The Stacks project

Remark 75.15.2. The proof of Lemma 75.15.1 shows that

\[ R|_ W = P \oplus P^{\oplus n_1}[1] \oplus \ldots \oplus P^{\oplus n_ m}[m] \]

for some $m \geq 0$ and $n_ j \geq 0$. Thus the highest degree cohomology sheaf of $R|_ W$ equals that of $P$. By repeating the construction for the map $P^{\oplus n_1}[1] \oplus \ldots \oplus P^{\oplus n_ m}[m] \to R|_ W$, taking cones, and using induction we can achieve equality of cohomology sheaves of $R|_ W$ and $P$ above any given degree.

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