Let $U$ be an arbitrary scheme and we consider the schematic fiber product $\mathbb{A}^1 _U = \mathbb{A}^1 _{\mathbb{Z}} \times_{\mathbb{Z}} U$.
My question referer to Bosch's "linear morphisms" (of schemes) $\psi: \mathbb{A}^1 _U \to \mathbb{A}^1 _U$ as desecribed below.
If we firstly recall the case of vecor bundles $V$ over base $B$ then the morphisms are locally linear on trivialisations $V \vert_{p^{-1}(U_i)} \cong \mathbb{K}^n \times U_i$
via $\phi_{ij}: \mathbb{K}^n \times U_i \cap U_j \to \mathbb{K}^n \times U_i \cap U_j$, $(v, u) \mapsto (A_{ij}(u)v, u)$ with linear $A_{ij}(u)$.
For trivial bundles $V \cong \mathbb{K}^n \times B$ we set $U_i =B$.
MY QUESTION:
In Bosch's "Algebraic Geometry and Commutative Algebra" (see pages 428/ 429) there was suggested an attempt to "transfer" this formalism to schemes by do following (see below why this construction doesn't convince me). Here the excerpt:
We define a morphism $\psi: \mathbb{A}^1 _U \to \mathbb{A}^1 _U $ via
$$(t,x) \mapsto (\eta(x) \cdot t, x)$$
for a section $\eta \in \mathcal{O}_U(U)$, $t \in \mathbb{A}^1 _{\mathbb{Z}}, x \in U$.
Below it is explained how $\psi$ is defined on level of rings (=local sections) namely as a map of ringed spaces we have by definition for an open affine subset $Spec(R)= V \subset U$ the description as ring map
$$\psi^{\#} _{\mathbb{A}^1 _{\mathbb{Z}} \times V}: R[\zeta]=\mathbb{A}^1 _U(\mathbb{A}^1 _{\mathbb{Z}} \times V) \to \mathbb{A}^1 _U(\mathbb{A}^1 _{\mathbb{Z}} \times V)$$
$$\zeta \mapsto (\eta \vert _{Spec(R)}) \cdot \zeta$$
Take into account that $\mathbb{A}^1 _U(\mathbb{A}^1 _{\mathbb{Z}} \times V)= \mathbb{A}^1 _{\mathbb{Z}}(\mathbb{A}^1 _{\mathbb{Z}} ) \otimes _{\mathbb{Z}} R = R[\zeta]$.
So on level of local sections the definition of $\psi$ is clear. WHAT I DON'T UNDERSTAND is how to interpret $(t,x) \mapsto (\eta(x) \cdot t, x) \in \mathbb{A}^1 _U = \mathbb{A}^1 _{\mathbb{Z}} \times_{\mathbb{Z}} U$ on level of Specs. Indeed, $t \in \mathbb{A}^1 _{\mathbb{Z}}, x \in U$. What is $\eta(x) \cdot t$? Namely what is here the product operation "$\cdot$"?
Set theoretically since $\mathbb{A}^1 _U = \mathbb{A}^1 _{\mathbb{Z}} \times_{\mathbb{Z}} U$ the $\eta(x) \cdot t$ lives in first factor $\mathbb{A}^1 _{\mathbb{Z}}$.
So mathematically the object with the "multiplication"/action $\eta(x) \cdot t$ doesn't make any sense to me.
Could anybody explain the meaning of it? Is $(\eta(x) \cdot t, x) $ just a symbolical notation?
