The Stacks project

Lemma 52.6.10. There exist a way to construct

  1. for every pair $(A, I)$ consisting of a ring $A$ and a finitely generated ideal $I \subset A$ a complex $K(A, I)$ of $A$-modules,

  2. a map $K(A, I) \to A$ of complexes of $A$-modules,

  3. for every ring map $A \to B$ and finitely generated ideal $I \subset A$ a map of complexes $K(A, I) \to K(B, IB)$,

such that

  1. for $A \to B$ and $I \subset A$ finitely generated the diagram

    \[ \xymatrix{ K(A, I) \ar[r] \ar[d] & A \ar[d] \\ K(B, IB) \ar[r] & B } \]

    commutes,

  2. for $A \to B \to C$ and $I \subset A$ finitely generated the composition of the maps $K(A, I) \to K(B, IB) \to K(C, IC)$ is the map $K(A, I) \to K(C, IC)$.

  3. for $A \to B$ and a finitely generated ideal $I \subset A$ the induced map $K(A, I) \otimes _ A^\mathbf {L} B \to K(B, IB)$ is an isomorphism in $D(B)$, and

  4. if $I = (f_1, \ldots , f_ r) \subset A$ then there is a commutative diagram

    \[ \xymatrix{ (A \to \prod \nolimits _{i_0} A_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} A_{f_{i_0}f_{i_1}} \to \ldots \to A_{f_1\ldots f_ r}) \ar[r] \ar[d] & K(A, I) \ar[d] \\ A \ar[r]^1 & A } \]

    in $D(A)$ whose horizontal arrows are isomorphisms.

Proof. Let $S$ be the set of rings $A_0$ of the form $A_0 = \mathbf{Z}[x_1, \ldots , x_ n]/J$. Every finite type $\mathbf{Z}$-algebra is isomorphic to an element of $S$. Let $\mathcal{A}_0$ be the category whose objects are pairs $(A_0, I_0)$ where $A_0 \in S$ and $I_0 \subset A_0$ is an ideal and whose morphisms $(A_0, I_0) \to (B_0, J_0)$ are ring maps $\varphi : A_0 \to B_0$ such that $J_0 = \varphi (I_0)B_0$.

Suppose we can construct $K(A_0, I_0) \to A_0$ functorially for objects of $\mathcal{A}_0$ having properties (a), (b), (c), and (d). Then we take

\[ K(A, I) = \mathop{\mathrm{colim}}\nolimits _{\varphi : (A_0, I_0) \to (A, I)} K(A_0, I_0) \]

where the colimit is over ring maps $\varphi : A_0 \to A$ such that $\varphi (I_0)A = I$ with $(A_0, I_0)$ in $\mathcal{A}_0$. A morphism between $(A_0, I_0) \to (A, I)$ and $(A_0', I_0') \to (A, I)$ are given by maps $(A_0, I_0) \to (A_0', I_0')$ in $\mathcal{A}_0$ commuting with maps to $A$. The category of these $(A_0, I_0) \to (A, I)$ is filtered (details omitted). Moreover, $\mathop{\mathrm{colim}}\nolimits _{\varphi : (A_0, I_0) \to (A, I)} A_0 = A$ so that $K(A, I)$ is a complex of $A$-modules. Finally, given $\varphi : A \to B$ and $I \subset A$ for every $(A_0, I_0) \to (A, I)$ in the colimit, the composition $(A_0, I_0) \to (B, IB)$ lives in the colimit for $(B, IB)$. In this way we get a map on colimits. Properties (a), (b), (c), and (d) follow readily from this and the corresponding properties of the complexes $K(A_0, I_0)$.

Endow $\mathcal{C}_0 = \mathcal{A}_0^{opp}$ with the chaotic topology. We equip $\mathcal{C}_0$ with the sheaf of rings $\mathcal{O} : (A, I) \mapsto A$. The ideals $I$ fit together to give a sheaf of ideals $\mathcal{I} \subset \mathcal{O}$. Choose an injective resolution $\mathcal{O} \to \mathcal{J}^\bullet $. Consider the object

\[ \mathcal{F}^\bullet = \bigcup \nolimits _ n \mathcal{J}^\bullet [\mathcal{I}^ n] \]

Let $U = (A, I) \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_0)$. Since the topology in $\mathcal{C}_0$ is chaotic, the value $\mathcal{J}^\bullet (U)$ is a resolution of $A$ by injective $A$-modules. Hence the value $\mathcal{F}^\bullet (U)$ is an object of $D(A)$ representing the image of $R\Gamma _ I(A)$ in $D(A)$, see Dualizing Complexes, Section 47.9. Choose a complex of $\mathcal{O}$-modules $\mathcal{K}^\bullet $ and a commutative diagram

\[ \xymatrix{ \mathcal{O} \ar[r] & \mathcal{J}^\bullet \\ \mathcal{K}^\bullet \ar[r] \ar[u] & \mathcal{F}^\bullet \ar[u] } \]

where the horizontal arrows are quasi-isomorphisms. This is possible by the construction of the derived category $D(\mathcal{O})$. Set $K(A, I) = \mathcal{K}^\bullet (U)$ where $U = (A, I)$. Properties (a) and (b) are clear and properties (c) and (d) follow from Dualizing Complexes, Lemmas 47.10.2 and 47.10.3. $\square$

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