be called calculus ...
yeah, my question was definitely inspired by "algebraic generalizations" of calculus - i had encountered "differential calculus on commutative algebra" although i think this might be a bit different
math wikipedia is very eclectic. there's really good stuff and really mediocre stuff with no pattern (e.g. interest v. obscurity) to explain it. entries will have definitions, examples, sometimes theorems, sometimes cites to papers or history, sometimes not. i guess because most users and editors only zero in on the thing they are interested in. but from a step back it looks pretty weird.
most of wikipedia is kind of like this by its nature but it seems particularly acute in an relatively organized field with lots of talented enthusiast wikipedians.
