$m{\vec x}_1+\ell{\vec c}\in[0,d{\vec c}]$. Since $\Lambda e'=P(m{\vec x}_1)$, it follows from Lemma \[cohomology\] that $$H^i({\mathbb{P}}^d,e'X(-\ell))={\operatorname{Ext}\nolimits}^i_\Lambda(\Lambda e'(\ell{\vec c}),X)={\operatorname{Ext}\nolimits}^i_\Lambda(P(m{\vec x}_1+\ell{\vec c}),X)\subset{\operatorname{Ext}\nolimits}^i_\Lambda(T,X)=0$$ for all $i\geq0$. Thus we have $$H^i({\mathbb{P}}^{d-1},e'X(-\ell))=H^i({\mathbb{P}}^d,e'X(-\ell))=0$$ for all $\ell$ with $0\le \ell<d$. By Theorem \[beilinson\] (replace $d$ there by $d-1$), we have $e'X=0$.

The case $N\ge2$ can be shown similarly:

By Assumption \[general position\] and Lemma \[killebyideal\] if $d-N\geq 0$ we can regard $e'X$ as a sheaf on ${\mathbb{P}}^{d-N}$ and if $d-N<0$ it is follows that $e'X=0$.

On the other hand, by ${\operatorname{Ext}\nolimits}^i_\Lambda(T,X)=0$ for all $i\ge0$, we have $$H^i({\mathbb{P}}^d, e'X(-k))=H^i({\mathbb{P}}^{d-N}, e'X(-k))=0$$ for all $i\ge0$ and all $k$ with $0\le k\le d-N$. By Theorem \[beilinson\] (replace $d$ there by $d-N$) we have $e'X=0$.

We will now show that Proposition \[generate\] implies the generation condition. To do this, we first show that $\Lambda$ has global dimension $d$.

\[gldimmat\] Let $(R,{\mathfrak m})$ be a regular local ring of dimension $d$ and $(a_1,\ldots,a_\ell)$ a regular sequence for $R$. Then the ring $$B:={\rm T}_{p_1}(R,(a_1))\otimes_R\cdots\otimes_R{\rm T}_{p_\ell}(R,(a_\ell))$$ has global dimension $d$.

It is enough to show that any simple $B$-module $S$ has projective dimension $d$. Up to an automorphism of $B$, we can assume that $$S=\left[\begin{array}{c}
R/{\mathfrak m}\\ 0\\ \vdots\\ 0\\ 0
\end{array}\right]$$ where we regard $B$ 