I'm trying to build a better understanding of the spectrum of a ring, and this leads me to wonder: can we learn about the base ring $R$ by studying $R[X]$? So, let's consider the most "dramatic" case, when $R$ is a ring that is not a field, versus a field $F$. Can we say more about $F[X]$ than about $R[X]$?
In general, I don't know where I should read more about this: a text book that focuses on the geometry of the spectrum of rings would be greatly appreciated!
Not sure this will be a complete answer. I suppose for simplicity that the ring $R$ contains a field $F$ (so say either $\mathbb{F}_p$ or $\mathbb{Q}_p$, or preferably a field over which it is a finitely generated algebra). Then $Spec(R[X]) = Spec(R \otimes_F F[X] ) = Spec(R) \times_F Spec(F[X]) = Spec(R) \otimes_F \mathbb{A}^1_F$. In other words, in most cases (whenever $R$ contains a field $F$, preferably finitely generated over $R$, and so we can think about geometry over the field $F$ when dealing with $R$), the entity $Spec(R[X])$ "splits" into a product of $Spec(R)$ with the affine line over $F$, so not really a big change - you imagine the affine line over $F$ sitting along the horizontal axis, you imagine $Spec(R)$ sitting along the vertical axis, and you imagine the product.
Slightly more generally, you can always try to imagine what are fibers of $Spec(R[X])$ over $Spec(R)$ (w.r.t. the natural map $p : Spec(R[X]) \to Spec(R)$). Thus, for example working with geometric points, you choose a map $R \to K$ where $K$ is an algebraically closed field, and then you obtain $K \otimes_R R[X] \cong K[X]$ or in different words the geometric fiber of $p$ over $Spec(K) \to Spec(R)$ is the affine line $\mathbb{A}^1_K$. So $p$ is kind of the affine line over $Spec(R)$ - all its fibers are affine lines.
Anyway, maybe someone can come up with a better explanation (perhaps the key case to look at is $R = \mathbb{Z}$, for example analyzing what are the prime ideals in the ring $\mathbb{Z}[X]$ will give some picture (I vaguely remember that maybe the red book of Mumford, or perhaps a book with title something like "the geometry of schemes" may have this), but I will not try now).