Kurs:Vector bundles, forcing algebras and local cohomology (Tehran 2012)/Lecture 1

Systems of linear equations

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.

These objects ${}T$ 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

${}f_{1}T_{1}+\cdots +f_{n}T_{n}=f\,$

is given. Then in the homogeneous case (${}f=0$) the fibers are vector spaces of dimension ${}n-1$ or ${}n$, and the later holds exactly for the points ${}P\in X$ where ${}f_{1}(P)=\ldots =f_{n}(P)=0$. In the inhomogeneous case the fibers are either empty or of dimension ${}n-1$ or ${}n$. We give some typical examples.

Example

We consider the line ${}X={\mathbb {A} }_{K}^{1}$ (or ${}X=K,\mathbb {R} ,{\mathbb {C} }$ etc.) with the (identical) function ${}x$. For ${}f_{1}=x$ and ${}f=0$, i.e. for the homogeneous equation ${}xt=0$, the geometric object ${}V$ consists of a horizontal line (corresponding to the zero-solution) and a vertical line over ${}x=0$. So all fibers except one are zero-dimensional vector spaces. For the inhomogeneous equation ${}xt=1$, ${}T$ is a hyperbola, and all fibers are zero-dimensional with the exception that the fiber over ${}x=0$ is empty.

For the homogeneous equation ${}0t=0$, ${}V$ is just the affine cylinder over the base line. For the inhomogeneous equation ${}0t=x$, ${}T$ consists of one vertical line, almost all fibers are empty.

Example

Let ${}X$ denote a plane (like $K^{2},\mathbb {R} ^{2},{\mathbb {A} }_{K}^{2}$) with coordinate functions ${}x$ and ${}y$. We consider an inhomogeneous linear equation of type

${}x^{a}t_{1}+y^{b}t_{2}=x^{c}y^{d}\,.$

The fiber of the solution set ${}T$ over a point ${}\neq (0,0)$ is one-dimensional, whereas the fiber over ${}(0,0)$ has dimension two (for ${}a,b,c,d\geq 1$). Many properties of ${}T$ 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 ${}n-1$, whereas the fiber over some special points degenerates to an ${}n$-dimensional solution set (or becomes empty).

Forcing algebras

We describe now the algebraic setting of systems of linear equations depending on a base space. For a commutative ring ${}R$, its spectrum ${}X=\operatorname {Spec} _{}^{}{\left(R\right)}$ is a topological space on which the ring elements can be considered as functions. The value of ${}f\in R$ at a prime ideal ${}P\in \operatorname {Spec} _{}^{}{\left(R\right)}$ is just the image of ${}f$ under the morphism ${}R\rightarrow R/P\rightarrow \kappa (P)=Q(R/P)$. In this interpretation, a ring element is a function with values in different fields. Suppose that ${}R$ contains a field ${}K$. Then an element ${}f\in R$ gives rise to the ring homomorphism

$K[Y]\longrightarrow R,\,Y\longmapsto f,$

which itself gives rise to a scheme morphism

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

This is another way to consider ${}f$ as a function on ${}\operatorname {Spec} _{}^{}{\left(R\right)}$.

Definition

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

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

is called the forcing algebra of these elements (or these data).

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

The forcing algebra induces the spectrum morphism

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

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

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

and we can write

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

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

(empty or) affine spaces.

But not only the fibers are affine spaces: If we localize the forcing algebra at ${}f_{i}$ we get

$(R[T_{1},\ldots ,T_{n}]/(f_{1}T_{1}+\cdots +f_{n}T_{n}-f))_{f_{i}}\cong R_{f_{i}}[T_{1},\ldots ,T_{i-1},T_{i+1},\ldots ,T_{n}]\,,$

since we can write

$T_{i}=-\sum _{j\neq i}{\frac {f_{j}}{f_{i}}}T_{j}+{\frac {f}{f_{i}}}.$

So over every ${}D(f_{i})$ the spectrum of the forcing algebra is an ${}(n-1)$-dimensional affine space over the base. So locally, restricted to ${}D(f_{i})$, we have isomorphisms

$T{|}_{D(f_{i})}\cong D(f_{i})\times {\mathbb {A} }^{n-1}.$

On the intersections ${}D(f_{i})\cap D(f_{j})$ we get two identifications with affine space, and the transition morphisms are linear if ${}f=0$, but only affine-linear in general (because of the translation with ${\frac {f}{f_{i}}}$).

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