all $\ell$ with $-d\le \ell\le d$.

-   ${\mathsf{D}}^{{\operatorname{b}\nolimits}}(\operatorname{\mathsf{coh}}{\mathbb{P}}^d)=\operatorname{\mathsf{thick}}\bigoplus_{i=0}^d{{\mathcal O}}(i)$.

The second condition implies that, if $X\in\operatorname{\mathsf{coh}}{\mathbb{P}}^d$ satisfies $H^i({\mathbb{P}}^d, X(-\ell))=0$ for all $i\ge0$ and all $\ell$ with $0\leq \ell\leq d$, then $X=0$.

Proof the rigidity condition
----------------------------

In this section we prove the following:

\[ext vanishing\] We have ${\operatorname{Ext}\nolimits}^i_\Lambda(T,T)=0$ for any $i>0$.

First we need the following observation:

\[cohomology\] For any idempotent $e$ of $H^0({\mathbb{P}}^d, \Lambda)$, we have ${\operatorname{Ext}\nolimits}^i_\Lambda(\Lambda e,X)\simeq H^i({\mathbb{P}}^d,eX)$ for all $X\in{\operatorname{\mathsf{mod}}}\Lambda$ and $i\ge0$.

We only have to show the case $i=0$ since both sides are the right derived functors. The case $i=0$ follows from the following isomorphism of $k$-vector spaces. $${\operatorname{Hom}\nolimits}_\Lambda(\Lambda e,-)\simeq e{\operatorname{Hom}\nolimits}_\Lambda(\Lambda,-)\simeq eH^0({\mathbb{P}}^d, -)\simeq H^0({\mathbb{P}}^d, e-),$$ where the middle equality follows from a natural isomorphism ${\operatorname{Hom}\nolimits}_\Lambda(\Lambda,-)\simeq H^0({\mathbb{P}}^d, -)$.

Next we show the following:

\[first entry 2\]

-   Let $\ell\in{\mathbb{Z}}$. Then $e_i(P_i(\ell{\vec x}_i))\simeq{{\mathcal O}}(\lfloor\ell/p_i\rfloor)$ where $\lfloor x\rfloor$ is the largest integer $a$ satisfying $a\leq x$.

-   For any ${\vec x},{\vec y}\in[0,d{\vec c}]$, we have $e(P({\vec y}-{\vec x}))\simeq{{\mathcal O}}(\ell)$ for some $\ell$ with $-d\le \ell\le d$.

Notice that $e_i(P_i({\vec x}_i))$ can not be written as $(e_iP_i)({\vec x}_i)$ since $({\vec x}_i)$ is defined for $\Lambda$-modules, not for ${{\mathcal O}}$-modules.

\(a) Follows from Observation \[obs:what J does\] and .

\(b) By 