§ Nullstellensatz for schemes
§ System of equations
Consider a set of polynomials subset .
A system of equations for unknowns is the tuple
We abbreviate this to where the bolded version
implies that these are vectors of values.
§ Solutions to system of equations
Note that we often define equations, for example over a ring
such as . But its solutions live elsewhere: In this case, the solutions
live in , as well as in . Hence, we should not restrict
our solution space to be the ring where we defined our coefficients from!
Rather, as long as we are able to interpret the polynomial
in some other ring , we can look for solutions in the ring . Some thought
will tell us that all we need is a ring homomorphism .
Alternatively/equivalently, we need to be a -algebra.
Let us consider the single-variable case with . This naturally
extends to the multivariate case.
Using , we can map to by taking
to . This clearly extends
to the multivariate case. Thus, we can interpret solutions to an equation
Formally, the solution to a system
in ring , written as is a set of elements
such that for all in and for all in .
§ Equivalent systems of equations
Two systems of equations over the same ring are said to be
equivalent over iff for all -algebras , we have
§ Biggest system of equations
For a given system of equations over the ring ,
we can generate the largest system of equations that still has the same solution:
generate the ideal , and consider the system
of equations .
§ Varieties and coordinate rings
Let . The polynomial is also a function which maps
to through evaluation .
Let us have a variety defined by some set of polynomials
. So the variety is the vanishing set of ,
and is the largest such set of polynomials.
Now, two functions are equal on the variety
iff they differ by a function whose value is zero on the variety . Said
differently, we have that iff where vanishes on .
We know that the polynomials in vanish on , and is the
largest set to do so. Hence we have that .
To wrap up, we have that two functions are equal on , that is,
So we can choose to build a ring where are "the same function". We do
this by considering the ring .
This ring is called as the coordinate ring of the variety .
§ An aside: why is it called the "coordinate ring"?
We can consider the ith coordinate function as one that takes to
So we have which defines a function
which extracts the th coordinate.
Now the quotienting from the variety to build , the coordinate ring
of the variety will make sure to "modulo out" the coordinates that
"do not matter" on the variety.
§ Notation for coordinate ring of solutions:
For a system , we are interested in the
solutions to , which forms a variety . Furthermore,
we are interested in the algebra of this variety, so we wish to talk about
the coordinate ring . We will
denote the ring as .
§ Solutions for in : -algebra morphisms
Let's simplify to the single variable case. Multivariate
follows similarly by recursing on the single
variable case. .
There is a one-to-one coorespondence between solutions to in and
elements in where is the set of -algebra
Expanding definitions, we need to establish a correspondence between
- Points such that for all .
- Morphisms .
§ Forward: Solution to morphism
A solution for in is a point such that
vanishes on . Thus, the evaluation map
has kernel . Hence, forms an honest to god morphism
between and .
§ Backward: morphism to solution
Assume we are given a morphism . Expanding
definitions, this means that .
We need to build a solution. We build the solution .
Intuitively, we are thinking of as .
If we had an , then we would learn the point
by looking at , since .
We can show that this point exists in the solution as follows:
§ Consistent and inconsistent system over ring
Fix a -algebra . The system is consistent over iff
. the system over is inconsistent iff
§ Geometric Language: Points
Let be the main ring, a system of
equations in unknowns .
For any -algebra , we consider the set as a collection
of points in . These points are solutions to the system .
§ The points of -algebra