Proof.
Proof of (1). Assume that K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I} is bounded above, say H^ i(K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}) = 0 for i > b. Note that we have distinguished triangles
K \otimes _\mathcal {O}^\mathbf {L} \mathcal{I}^ n/\mathcal{I}^{n + 1} \to K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^{n + 1} \to K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n \to K \otimes _\mathcal {O}^\mathbf {L} \mathcal{I}^ n/\mathcal{I}^{n + 1}[1]
and that
K \otimes _\mathcal {O}^\mathbf {L} \mathcal{I}^ n/\mathcal{I}^{n + 1} = \left( K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}\right) \otimes _{\mathcal{O}/\mathcal{I}}^\mathbf {L} \mathcal{I}^ n/\mathcal{I}^{n + 1}
By induction we conclude that H^ i(K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n) = 0 for i > b for all n.
Proof of (2). Assume K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I} as an object of D(\mathcal{O}/\mathcal{I}) has tor amplitude in [a, b]. Let \mathcal{F} be a sheaf of \mathcal{O}/\mathcal{I}^ n-modules. Then we have a finite filtration
0 \subset \mathcal{I}^{n - 1}\mathcal{F} \subset \ldots \subset \mathcal{I}\mathcal{F} \subset \mathcal{F}
whose successive quotients are sheaves of \mathcal{O}/\mathcal{I}-modules. Thus to prove that K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n has tor amplitude in [a, b] it suffices to show H^ i(K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n \otimes _{\mathcal{O}/\mathcal{I}^ n}^\mathbf {L} \mathcal{G}) is zero for i \not\in [a, b] for all \mathcal{O}/\mathcal{I}-modules \mathcal{G}. Since
\left(K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n\right) \otimes _{\mathcal{O}/\mathcal{I}^ n}^\mathbf {L} \mathcal{G} = \left(K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}\right) \otimes _{\mathcal{O}/\mathcal{I}}^\mathbf {L} \mathcal{G}
for every sheaf of \mathcal{O}/\mathcal{I}-modules \mathcal{G} the result follows.
\square
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