Let $X$ be a nonsingular variety over an algebraically closed field $k$, and let $Y \subset X$ be a nonsingular subvariety. Consider the blowup $p: \tilde{X} \to X$ of $X$ along $Y$, with exceptional divisor $i: D \hookrightarrow \tilde{X}$.
In this context, one can show that the sequence $$ 0 \to p^* \Omega^1_{X} \to \Omega^1_{\tilde{X}} \to i_* \Omega^1_{D/Y} \to 0 $$ is short exact. In particular, there is a natural quasi-isomorphism of cotangent complexes $L_{\tilde{X}/X} \simeq \mathbf{R} i_* L_{D/Y}$.
Does this latter statement (phrased in terms of cotangent complexes) generalize to the case when we allow our varieties to be possibly singular? If so, what hypotheses are necessary?
