The Stacks project

91.15 Deformations of complexes

This section is a warmup for the next one. We will use as much as possible the material in the chapters on commutative algebra.

Lemma 91.15.1. Let $R' \to R$ be a surjection of rings whose kernel is an ideal $I$ of square zero. For every $K \in D^-(R)$ there is a canonical map

\[ \omega (K) : K \longrightarrow K \otimes _ R^\mathbf {L} I[2] \]

in $D(R)$ with the following properties

  1. $\omega (K) = 0$ if and only if there exists $K' \in D(R')$ with $K' \otimes _{R'}^\mathbf {L} R = K$,

  2. given $K \to L$ in $D^-(R)$ the diagram

    \[ \xymatrix{ K \ar[d] \ar[rr]_-{\omega (K)} & & K \otimes ^\mathbf {L}_ R I[2] \ar[d] \\ L \ar[rr]^-{\omega (L)} & & L \otimes ^\mathbf {L}_ R I[2] } \]

    commutes, and

  3. formation of $\omega (K)$ is compatible with ring maps $R' \to S'$ (see proof for a precise statement).

Proof. Choose a bounded above complex $K^\bullet $ of free $R$-modules representing $K$. Then we can choose free $R'$-modules $(K')^ n$ lifting $K^ n$. We can choose $R'$-module maps $(d')^ n_ K : (K')^ n \to (K')^{n + 1}$ lifting the differentials $d^ n_ K : K^ n \to K^{n + 1}$ of $K^\bullet $. Although the compositions

\[ (d')^{n + 1}_ K \circ (d')^ n_ K : (K')^ n \to (K')^{n + 2} \]

may not be zero, they do factor as

\[ (K')^ n \to K^ n \xrightarrow {\omega ^ n_ K} K^{n + 2} \otimes _ R I = I(K')^{n + 2} \to (K')^{n + 2} \]

because $d^{n + 1} \circ d^ n = 0$. A calculation shows that $\omega ^ n_ K$ defines a map of complexes. This map of complexes defines $\omega (K)$.

Let us prove this construction is compatible with a map of complexes $\alpha ^\bullet : K^\bullet \to L^\bullet $ of bounded above free $R$-modules and given choices of lifts $(K')^ n, (L')^ n, (d')^ n_ K, (d')^ n_ L$. Namely, choose $(\alpha ')^ n : (K')^ n \to (L')^ n$ lifting the components $\alpha ^ n : K^ n \to L^ n$. As before we get a factorization

\[ (K')^ n \to K^ n \xrightarrow {h^ n} L^{n + 1} \otimes _ R I = I(L')^{n + 1} \to (L')^{n + 2} \]

of $(d')^ n_ L \circ (\alpha ')^ n - (\alpha ')^{n + 1} \circ (d')_ K^ n$. Then it is an pleasant calculation to show that

\[ \omega ^ n_ L \circ \alpha ^ n = (d_ L^{n + 1} \otimes \text{id}_ I) \circ h^ n + h^{n + 1} \circ d_ K^ n + (\alpha ^{n + 2} \otimes \text{id}_ I) \circ \omega ^ n_ K \]

This proves the commutativity of the diagram in (2) of the lemma in this particular case. Using this for two different choices of bounded above free complexes representing $K$, we find that $\omega (K)$ is well defined! And of course (2) holds in general as well.

If $K$ lifts to $K'$ in $D^-(R')$, then we can represent $K'$ by a bounded above complex of free $R'$-modules and we see immediately that $\omega (K) = 0$. Conversely, going back to our choices $K^\bullet $, $(K')^ n$, $(d')^ n_ K$, if $\omega (K) = 0$, then we can find $g^ n : K^ n \to K^{n + 1} \otimes _ R I$ with

\[ \omega ^ n = (d_ K^{n + 1} \otimes \text{id}_ I) \circ g^ n + g^{n + 1} \circ d_ K^ n \]

This means that with differentials $(d')^ n_ K - g^ n : (K')^ n \to (K')^{n + 1}$ we obtain a complex of free $R'$-modules lifting $K^\bullet $. This proves (1).

Finally, part (3) means the following: Let $R' \to S'$ be a map of rings. Set $S = S' \otimes _{R'} R$ and denote $J = IS' \subset S'$ the square zero kernel of $S' \to S$. Then given $K \in D^-(R)$ the statement is that we get a commutative diagram

\[ \xymatrix{ K \otimes _ R^\mathbf {L} S \ar[d] \ar[rr]_-{\omega (K) \otimes \text{id}} & & (K \otimes ^\mathbf {L}_ R I[2]) \otimes _ R^\mathbf {L} S \ar[d] \\ K \otimes _ R^\mathbf {L} S \ar[rr]^-{\omega (K \otimes _ R^\mathbf {L} S)} & & (K \otimes _ R^\mathbf {L} S) \otimes ^\mathbf {L}_ S J[2] } \]

Here the right vertical arrow comes from

\[ (K \otimes ^\mathbf {L}_ R I[2]) \otimes _ R^\mathbf {L} S = (K \otimes _ R^\mathbf {L} S) \otimes _ S^\mathbf {L} (I \otimes _ R^\mathbf {L} S)[2] \longrightarrow (K \otimes _ R^\mathbf {L} S) \otimes _ S^\mathbf {L} J[2] \]

Choose $K^\bullet $, $(K')^ n$, and $(d')^ n_ K$ as above. Then we can use $K^\bullet \otimes _ R S$, $(K')^ n \otimes _{R'} S'$, and $(d')^ n_ K \otimes \text{id}_{S'}$ for the construction of $\omega (K \otimes _ R^\mathbf {L} S)$. With these choices commutativity is immediately verified on the level of maps of complexes. $\square$

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