Lemma 57.16.1. Let $(R, \mathfrak m, \kappa ) \to (A, \mathfrak n, \lambda )$ be a flat local ring homorphism of local rings which is essentially of finite presentation. Let $\overline{f}_1, \ldots , \overline{f}_ r \in \mathfrak n/\mathfrak m A \subset A/\mathfrak m A$ be a regular sequence. Let $K \in D(A)$. Assume

$K$ is perfect,

$K \otimes _ A^\mathbf {L} A/\mathfrak m A$ is isomorphic in $D(A/\mathfrak m A)$ to the Koszul complex on $\overline{f}_1, \ldots , \overline{f}_ r$.

Then $K$ is isomorphic in $D(A)$ to a Koszul complex on a regular sequence $f_1, \ldots , f_ r \in A$ lifting the given elements $\overline{f}_1, \ldots , \overline{f}_ r$. Moreover, $A/(f_1, \ldots , f_ r)$ is flat over $R$.

**Proof.**
Let us use chain complexes in the proof of this lemma. The Koszul complex $K_\bullet (\overline{f}_1, \ldots , \overline{f}_ r)$ is defined in More on Algebra, Definition 15.28.2. By More on Algebra, Lemma 15.75.5 we can represent $K$ by a complex

\[ K_\bullet : A \to A^{\oplus r} \to \ldots \to A^{\oplus r} \to A \]

whose tensor product with $A/\mathfrak mA$ is equal (!) to $K_\bullet (\overline{f}_1, \ldots , \overline{f}_ r)$. Denote $f_1, \ldots , f_ r \in A$ the components of the arrow $A^{\oplus r} \to A$. These $f_ i$ are lifts of the $\overline{f}_ i$. By Algebra, Lemma 10.128.6 $f_1, \ldots , f_ r$ form a regular sequence in $A$ and $A/(f_1, \ldots , f_ r)$ is flat over $R$. Let $J = (f_1, \ldots , f_ r) \subset A$. Consider the diagram

\[ \xymatrix{ K_\bullet \ar[rd] \ar@{..>}[rr]_{\varphi _\bullet } & & K_\bullet (f_1, \ldots , f_ r) \ar[ld] \\ & A/J } \]

Since $f_1, \ldots , f_ r$ is a regular sequence the south-west arrow is a quasi-isomorphism (see More on Algebra, Lemma 15.30.2). Hence we can find the dotted arrow making the diagram commute for example by Algebra, Lemma 10.71.4. Reducing modulo $\mathfrak m$ we obtain a commutative diagram

\[ \xymatrix{ K_\bullet (\overline{f}_1, \ldots , \overline{f}_ r) \ar[rd] \ar[rr]_{\overline{\varphi }_\bullet } & & K_\bullet (\overline{f}_1, \ldots , \overline{f}_ r) \ar[ld] \\ & (A/\mathfrak m A)/(\overline{f}_1, \ldots , \overline{f}_ r) } \]

by our choice of $K_\bullet $. Thus $\overline{\varphi }$ is an isomorphism in the derived category $D(A/\mathfrak m A)$. It follows that $\overline{\varphi } \otimes _{A/\mathfrak m A}^\mathbf {L} \lambda $ is an isomorphism. Since $\overline{f}_ i \in \mathfrak n / \mathfrak m A$ we see that

\[ \text{Tor}_ i^{A/\mathfrak m A}( K_\bullet (\overline{f}_1, \ldots , \overline{f}_ r), \lambda ) = K_ i(\overline{f}_1, \ldots , \overline{f}_ r) \otimes _{A/\mathfrak m A} \lambda \]

Hence $\varphi _ i \bmod \mathfrak n$ is invertible. Since $A$ is local this means that $\varphi _ i$ is an isomorphism and the proof is complete.
$\square$

## Comments (3)

Comment #5408 by Shogōki on

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Comment #5639 by Johan on