We describe now the algebraic setting of systems of linear equations depending on a base space. For a commutative ring
, its spectrum
is a topological space on which the ring elements can be considered as functions. The value of
at a prime ideal
is just the image of
ander the ring homomorphism
. In this interpretation, a ring element is a function with values in different fields. Suppose that
contains a field
. Then an element
gives rise to the ring homomorphism
-
which gives rise to a scheme morphism
-
This is another way to consider
as a function on
with values in the affine line.
The following construction appeared first in the work of Hochster in the context of solid closure.
The forcing algebra
forces
to lie inside the extended ideal
(hence the name).
For every
-algebra
such that
there exists a
(non unique)
ring homomorphism
by sending
to the coefficient
in an expression
.
The forcing algebra induces the spectrum morphism
-
Over a point
,
the fiber of this morphism is given by
-
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)}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa1641633ecbabaad28b64b143aab5ac02c173b0)
where
means the evaluation of the
in the residue class field. Hence the
-points in the fiber are exactly the solutions to the inhomogeneous linear equation
.
In particular, all the fibers are
(empty or)
affine spaces.