Morphism and Composing morphisms of Varieties Hi guys I am trying to convince myself that composition of morphisms is again a morphism.
If $\phi: V \rightarrow W$ and $\psi : W \rightarrow Z$ are morphisms of varieties. Then $\psi \circ \phi : V \rightarrow Z$ is a morphism and $(\psi \circ \phi)^* = \phi^* \circ \psi ^*$
I think the first part is easy to show something is a morphism we just need to show the image of the variety is in the second variety. Thus let $a \in V$ by $\phi$ been morpism $\phi (a) \in W$ now by $\psi$ been a morphism $\psi (\phi(a)) \in Z$ this we have shown the first part.
For the second part I was thinking. Say 
$V \subset K^n \rightarrow W \subset K^m \rightarrow Z \subset K^p$ where we define $\phi (x_1,..x_n)= (f_1,...f_m)$ and $f_i \in K[x_1,..,x_n]$ and $\psi (y_1,...,y_m)= (g_1,...,g_p)$ and $g_i \in K[y_1,..,y_m]$ so now consider:
$(\psi \circ \phi)^*=h \circ (\psi \circ \phi)=h(\psi(f_1,..f_m)=h(g_1(f_1,..f_m),..g_p(f_1,..f_m))$ but the problem is when I start to expand $\phi ^* \circ \psi ^*=h_1(\phi) \circ h_2(\psi)=h_1(f_1,..f_m) \circ h_2(g_1,..,g_p)$ and this is where I am stuck inputs are appreciated. 
Also a side question I was reading in the book and it said it is "To check that it is morphism, we need to check that the pullback maps zero to zero" I am confused why is that enough
 A: ahh, I see.  So, by definition your functions are polynomials, and the composition of polynomials is again a polynomial, so that part is easy, as you say.   
For the pullback, your use of the definition of the pullback is not quite right.  What you wrote doesn't actually make sense, in fact, which is basically what you discovered.  Instead, you need to apply the pullback to a function, so
$$
\left(\phi^* \circ \psi^*\right)h = \phi^*\left(\psi^*h\right)
=\phi^*\left(h\circ\psi\right)
=h\circ\psi\circ\phi
=h\circ\left(\psi\circ\phi\right)
=\left(\psi\circ\phi\right)^*h
$$
Since that's true for arbitrary $h$, it shows that $\phi^*\circ\psi^*=\left(\psi\circ\phi\right)^*$
I'm not sure about the comment from the book, but I have an idea.  Consider the map from the line $X$ with coordinate $x$ to the plane $Y$ with coordinates $y,z$ that sends a point $a$ to $(a,1)$.  Then the pullback of the function $z-1$ is the zero function, even though $z-1$ is not zero itself, so the pullback doesn't send only zero to zero.  
That doesn't quite match what the book said ( I added "only" ), but it's my best guess.  
