Forcing algebra/Linear equation/Introduction/Section

We describe now the algebraic setting of systems of linear equations depending on a base space. For a commutative ring ${}R$, its spectrum ${}X=\operatorname {Spec} {\left(R\right)}$ is a topological space on which the ring elements can be considered as functions. The value of ${}f\in R$ at a prime ideal ${}P\in \operatorname {Spec} {\left(R\right)}$ is just the image of ${}f$ ander the ring homomorphism ${}R\rightarrow R/P\rightarrow \kappa (P)=Q(R/P)$. In this interpretation, a ring element is a function with values in different fields. Suppose that ${}R$ contains a field ${}K$. Then an element ${}f\in R$ gives rise to the ring homomorphism

$K[Y]\longrightarrow R,Y\longmapsto f,$

which gives rise to a scheme morphism

$\operatorname {Spec} {\left(R\right)}\longrightarrow \operatorname {Spec} {\left(K[Y]\right)}\cong {\mathbb {A} }_{K}^{1}.$

This is another way to consider ${}f$ as a function on ${}\operatorname {Spec} {\left(R\right)}$ with values in the affine line.

The following construction appeared first in the work of Hochster in the context of solid closure.

Definition

Let ${}R$ be a commutative ring and let ${}f_{1},\ldots ,f_{n}$ and ${}f$ be elements in ${}R$. Then the ${}R$-algebra

$R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}-f\right)}$

is called the forcing algebra of these elements

(or these data).

The forcing algebra ${}B$ forces ${}f$ to lie inside the extended ideal ${}{\left(f_{1},\ldots ,f_{n}\right)}B$ (hence the name). For every ${}R$-algebra ${}S$ such that ${}f\in {\left(f_{1},\ldots ,f_{n}\right)}S$ there exists a (non unique) ring homomorphism ${}B\rightarrow S$ by sending ${}T_{i}$ to the coefficient ${}s_{i}\in S$ in an expression ${}f=s_{1}f_{1}+\cdots +s_{n}f_{n}$.

The forcing algebra induces the spectrum morphism

$\operatorname {Spec} {\left(B\right)}\longrightarrow \operatorname {Spec} {\left(R\right)}.$

Over a point ${}P\in X=\operatorname {Spec} {\left(R\right)}$, the fiber of this morphism is given by

$\operatorname {Spec} {\left(B\otimes _{R}\kappa (P)\right)},$

and we can write

${}B\otimes _{R}\kappa (P)=\kappa (P)[T_{1},\ldots ,T_{n}]/{\left(f_{1}(P)T_{1}+\cdots +f_{n}(P)T_{n}-f(P)\right)}\,,$

where ${}f_{i}(P)$ means the evaluation of the ${}f_{i}$ in the residue class field. Hence the ${}\kappa (P)$-points in the fiber are exactly the solutions to the inhomogeneous linear equation ${}f_{1}(P)T_{1}+\cdots +f_{n}(P)T_{n}=f(P)$. In particular, all the fibers are (empty or) affine spaces.