Lemma 23.7.3. Let $R' \to R$ be a surjection of Noetherian rings whose kernel has square zero and is generated by one element $f$. Let $S = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$. Let $\sum r_ j f_ j = 0$ be a relation in $R[x_1, \ldots , x_ n]$. Assume that

each $r_ j$ has coefficients in the annihilator $I$ of $f$ in $R$,

for some lifts $r'_ j, f'_ j \in R'[x_1, \ldots , x_ n]$ we have $\sum r'_ j f'_ j = gf$ where $g$ is not nilpotent in $S/IS$.

Then $S$ does not have finite tor dimension over $R$ (i.e., $S$ is not a perfect $R$-algebra).

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