# Trace of a morphism of sheaves

$$\underline {Background}$$:Suppose $$X$$ be a scheme and $$\mathcal F$$ and $$\mathcal G$$ are two sheaves of $$\mathcal O_X$$ modules.

Also assume that $$\exists$${U} a cover of $$X$$ by open sets such that each $$\mathcal F(U),\mathcal G(U)$$ are finitely free $$\mathcal O(U)$$ modules.

Let $$\exists \phi:\mathcal F\to\mathcal G$$ a sheaf of $$\mathcal O_X$$ modules.

$$\underline {Question}$$:what is the meaning of the statement “trace$$\phi=0$$

$$\underline {Guess}$$:Does it mean for all open set $$V$$ of $$X$$

$$\phi(V):\mathcal F(V)\to \mathcal G(V)$$ has trace of the matrix of $$\phi(V)=0$$

But this makes sense only for $$V=U$$ because,we only have $$\phi(U)$$ is a morphism between 2 finitely free $$\mathcal O(U)$$ modules,and hence its matrix w.r.to canonical bases exists.

So,I would like to know the appropriate definition of trace being $$0$$ and any text which mentions it clearly.

Any help from anyone is welcome.