The Stacks project

Lemma 109.14.2. There exists a local homomorphism of local rings $A \to B$ and a regular sequence $x, y$ in the maximal ideal of $B$ such that $B/(x, y)$ is flat over $A$, but such that the images $\overline{x}, \overline{y}$ of $x, y$ in $B/\mathfrak m_ AB$ do not form a regular sequence, nor even a Koszul-regular sequence.

Proof. Set $A = k[z]_{(z)}$ and let $B = (k[x, y, z] \oplus E)_{(x, y, z, E)}$. Since $x, y, z$ is a regular sequence in $B$, see proof of Lemma 109.14.1, we see that $x, y$ is a regular sequence in $B$ and that $B/(x, y)$ is a torsion free $A$-module, hence flat. On the other hand, there exists a nonzero element $\delta \in B/\mathfrak m_ AB = B/zB$ which is annihilated by $\overline{x}, \overline{y}$. Hence $H_2(K_\bullet (B/\mathfrak m_ AB, \overline{x}, \overline{y})) \not= 0$. Thus $\overline{x}, \overline{y}$ is not Koszul-regular, in particular it is not a regular sequence, see More on Algebra, Lemma 15.30.2. $\square$


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