The Stacks project

Remark 88.11.2. Let $A \to B$ be an arrow of $\text{WAdm}^{adic*}$ which is adic and topologically of finite type (see Lemma 88.11.1). Write $B = A\{ x_1, \ldots , x_ r\} /J$. Then we can set1

\[ \mathop{N\! L}\nolimits _{B/A}^\wedge = \left(J/J^2 \longrightarrow \bigoplus B\text{d}x_ i\right) \]

Exactly as in the proof of Lemma 88.3.1 the reader can show that this complex of $B$-modules is well defined up to (unique isomorphism) in the homotopy category $K(B)$. Now, if $A$ is Noetherian and $I \subset A$ is an ideal of definition, then this construction reproduces the naive cotangent complex of $B$ over $(A, I)$ defined by Equation (88.3.0.1) in Section 88.3 simply because $A[x_1, \ldots , x_ n]^\wedge $ agrees with $A\{ x_1, \ldots , x_ r\} $ by Formal Spaces, Remark 87.28.2. In particular, we find that, still when $A$ is an adic Noetherian topological ring, the object $\mathop{N\! L}\nolimits _{B/A}^\wedge $ is independent of the choice of the ideal of definition $I \subset A$.

[1] In fact, this construction works for arrows of $\text{WAdm}^{count}$ satisfying the equivalent conditions of Formal Spaces, Lemma 87.29.6.

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