I am currently studying the book of Daniel Perrin in Algebraic Geometry. Particularly, I am having trouble understanding some of the definitions in the chapter for Tangent Space and singular points.
He uses the "point scheme" $P$ consisting of one point and denotes it by $Spec\ k$, correspondingly he uses the "fat point" $P_ε$ denoted by $Spec\ k[ε]$. He says that there is an obvious morphism $i:P \to P_ε$ which corresponds to $p:k[ε] \to k$, sending $(a+bε) \to a$ , i.e. $ε \to 0$. Which makes sense from the equivalence between ring morphisms and morphisms in affine schemes.
So, he defines a deformation of $V$ (affine algebraic variety) at $x$ being a morphism $t:P_ε \to V$ such that $t \circ i = x$, where $ x:P \to V$ is the map that sends $P$ (single element) to the point $x$ in the affine algebraic variety $V$.
Then, he says that this is equivalent to a $k$-algebra homomorphism $t^*:Γ(V) \to k[ε]$ such that $p \circ t^* = χ_x$ where the latter is the character given by $f \to f(x)$, namely evaluation homomorphism.
Omitting the affine algebraic variety term and defining $V := Spec\ Γ(V)$ (which is a good representation for $V$), everything makes sense for me except for the fact that $t$ is equivalent to a $k$-algebra homomorphism $t^*$ and not just a ring homomorphism. So, my first question is, is there any proposition or reason why should $t^*$ be also linear?
On the other hand Perrin defines the abovementioned morphisms to the category of ringed spaces of $k$-valued functions and he only mentions the generall form of the ringed spaces (i.e continuous functions and family of homomorphisms) let alone the morphisms in Locally Ringed Spaces.
In any case, I think that something is missing, can anyone help me or give me any reference to understand how the above definitions and equivalences work?