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A Non-reduced Characteristic-zero Algebra with Vanishing Cotangent Complex

Author: Wei Yang · Added: September 25, 2026

Abstract

For every field $k$ of characteristic zero, we construct a local algebraic $k$ -algebra $A$ with $L_{A/k}\simeq0$ , $A_{\mathrm{red}}\cong k$ , and a nonzero square-zero element. In particular, $A$ is not ind-étale over $k$ . The construction is a sequential filtered colimit of finite local complete intersections: the transition maps annihilate cotangent homology, while an explicit exponent bound preserves a chosen nilpotent. It answers Bhatt’s existence question and the non-reduced formulation of Mondal and Mukhopadhyay affirmatively. For $k=\mathbb{Q}$ , it also realizes the case $A_{\mathrm{red}}\cong\mathbb{Q}$ singled out in Antieau’s Question 6.