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?

Restriction of Scalars and Functoriality of Presheaves.

Let $\phi\colon R\longrightarrow S$ be a morphism of rings. There is associated to $\phi$ a natural functor from $\mathrm{Alg}_S$ to $\mathrm{Alg}_R$, called the restriction of scalars functor: $$f\colon\mathrm{Alg}_S\longrightarrow\mathrm{Alg}_R.$$ In detail, this is the functor taking an $S$-algebra $S\rightarrow A$ to the $R$-algebra $R\rightarrow S\rightarrow A$, which we denote $A_R$.

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

Base Change of Schemes.

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

This functor takes an $R$-scheme $h_A$ to the presheaf $$f^*h_A\colon\mathrm{Alg}_S\longrightarrow\mathrm{Alg}_R\longrightarrow\mathrm{Sets}$$ defined by $$B\mapsto\mathrm{Hom}_{\mathrm{Alg}_R}(B_R,A)\cong\mathrm{Hom}_{\mathrm{Alg_S}}(B,A\otimes_RS),$$ where the isomorphism comes from the adjunction between restriction and extension of scalars.

That is, $f^*h_A=h_A\times_R h_S$ and the functor $f^*$ is therefore base change of schemes.

Adjoints to Base Change.

As R. van Dobben de Bruyn points in the comments, the right adjoint $f_*$ of $f^*$ is called Weil restriction. While it can fail to be a scheme in general, it is representable by schemes under nice conditions. One may then ask about the left adjoint $f_!$:

Question 1. Is the left adjoint $f_!$ of $f^*$ representable by schemes? Moreover, if it isn't, are there conditions we can require of an $S$-scheme $X$ guaranteeing the presheaf $f_!X$ to be a scheme?

‎

Question 2. What about the non-affine case?

(I gather from the Wikipedia page on Weil restriction that these have been studied in the very general case of schemes over ringed topoi. What reference/s is Wikipedia alluding to?)

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:

adjunction

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?

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?

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 is an induced adjoint triple of functors between the corresponding presheaf categories:

adjunction

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.

This 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 $\mathrm{Spec}(A)\rightarrow\mathrm{Spec}(R)\rightarrow\mathrm{Spec}(S)$.

This 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? (Maybe base change or something like that)
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?

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:

adjunction

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?