The Stacks project

Ind-smoothness of an algebra is stable under infinitesimal deformations

Proposition 16.5.3. Let $R \to \Lambda $ be a ring map. Let $I \subset R$ be an ideal. Assume that

  1. $I$ is nilpotent,

  2. $\Lambda /I\Lambda $ is a filtered colimit of smooth $R/I$-algebras, and

  3. $R \to \Lambda $ is flat.

Then $\Lambda $ is a filtered colimit of smooth $R$-algebras.

Proof. Since $I^ n = 0$ for some $n$, it follows by induction on $n$ that it suffices to consider the case where $I^2 = 0$. Let $\varphi : A \to \Lambda $ be an $R$-algebra map with $A$ of finite presentation over $R$. We have to find a factorization $A \to B \to \Lambda $ with $B$ smooth over $R$, see Algebra, Lemma 10.127.4. By Lemma 16.5.1 we may assume that $A = B/J$ with $B$ smooth over $R$ and $J \subset IB$ a finitely generated ideal. By Lemma 16.5.2 we can find a (possibly noncommutative) diagram

\[ \xymatrix{ B \ar[rr]_\alpha \ar[rd]_\varphi & & B' \ar[ld]^\beta \\ & \Lambda } \]

of $R$-algebras which commutes modulo $I$ and such that $\alpha (J) = 0$. The map

\[ D : B \longrightarrow I\Lambda ,\quad b \longmapsto \varphi (b) - \beta (\alpha (b)) \]

is a derivation over $R$ hence we can write it as $D = \xi \circ \text{d}_{B/R}$ for some $B$-linear map $\xi : \Omega _{B/R} \to I\Lambda $. Since $\Omega _{B/R}$ is a finite projective $B$-module we can write $\xi = \sum _{i = 1, \ldots , n} \epsilon _ i \Xi _ i$ for some $\epsilon _ i \in I$ and $B$-linear maps $\Xi _ i : \Omega _{B/R} \to \Lambda $. (Details omitted. Hint: write $\Omega _{B/R}$ as a direct sum of a finite free module to reduce to the finite free case.) We define

\[ B'' = \text{Sym}^*_{B'}\left(\bigoplus \nolimits _{i = 1, \ldots , n} \Omega _{B/R} \otimes _{B, \alpha } B'\right) \]

and we define $\beta ' : B'' \to \Lambda $ by $\beta $ on $B'$ and by

\[ \beta '|_{i\text{th summand }\Omega _{B/R} \otimes _{B, \alpha } B'} = \Xi _ i \otimes \beta \]

and $\alpha ' : B \to B''$ by

\[ \alpha '(b) = \alpha (b) \oplus \sum \epsilon _ i \text{d}_{B/R}(b) \otimes 1 \oplus 0 \oplus \ldots \]

At this point the diagram

\[ \xymatrix{ B \ar[rr]_{\alpha '} \ar[rd]_\varphi & & B'' \ar[ld]^{\beta '} \\ & \Lambda } \]

does commute. Moreover, it is direct from the definitions that $\alpha '(J) = 0$ as $I^2 = 0$. Hence the desired factorization. $\square$

Comments (3)

Comment #3752 by slogan_bot on

Suggested slogan: Ind-smoothness of an algebra is stable under infinitesimal deformations.

Comment #4040 by Kęstutis Česnavičius on

At the end of the statement 'colimit' should be 'filtered colimit.'

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