1 Preliminaries.

1.1 Functoriality of Presheaves.

Let $\phi\colon R\longrightarrow S$ be a morphism of rings. There is a natural functor $$f\colon\mathbf{Alg}_S\longrightarrow\mathbf{Alg}_R$$ taking an $S$-algebra $S\rightarrow A$ to the $R$-algebra $R\rightarrow S\rightarrow A$.

As remarked in this nLab page (and developed in detail in SGA IV, Exposé I, Section 5), there exists an induced adjoint triple of functors between the corresponding presheaf categories:

where $f^*\colon\mathrm{PSh}(\mathbf{Alg}_R)\longrightarrow\mathrm{PSh}(\mathbf{Alg}_S)$ is given by precomposition with $f$.

1.2 Relation to Schemes.

We may restrict $f^*$ to the full subcategory $\mathbf{Aff}_R$ of $\mathrm{PSh}(\mathbf{Alg}_R)$ spanned by the representable presheaves on $\mathbf{Alg}_R$, i.e. by affine $R$-schemes.

Restriction then gives a functor $$f^*|_{\mathbf{Aff}_R}\colon\mathbf{Aff}_R\longrightarrow\mathbf{Aff}_S$$ sending an affine $R$-scheme $\mathrm{Spec}(A)\rightarrow\mathrm{Spec}(R)$ to the affine $S$-scheme $\mathrm{Spec}(A)\rightarrow\mathrm{Spec}(R)\rightarrow\mathrm{Spec}(S)$.

The fact that $f^*$ sends schemes to schemes (rather than just presheaves on $\mathbf{Alg}_S$) leads us to the following questions:

2 Questions.

1. Is the image of the restriction of the functors $f_!$ and $f_*$ to $\mathbf{Aff}_R$ contained in the category $\mathbf{Aff}_S$? That is, do $f_!$ and $f_*$ send $R$-schemes to $S$-schemes?
2. If yes, do they admit a nice description in terms of standard scheme-theoretic constructions?
3. Do we also get a corresponding adjoint triple of functors between $\mathbf{Aff}_R$ and $\mathbf{Aff}_S$ from the triple $(f_!\dashv f^*\dashv f_*)$?
4. (Assuming this procedure works), can we globalise it to non-affine schemes?