I asked this question on Mathematics Stackexchange, but got no answer. Let $B$ be a commutative ring with $1$, let $A$ be a subring such that any unit of $B$ which belongs to $A$ is a unit of $A$, and let $\phi:F\to F$ be an endomorphism of a free $A$-modules $F$ such that $B\otimes_A\phi:B\otim...

Here are two beliefs. I think everybody will agree that one of them, at least, is false. I adhere to the second one. Belief 1. The simplest way to compute the exponential $e^A$ of a complex square matrix $A$ is to use the Jordan decomposition. Belief 2. It's simpler and more efficient to use t...

Here are two beliefs. I think everybody will agree that one of them, at least, is false. I adhere to the second one. Belief 1. There is no simple generalization of the Hodge Theorem to noncompact manifolds. Belief 2. The most naive statement which would, if true, generalize the Hodge Theorem ...

EDIT. Here is the part of the answer that has been rewritten: We give below a short proof of the Fundamental Theorem of Galois Theory (FTGT) for finite degree extensions. We derive the FTGT from two statements, denoted (a) and (b). These two statements, and the way they are proved here, go back ...

Here is a minor variation of the proof. We show that there is a subset $Z$ of $\mathbb R^2$ which intersects each line exactly twice. We define the ordinals in such a way that each ordinal $a$ is the set of those ordinals $< a$. For any set $S$ we write $|S|$ for the least ordinal equipotent to...

The answer is no. There are counterexamples already for surfaces, due to X. Gang and F. Campana (unpublished). The link to Campana's article is here, and the relevant result is Proposition 0.1 Il existe des surfaces projectives complexes $S$, $S_0$ simplement connexes et de type général ho...