Need help understanding morphisms of algebraic varieties So far, in my course we have talked about affine algebraic varieties, then about prevarieties, and finally about algebraic varieties and projective varieties. Here is what I understand so far (please correct me if anything I say here is wrong):

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*An affine algebraic variety over $k$ is a solution set of some set of polynomial equations over a field $k$ (eg. $V = \{ (a_1 , \dots , a_n) \in k^n ; f(a_1 , \dots , a_n) = 0 \ \forall f \in a \}$ for some ideal $\mathcal{a} \subset k[x_1 , \dots , x_n]$. A morphism of affine varieties $V \to W$ corresponds to a ring homomorphism of their respective coordinate rings $k[W] \to k[V]$. We were given some analogous definitions of affine varieties (eg. description as the space of maximal ideals of a $k$-algebra) which I do not understand very well.


*A prevariety is a $k$-ringed space $(V, \mathcal{O}_V)$ with a finite open (with respect to Zariski topology) covering $V =\bigcup U_i$ such that $(U_i, \mathcal{O}_V |_{U_i})$ is an affine algebraic variety over $k$.


*An algebraic variety is a prevariety with some additional separation type condition.


*A projective variety is an algebraic variety which is a closed (with respect to Zariski topology) subset of the projective space $\mathbf{P}_k^n$ ($n$-dimensional projective space over field $k$).
Now in our lecture notes, my professor states that a morphism of algebraic varieties $V \to W$ is simply a morphism of ringed spaces, which we defined to be a regular map $\phi: V \to W$ and a family of $k$-algebra homomorphisms:
\begin{equation}
\phi_U : \mathcal{O}_W(U) \to \mathcal{O}_V(\phi^{-1}(U))
\end{equation}
for every open subset $U$ of $W$, which commute with the restriction maps (?).
However, this definition is very abstract and was given with very few concrete examples. For example, let us say I wanted to define an automorphism of the projective spaces $\phi: \mathbf{P}_\mathbb{R}^1 \to \mathbf{P}_\mathbb{R}^1$. How would I do so using this definition? I know it has to be a bijective map on the points, but how can I make sense of the 2nd condition? What confuses me here is that we have no notion of coordinate rings like we do in the case of affine varieties.
 A: Question: "For example, let us say I wanted to define an automorphism of the projective spaces ϕ:P1R→P1R. How would I do so using this definition?"
Answer: There is a calculation in Hartshorne Ex.II.7.1.1 of the automorphism group of the projective line. Let $k$ be the real numbers and $V^*:=k\{x_0,x_1\}$ and $C:=Proj(k[x_0,x_1])$. Any matrix $\phi^* \in GL(2,k)$ induce an automorphism
$$\phi^*: k[x_0,x_1] \rightarrow k[x_0,x_1].$$
You must check that the automorphism $\phi$ induce an automorphism
$$\phi: C \rightarrow C$$
defined over $k$. This is because the map $\phi^*$ is an isomorphism at the graded components
$$\phi^*_d:k[x_0,x_1]_d \rightarrow k[x_0,x_1]_d$$
for all $d \geq 0$. Hence by HH.Ex.II.2.14 it follows the induced map $\phi$ is an automorphism.
If $\psi:= \lambda Id(2)$ with $\lambda \in k^*$ and $Id(2)$ the $2\times 2$-identity matrix, it follows $\psi$ induce the identity automorphism of $C$. The automorphisms $\phi$ and $\phi + \lambda Id(2)$ induce the same automorphism of $C$, hence the quotient group $PGL(2,k):=GL(2,k)/k^*$ is a subgroup of the group $Aut_k(C)$ of automorphisms of $C$. In Ex.II.7.1.1 they prove that these are all automorphisms of the projective line. Any automorphism $\phi$ over $k$ must have the property that $\phi^*(\mathcal{O}(1))=\mathcal{O}(1)$ since $\phi$ induce an automorphism of the Picard group $Pic(C)$. The map $\phi$ must induce an automorphism of $H^0(C, \mathcal{O}(1))$ which is a 2-dimensional vector space over $k$ and it follows from Theorem II.7.1 that $\phi$ is induced by an element of $PGL(2,k)$.
There is a relation between maps $\phi: C \rightarrow C$ and line bundle quotients
$$\phi_L:\pi^*(V^*) \rightarrow L \rightarrow 0$$
(see Proposition II.7.12) and this result and Thm II.7.1 is needed for the calculation of $Aut_k(C)$.
Note: The projective line $C$ has an open cover
$$D(x_0):=Spec(k[t])\text{ and }D(x_1):=Spec(k[s])$$
with $t:=x_1/x_0, s:=1/t$ and $$D(x_0) \cap D(x_1) \cong Spec(k[t,s]).$$
Any map $\psi:Spec(A) \rightarrow Spec(A)$ induce for any element $a\in A$ a map
$$\phi_a: D(a):=Spec(A_a) \rightarrow D(\phi(a)):=Spec(A_{\phi(a)})$$
If $A$ is a $k$-algebra and $S:=Spec(A)$ it follows by construction
$$Aut_k(S) \cong Aut_{k-alg}(A),$$
hence to construct an automorphism of $k$-schemes of $S$ is equivalent to constructing a $k$-algebra automorphism of $A$. It is not immediate how to calculate the automorphism group $Aut_k(C)$ of $C$ using the open cover above: An automorphism $\phi$ may not fix the open sets $D(x_i)$.
This calculation requires knowledge about sheaves and linebundles.
Coordinatewise: If you define the real projective line $C(k)$ as the set of equivalence classes $x:=(a_0:a_1) \in k^2-\{(0,0)\}$ with $(a_0:a_1) \cong (\lambda a_0:\lambda a_1)$ with $\lambda \in k^*$, given any matrix $g \in GL(2,k)$ you may define
$$gx:=(aa_0+ba_1:ca_0+da_1)$$
and since $g$ is an invertible matrix it follows $gx\in C(k)$ is a well defined element. This defines for any matrix $g\in GL(2,k)$ an automorphism (over $k$)
$$g: C(k) \rightarrow C(k)$$
with inverse $g^{-1}$. The maps $g$ and $g+ \lambda Id(2)$ where $\lambda \in k^*$ and $Id(2)$ is the $2 \times 2$ identity matrix induce the same map, and hence you get a map
$$ \rho: PGL(2,k):=GL(2,k)/\{\lambda Id(2)\} \rightarrow Aut_k(C(k))$$
If you are to do everything rigorously in the language of "schemes" things become more complicated.
