Generalisation of a result on Kahler differentials

Question

Let $B$ be a local ring which contains a field $k$ of characteristic zero, isomorphic to its residue field $B/\mathfrak{m}$. We know that the map $\delta:\mathfrak{m}/\mathfrak{m}^2 \to \Omega^1_{B/k} \otimes_B k$ is an isomorphism, where $\delta(\overline{b})=db \otimes 1$. Assume that $B$ is regular. Is it true that for any $n > 2$, we have an isomorphism induced by $\delta:\mathfrak{m}/\mathfrak{m}^n \to \Omega^1_{B/k} \otimes_B B/\mathfrak{m}^{n-1}$?

Answer 1

First of all: the answer, in general, is no. See below for a counterexample.

Second: I'll tell you why I don't think this is a 'natural' or correct thing to expect, at least in algebraic geometry, say if $k$ is algebraically closed and $B$ is a localization of a finitely-generated $k$-algebra.

Since $B$ is regular, $\Omega^1_{B/k}$ is a free module of rank equal to the dimension, say $\dim B = d$. This is Hartshorne Theorem II.8.8. (Your stated isomorphism for $n=2$ is Proposition II.8.7).

So, fix a minimal generating set $x_1, \ldots, x_d$ for $\mathfrak{m}$; then two things are true: first, $dx_1, \ldots, dx_d$ freely generate $\Omega^1_{B/k}$. Second, both of the modules you're asking about are unchanged if we pass to the completion $\widehat{B}$, which is isomorphic to a power series ring in $x_1, \ldots, x_d$.

Now $B / \mathfrak{m}^{n-1} = \widehat{B} / \widehat{m}^{n-1}$ has a $k$-basis given by "monomials of degree $\leq n-2$ in $x_1, \ldots, x_d$". So then $\Omega^1_{B/k} \otimes B / \mathfrak{m}^{n-1}$ is $d$ copies of this, for $d \cdot {d+n-1 \choose d+1}$ monomials in all.

On the other hand, $m / m^n$ has a $k$-basis given by "monomials of degree between $1$ and $n-1$ in $x_1, \ldots, x_d$", or ${d+n \choose d+1} - 1$ monomials in all.

These are just not the same sets of monomials! For instance, if $B$ is the localization of $k[x,y]$ at $(x,y)$, then $\mathfrak{m} / \mathfrak{m}^3$ has a $k$-basis given by $\{x, y, x^2, y^2, xy\}$ (5 terms in all). On the other hand, $(B/\mathfrak{m}^2)^{\oplus 2}$ has 6 basis elements (two copies of $\{1,x,y\}$ -- an even number, incidentally).