is a coherent $\mathcal O_T$-module. A *morphism* (*isomorphism*) *of monads* is a morphism (isomorphism) of complexes.

The *display* of the monad is the commutative diagram (with exact rows and columns) $$\begin{gathered}
\xymatrix{
  & & 0 \ar[d] & 0 \ar[d] &  \\
0 \ar[r] & \mathcal{U} \ar@{=}[d] \ar[r]^a & \mathcal{B} \ar[d] \ar[r] & \mathcal{E} \ar[d] \ar[r] & 0 \\
0 \ar[r] & \mathcal{U} \ar[r]^a & \mathcal{V} \ar[d]^b \ar[r] & \mathcal{A} \ar[d]^{\nu} \ar[r] & 0 \\  \
  & & \mathcal{W} \ar@{=}[r] \ar[d] & \mathcal{W} \ar[d] & \\
  & & 0 & 0 & 
}
\end{gathered}
\label{eq44}$$ where $\mathcal{A}:={\operatorname{coker}}a $, $\mathcal{B}:=\ker b $, $\mathcal{E}=\mathcal{E}(M) $, and all morphisms are naturally induced.

Let us suppose that $T$ is a product $T=X\times S$, where $X$ is a smooth connected projective variety and $S$ a scheme; we denote by $t_i$, $i=1,2$ the canonical projections onto the first and second factor, respectively.

\[lmInjSurj\] Let $\varphi:{\mathcal{F}}\longrightarrow {\mathcal{G}}$ be a morphism between two coherent sheaves on $T$. If for all *closed* points $s\in S$ the restricted morphism $\varphi_s$ is surjective (resp. injective), then $\varphi$ is surjective (resp. injective).

For all closed points $s\in S$ and for all closed points $x\in T_{s}\simeq X$ the germs $(\varphi_{s})_{x}$ are surjective (resp. injective) 