Lemma 10.134.6 (07BP)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.134: The naive cotangent complex
Lemma 10.134.6 (cite)

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Lemma 10.134.6. Let $A \to B$ be a surjective ring map with kernel $I$ . Then $\mathop{N\! L}\nolimits _{B/A}$ is homotopy equivalent to the chain complex $(I/I^2 \to 0)$ with $I/I^2$ in degree $1$ . In particular $H_1(L_{B/A}) = I/I^2$ .

Proof. Follows from Lemma 10.134.2 and the fact that $A \to B$ is a presentation of $B$ over $A$ . $\square$

Comments (2)

Comment #5946 by Zhouhang MAO on February 23, 2021 at 12:25

"In particular, $H_1(L_{B/A})=I/I^2$ . Here it should be $H_1(NL_{B/A})=I/I^2$ (although the statement for the cotangent complex is correct, this is not what you want to say here).

Comment #6132 by Johan on April 30, 2021 at 13:46

See Definiition 10.134.1. I agree this may be confusing., but I am going to leave it alone for now.

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