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

${\displaystyle K[Y]\longrightarrow R,Y\longmapsto f,}$

which gives rise to a scheme morphism

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

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

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

## Definition

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

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

is called the forcing algebra of these elements

(or these data).

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

The forcing algebra induces the spectrum morphism

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

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

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

and we can write

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

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