Inhomogeneous linear equation/System/Geometric situation/description

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

${}f_{11}t_{1}+\cdots +f_{1n}t_{n}=0\,,$

${}f_{21}t_{1}+\cdots +f_{2n}t_{n}=0\,,$

$\vdots$

${}f_{m1}t_{1}+\cdots +f_{mn}t_{n}=0\,,$

where the ${}f_{ij}$ are elements in ${}K$. The solution set to this system of homogeneous equations is a vector space ${}V$ over ${}K$ (a subvector space of $K^{n}$), its dimension is ${}n-\operatorname {rk} (A)$, where

${}A=(f_{ij})_{ij}\,$

is the matrix given by these elements. Additional elements ${}f_{1},\ldots ,f_{m}\in K$ give rise to the system of inhomogeneous linear equations,

${}f_{11}t_{1}+\cdots +f_{1n}t_{n}=f_{1}\,,$

${}f_{21}t_{1}+\cdots +f_{2n}t_{n}=f_{2}\,,$

$\vdots$

${}f_{m1}t_{1}+\cdots +f_{mn}t_{n}=f_{m}\,.$

The solution set ${}T$ 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

$V\times T\longrightarrow T,\,(v,t)\longmapsto v+t,$

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 ${}(V,+,0)$ on the set ${}T$. Moreover, if we fix one solution ${}t_{0}\in T$

(supposing that at least one solution exists), then there exists a bijection

$V\longrightarrow T,\,v\longmapsto v+t_{0}.$

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

Suppose now that ${}X$ 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 ${}K$ we have functions

$f_{ij}\colon X\longrightarrow K$

on ${}X$ (which are continuous, or differentiable, or algebraic). We form the matrix of functions ${}A={\left(f_{ij}\right)}_{ij}$, which yields for every point ${}P\in X$ a matrix ${}A(P)$ over ${}K$. Then we get from these data the space

${}V={\left\{(P;t_{1},\ldots ,t_{n})\mid A(P){\begin{pmatrix}t_{1}\\\vdots \\t_{n}\end{pmatrix}}=0\right\}}\subseteq X\times K^{n}\,$

together with the projection to ${}X$. For a fixed point ${}P\in X$, the fiber ${}V_{P}$ of ${}V$ over ${}P$ is the solution space to the corresponding system of homogeneous linear equations given by inserting ${}P$ into ${}f_{ij}$. In particular, all fibers of the map

$V\longrightarrow X,$

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

$V\times _{X}V\longrightarrow V,\,(P;s_{1},\ldots ,s_{n};t_{1},\ldots ,t_{n})\longmapsto (P;s_{1}+t_{1},\ldots ,s_{n}+t_{n}),$

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

$X\longrightarrow V,\,P\longmapsto (P;0,\ldots ,0),$

is called the zero-section.

Suppose now that additional functions

$f_{1},\ldots ,f_{m}\colon X\longrightarrow K$

are given. Then we can form the set

${}T={\left\{(P;t_{1},\ldots ,t_{n})\mid A(P){\begin{pmatrix}t_{1}\\\vdots \\t_{n}\end{pmatrix}}={\begin{pmatrix}f_{1}(P)\\\vdots \\f_{n}(P)\end{pmatrix}}\right\}}\subseteq X\times K^{n}\,$

with the projection to ${}X$. Again, every fiber ${}T_{P}$ of ${}T$ over a point ${}P\in X$ is the solution set to the system of inhomogeneous linear equations which arises by inserting ${}P$ into ${}f_{ij}$ and ${}f_{i}$. The actions of the fibers ${}V_{P}$ on ${}T_{P}$ (coming from linear algebra) extend to an action

$V\times _{X}T\longrightarrow T,\,(P;t_{1},\ldots ,t_{n};s_{1},\ldots ,s_{n})\longmapsto (P;t_{1}+s_{1},\ldots ,t_{n}+s_{n}).$

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

$s\colon X\longrightarrow T$

with ${}s(P)\in T_{P}$ exists, then we can construct a (continuous, differentiable, algebraic) isomorphism between ${}V$ and ${}T$. However, different from the situation in linear algebra (which corresponds to the situation where ${}X$ is just one point), such a section does rarely exist.