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Given any set $S$ of complex numbers without accumulation points, you can find an entire function with $S$ as its set of zeroes (this is known as Weierstrass's theorem). So the answer to your question …

This is a consequence of the following result: if $A \to B$ is a flat local homomorphism of local rings, $A$ and $B/\mathfrak{m}_AB$ are reduced, then $B$ is reduced. Keeping in mind that reduced is e …

I think so. It seems to me that the two definitions proposed by Brian are equivalent. Let us use the following: a coherent sheaf $F$ on a reduced space $X$ is torsion-free if whenever $a$ is a non-zer …

Consider a smooth surface $Y$ with a point $p\in Y$. Let $X$ be obtained by gluing two copies of $Y$ at $p$, with the obvious morphism $X \to Y$. This is surjective and finite, and has finite Tor-dime …

The Lagrange inversion formula allows to give a combinatorial interpretation of the Jacobian conjecture. See for example the classic paper of Bass, Connell and Write, The Jacobian conjecture: reductio …