I am currently reading Ravi's lecture notes on Algebraic Geometry, and in the introduction of group schemes (Section 6.6), he made a comment after 6.6N that the category of group schemes has a zero object.
I can understand in the category of $A-$schemes, Spec $A$ would be the terminal object, but how can this be a initial object?
I mean, what if we take $A=\mathbb{Z}$ and a group object $X=$ Spec $\mathbb{F}_p[t]$? I can't see a good morphism from Spec $\mathbb{Z}$ to $X$ here.
It is not claimed that the category of $A$-schemes has a zero object. But the category of group $A$-schemes has a zero object. This has nothing to do with schemes. If $C$ is any category with finite products, then $\mathsf{Grp}(C)$ has a zero object, the "trivial group" $T$. The underlying object is $1$, the final object, and the multiplication is the unique morphism $1 \times 1 \to 1$. It should be clear that it is final. To show that it is initial, recall that every group object $$G=(X,\eta : 1 \to X,\mu : X \times X \to X, \iota : X \to X)$$ has a morphism $\eta : 1 \to X$ in $C$ and observe that this is the unique morphism $T \to G$ of group objects.