Hello, this one has always been mysterious to me. The Kähler differentials $\Omega_{A/k}$ are definined, by the universal property $$Der_k(A,M)=A-Mod(\Omega_{A/k},M)$$ so for $M=A …

One way to define the algebra of differential forms $\Omega(M)$ on a smooth manifold $M$ (as explained by John Baez's week287) is as the exterior algebra of the dual of the module …

Define an "eventual counterexample" to be $P(a) = T $ for $a < n$ $P(n) = F$ $n$ is sufficiently large for $P(n) = T\ \ \forall n \in \mathbb{N}$ to be a 'reasonable' conjectu …

Somehow this question made me think of instances of small exceptions in general, and I remembered the statement I heard once that $S_5,A_6,S_6,A_7,A_8,S_8$ are the only instances o …

I'm a bit confused by the double role which sheaves play in the theory of stacks. On the one hand, sheaves on a site are the obvious generalization of a sheaf on a topological spa …

In complex projective geometry, we have a specified Kähler class $\omega$ and we have a Lefschetz operator $L:H^i(X,\mathbb{C})\to H^{i+2}(X,\mathbb{C})$ given by $L(\eta)=\omega\w …

Before you close for "homework problem", please note the tags. Last week, I gave my calculus 1 class the assignment to calculate the $n$-volume of the $n$-ball. They had finished …

I am no expert in the field. I hope the question is suitable for MO. Background/Motivation I once followed a quantum mechanics course aimed at mathematicians. Instead of the usua …

I'm looking for a "conceptual" explanation to the question in the title. The standard proofs that I've seen go as follows: use the Schubert cell decomposition to get a basis for co …

As the title says, please share colloquial statements that encode (in a non-rigorous way, of course) some nontrivial mathematical fact (or heuristic). Instead of giving examples h …

My apologies if this is too elementary, but it's been years since I heard of this paradox and I've never heard a satisfactory explanation. I've already tried it on my fair share o …

A standard example for demonstrating the need for genuinely weak n-categories is that a weak 3-category with unique 0- and 1-cells amounts to the same thing as a braided monoidal c …

In various contexts, I have come across results referred to as "big monodromy." A standard arithmetic example is the open image theorem for the image of Galois action on non-CM el …

The Riemann-Stieltjes-Integral $\int_a^bf(x)dg(x)$ is a generalization of the Riemann Integral. It is e.g. heavily used as a starting point for stochastic integration. The approxim …

The Poisson summation says, roughly, that summing a smooth $L^1$-function of a real variable at integral points is the same as summing its Fourier transform at integral points(afte …