# ag.algebraic geometry – Obstruction to deformation of composite morphism (Reference request + question)

Let $$f_0:X_0xrightarrow{g_0}Y_0xrightarrow{h_0}Z_0/S_0$$ be a morphism of smooth projective $$S_0$$-schemes such that $$g_0,h_0$$ are flat. Let $$S_0subset S$$ be a first-order thickening, and let $$X,Y,Z$$ be $$S$$-schemes restricting to $$X_0,Y_0,Z_0$$ over $$S_0$$. Let $$sigma_{f_0}$$ (resp. $$sigma_{g_0}$$ and $$sigma_{h_0}$$) denote the obstruction to lifting $$f_0$$ to $$f:Xlongrightarrow Z/S$$ (resp. $$g_0$$ to $$g:Xlongrightarrow Y/S$$ and $$h_0$$ to $$h:Ylongrightarrow Z/S$$)

(1) I read in some deformation theory notes that $$sigma_{f_0}=g_0^ast sigma_{h_0}$$. Is there a reference for this fact?

(2) Assuming that what I wrote in point (1) is correct, suppose furthermore that $$S_0$$ is the spectrum of an algebraically closed field, that $$sigma_{f_0}=0$$, and that $$g_0$$ a finite separable morphism. Is it true that $$sigma_{h_0}=0?$$