Linear systems of equations/Homogeneous and inhomogeneous/Torsor/Introduction/Section

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We start with some linear algebra. Let be a field. We consider a system of linear homogeneous equations over ,

where the are elements in . The solution set to this system of homogeneous equations is a vector space over (a subvector space of ), its dimension is , where

is the matrix given by these elements. Additional elements give rise to the system of inhomogeneous linear equations,

The solution set of this inhomogeneous system may be empty, but nevertheless it is tightly related to the solution space of the homogeneous system. First of all, there exists an action

because the sum of a solution of the homogeneous system and a solution of the inhomogeneous system is again a solution of the inhomogeneous system. This action is a group action of the group on the set . Moreover, if we fix one solution (supposing that at least one solution exists), then there exists a bijection

This means that the group acts simply transitive on , and so can be identified with the vector space , however not in a canonical way.

Suppose now that is a geometric object (a topological space, a manifold, a variety, a scheme, the spectrum of a ring) and that instead of elements in the field we have functions

on (which are continuous, or differentiable, or algebraic). We form the matrix of functions , which yields for every point a matrix over . Then we get from these data the space

together with the projection to . For a fixed point , the fiber of over is the solution space to the corresponding system of homogeneous linear equations given by inserting into . In particular, all fibers of the map

are vector spaces (maybe of non-constant dimension). These vector space structures yield an addition[1]

(only points in the same fiber can be added). The mapping

is called the zero-section.

Suppose now that additional functions

are given. Then we can form the set

with the projection to . Again, every fiber of over a point is the solution set to the system of inhomogeneous linear equations which arises by inserting into and . The actions of the fibers on (coming from linear algebra) extend to an action

Also, if a (continuous, differentiable, algebraic) map

with exists, then we can construct a (continuous, differentiable, algebraic) isomorphism between and . However, different from the situation in linear algebra (which corresponds to the situation where is just one point), such a section does rarely exist.

These objects have new and sometimes difficult global properties which we try to understand in these lectures. We will work mainly in an algebraic setting and restrict to the situation where just one equation

is given. Then in the homogeneous case () the fibers are vector spaces of dimension or , and the later holds exactly for the points where . In the inhomogeneous case the fibers are either empty or of dimension or . We give some typical examples.

We consider the line (or etc.) with the (identical) function . For and , i.e. for the homogeneous equation , the geometric object consists of a horizontal line (corresponding to the zero-solution) and a vertical line over . So all fibers except one are zero-dimensional vector spaces. For the inhomogeneous equation , is a hyperbola, and all fibers are zero-dimensional with the exception that the fiber over is empty.

For the homogeneous equation , is just the affine cylinder over the base line. For the inhomogeneous equation , consists of one vertical line, almost all fibers are empty.

Let denote a plane (like ) with coordinate functions and . We consider an inhomogeneous linear equation of type

The fiber of the solution set over a point is one-dimensional, whereas the fiber over has dimension two (for ). Many properties of depend on these four exponents.

In (most of) these example we can observe the following behavior. On an open subset, the dimension of the fibers is constant and equals , whereas the fiber over some special points degenerates to an -dimensional solution set (or becomes empty).

  1. is the fiber product of with itself.