Lemma 15.84.5. Let $R$ be a ring and let $I \subset R$ be an ideal. Let $K \in D(R)$. Assume $H^ i(K) = 0$ for $i \not\in \{ -1, 0\}$. The following are equivalent

1. $\mathop{\mathrm{Ext}}\nolimits ^1_ R(K, N)$ is annihilated by $I$ for all $R$-modules $N$,

2. $K$ can be represented by a complex $K^{-1} \to K^0$ with $K^0$ free such that for any $a \in I$ the map $a : K^{-1} \to K^{-1}$ factors through $d_ K^{-1} : K^{-1} \to K^0$,

3. whenever $K$ is represented by a two term complex $K^{-1} \to K^0$ with $K^0$ projective, then for any $a \in I$ the map $a : K^{-1} \to K^{-1}$ factors through $d_ K^{-1} : K^{-1} \to K^0$.

If $R$ is Noetherian and $H^ i(K)$ is a finite $R$-module for $i = -1, 0$, then these are also equivalent to

1. $\mathop{\mathrm{Ext}}\nolimits ^1_ R(K, N)$ is annihilated by $I$ for every finite $R$-module $N$,

2. $K$ can be represented by a complex $K^{-1} \to K^0$ with $K^0$ finite free and $K^{-1}$ finite such that for any $a \in I$ the map $a : K^{-1} \to K^{-1}$ factors through $d_ K^{-1} : K^{-1} \to K^0$.

Proof. Assume (1) and let $K^{-1} \to K^0$ be a two term complex representing $K$ with $K^0$ projective. We will use the description of maps in $D(R)$ out of $K^\bullet$ given in Lemma 15.84.4 without further mention. Choosing $N = K^{-1}$ consider the element $\xi$ of $\mathop{\mathrm{Ext}}\nolimits ^1_ R(K, N)$ given by $\text{id}_{K^{-1}} : K^{-1} \to K^{-1}$. Since is annihilated by $a \in I$ we see that we get the dotted arrow fitting into the following commutative diagram

$\xymatrix{ K^{-1} \ar[d]_ a \ar[r]_{d_ K^{-1}} & K^0 \ar@{..>}[ld]^ h \\ K^{-1} }$

This proves that (3) holds. Part (3) implies (2) in view of Lemma 15.84.3 part (1). Assume $K^\bullet$ is as in (2) and $N$ is an arbitrary $R$-module. Any element $\xi$ of $\mathop{\mathrm{Ext}}\nolimits ^1_ R(K, N)$ is given as the class of a map $\varphi : K^{-1} \to N$. Then for $a \in I$ by assumption we may choose a map $h$ as in the diagram above and we see that $a\varphi = \varphi \circ a = \varphi \circ h \circ \text{d}_ K^{-1}$ which proves that $a \xi$ is zero in $\mathop{\mathrm{Ext}}\nolimits ^1_ R(K, N)$. Thus (1), (2), and (3) are equivalent.

Assume $R$ is Noetherian and $H^ i(K)$ is a finite $R$-module for $i = -1, 0$. Part (3) implies (5) in view of Lemma 15.84.3 part (2). It is clear that (5) implies (2). Trivially (1) implies (4). Thus to finish the proof it suffices to show that (4) implies any of the other conditions. Let $K^{-1} \to K^0$ be a complex representing $K$ with $K^0$ finite free and $K^{-1}$ finite as in Lemma 15.84.3 part (2). The argument given in the proof of (2) $\Rightarrow$ (1) shows that if $\mathop{\mathrm{Ext}}\nolimits ^1_ R(K, K^{-1})$ is annihilated by $I$, then (1) holds. In this way we see that (4) implies (1) and the proof is complete. $\square$

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