The Stacks project

Lemma 15.83.10. Let $R$ be ring and let $I \subset R$ be an ideal. Let $K \in D(R)$ with $H^ i(K) = 0$ for $i \not\in \{ -1, 0\} $. The following are equivalent

  1. there exists a $c \geq 0$ such that the equivalent conditions (1), (2), (3) of Lemma 15.83.5 hold for $K$ and the ideal $I^ c$,

  2. there exists a $c \geq 0$ such that (a) $I^ c$ annihilates $H^{-1}(K)$ and (b) $H^0(K)$ is an $I^ c$-projective module (see Section 15.69).

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

  1. there exists a $c \geq 0$ such that the equivalent conditions (4), (5) of Lemma 15.83.5 hold for $K$ and the ideal $I^ c$,

  2. $H^{-1}(K)$ is $I$-power torsion and there exist $f_1, \ldots , f_ s \in R$ with $V(f_1, \ldots , f_ s) \subset V(I)$ such that the localizations $H^0(K)_{f_ i}$ are projective $R_{f_ i}$-modules,

  3. $H^{-1}(K)$ is $I$-power torsion and there exist $f_1, \ldots , f_ s \in I$ with $V(f_1, \ldots , f_ s) = V(I)$ such that the localizations $H^0(K)_{f_ i}$ are projective $R_{f_ i}$-modules.

Proof. The distinguished triangle $H^{-1}(K)[1] \to K \to H^0(K)[0] \to H^{-1}(K)[2]$ determines an exact sequence

\[ 0 \to \mathop{\mathrm{Ext}}\nolimits ^1_ R(H^0(K), N) \to \mathop{\mathrm{Ext}}\nolimits ^1_ R(K, N) \to \mathop{\mathrm{Hom}}\nolimits _ R(H^{-1}(K), N) \to \mathop{\mathrm{Ext}}\nolimits ^2_ R(H^0(K), N) \]

Thus (2) implies that $I^{2c}$ annihilates $\mathop{\mathrm{Ext}}\nolimits ^1_ R(K, N)$ for every $R$-module $N$. Assuming (1) we immediately see that $H^0(K)$ is $I^ c$-projective. On the other hand, we may choose an injective map $H^{-1}(K) \to N$ for some injective $R$-module $N$. Then this map is the image of an element of $\mathop{\mathrm{Ext}}\nolimits ^1_ R(K, N)$ by the vanishing of the $\mathop{\mathrm{Ext}}\nolimits ^2$ in the sequence and we conclude $H^{-1}(K)$ is annihilated by $I^ c$.

Assume $R$ is Noetherian and $H^ i(K)$ is a finite $R$-module for $i = -1, 0$. By Lemma 15.83.5 we see that (3) is equivalent to (1) and (2). Also, if (3) holds then for $f \in I$ the multiplication by $f$ on $H^0(K)$ factors through a projective module, which implies that $H^0(K)_ f$ is a summand of a projective $R_ f$-module and hence itself a projective $R_ f$-module. Choosing $f_1, \ldots , f_ s$ to be generators of $I$ we find the equivalent conditions (1), (2), and (3) imply (5). Of course (5) trivially implies (4).

Assume (4). Since $H^{-1}(K)$ is a finite $R$-module and $I$-power torsion we see that $I^{c_1}$ annihilates $H^{-1}(K)$ for some $c_1 \geq 0$. Choose a short exact sequence

\[ 0 \to M \to R^{\oplus r} \to H^0(K) \to 0 \]

which determines an element $\xi \in \mathop{\mathrm{Ext}}\nolimits ^1_ R(H^0(K), M)$. For any $f \in I$ we have $\mathop{\mathrm{Ext}}\nolimits ^1_ R(H^0(K), M)_ f = \mathop{\mathrm{Ext}}\nolimits ^1_{R_ f}(H^0(K)_ f, M_ f)$ by Lemma 15.64.4. Hence if $H^0(K)_ f$ is projective, then a power of $f$ annihilates $\xi $. We conclude that $\xi $ is annihilated by $(f_1, \ldots , f_ s)^{c_2}$ for some $c_2 \geq 0$. Since $V(f_1, \ldots , f_ s) \subset V(I)$ we have $\sqrt{I} \subset (f_1, \ldots , f_ s)$ (Algebra, Lemma 10.17.2). Since $R$ is Noetherian we find $I^{c_3} \subset (f_1, \ldots , f_ s)$ for some $c_3 \geq 0$ (Algebra, Lemma 10.32.5). Hence $I^{c2c3}$ annihilates $\xi $. This in turn says that $H^0(K)$ is $I^{c_2c_3}$-projective (as multiplication by $a \in I$ which annihilate $\xi $ factor through $R^{\oplus r}$). Hence taking $c = \max (c_1, c_2c_3)$ we see that (2) holds. $\square$


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