Forcing algebra/Linear equation/Introduction/Section

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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.

Let be a commutative ring and let and be elements in . Then the -algebra

is called the forcing algebra of these elements

(or these data).

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

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.