Last time, we estabslished the basics of classical algebraic geometry
over algebraically closed fields, and motivated the duality between
algebra and geometry via Hilbert’s Nullstellenstaz, and got our first
clues about how to generalise algebraic geometry via the correspondence
of maximal ideals to points.

Before defining MaxSpec outright, let us first see that it can indeed
generalise classical varieties (After all, that is what we are looking
to do). It’s worth noting at this point that we are currently only
dealing with affine varieties, but that projective varieties, whose
definition needs a bit more care, are immensely important to algebraic
geometry. We will eventually meet these objects, in the world of
schemes.

The polynomial ring
gives us a ring of algebraic functions on
, whose ideals correspond to points. Naturally, one may
seek the analogue of this for a general algebraic variety.

Let
be an algebraic variety for
a radical ideal. Since we are interested in the structure
defined by polynomials, the functions we want to consider are those that
arise from polynomial functions. We make the following definition:

Definition: The __Coordinate Ring__
of an algebraic variety
is defined as follows:
So as functions, the elements of
are functions
which are restrictions of polynomial functions (Note that
if we endow
with the Zariski topology, such functions are continuous,
but not all continuous functions are of this form), so they can be given
explicitly by polynomials.

Note that this presentation needn’t be unique. Consider, for example,
the
-axis in
, defined by
, and the polynomials
, then
as functions, since we are consider the set of points for
which
. In fact, these presentations needn’t even be the same on
! Take
, then
and
satisfy

There is a more algebraic description of the coordinate ring: There is a
natural map
given via considering a polynomial
as a function, and restricting it to
. The kernel of this map is the set of polynomials which
map to
, i.e. as functions they vanish on
, so this is exactly
(By the Nullstellensatz). By definition, this map is
surjective, so the first isomorphism theorem gives
This allows us to think of elements of
as polynomials, modulo the equivalence relation
iff
vanishes on
(i.e.
), so if
as functions. Now the correspondence theorem for rings in
addition to the final corollary from last time gives a bijective
correspondence between points of
and maximal ideals in
(Equivalently, maximal ideals in
containing
). This extends our points-ideals correspondence further to
any affine algebraic set.

The set of maximal ideals of
is again in correspondence with the points of
, so that
MaxSpec should be constructed such that it is homeomorphic (Or
isomorphic, for a correct notion of isomorphism) to
. This gives us the following useful perspective, which we
will cling onto for dear life for the next while to make sense of all of
this:

For a general ring, we would like to think of
as a "coordinate ring" of polynomial functions on our set
of points
MaxSpec. We will continue exploring this perspective soon, but
first we define our prototype of MaxSpec.

Definition: Let
be a ring. Define
as the topological space whose underlying set is the set
of maximal ideals of
, and whose **closed** sets are of the form
For an ideal
, i.e. the set of maximal ideals containing it.

Of course, we could have defined
for a random subset
of elements, and noted that
(Even more simply than before). This fits in with our
understanding of
as the coordinate ring:

For a polynomial
, evaluating at a point
is the same as looking at the image of
under the canonical map
Thus for a general ring, we can think of an element
as a function on
. In classical AG, since
is algebraically closed, every quotient by a maximal ideal
is canonically isomorphic to to
, but generally this is not the case, so this function will
have its range vary from point to point. In other words, for every
we a map
Although the range varies, it still makes sense to define
a ’vanishing set’ as the set of maximal ideals in which
, or equivalently
. This definition coincides with the definition we gave
above, i.e. if
is an ideal, then we can define
So now we have an operation
which sends rings to spaces, and we have an inkling of
intuition about how to think of this thing via the coordinate ring. The
next step is to understand the types of maps we are interested in
between these MaxSpecs. Obviously, any random continuous map won’t do,
since we are trying to do geometry, so we need extra structure. We must
harken back to classical AG once again to try and find a generalisable
notion of a map we are interested in.

For varities
, it is clear what the correct notion of function should
be: Since we are dealing with structure coming from polynomials,
similarly to the coordinate ring we shall define a __Morphism__ to be
a map
such that there are polynomial functions
for which
, where here we mean
Another reason these maps are natural is that it gives us
another correspondence between algebra and geometry: Suppose we were
looking for what the correct notion of a map of varieties is. Extending
the analogy between algebra and geometry, we would want this notion to
bijectively correspond to homomorphisms
. We are only considering
-Algebra homomorphisms, as this ’remembers’ the underlying
field
.

Given such a homomorphism
, it is a fact that if
is maximal, then
is maximal. Using the Nullstellensatz, this gives us a
function
, which in turn defines a function
. Oops! We reversed the arrows! Indeed, the correspondence
will be between ’nice’ maps
and
-Algebra homomorphisms
.

So now given a map
we get an induced map
, given by
, where
. It turns out that this map
is a morphism in the sense previously defined! So is given
by polynomials. A morphism
also gives rise to a map
, which is actually given by a precomposition with
polynomials.

For the category theory inclined, we have defined a functor
Where the morphisms are as defined previously. We restrict
to finitely generated reduced
-Algebras because these are exactly the algebras given by a
quotient of
for any
by a radical ideal. This functor is actually an
equivalence of categories, so Finitely generated reduced
-Algebras are *dual* to Affine Varieties.

Welcome back, everyone who skipped the last prargraph. The bottom line
is that the notion of "is given by polynomials" is not easy to capture.
We return to our perspective of
as the coordinate ring of
. With this perspective in mind, we can take the
’Algebraic’ analogue, i.e. a map of
-Algebras, or in the more general case, a map of rings
, and try defining a morphism to be a map of the form
for
.

Unfortunately, for a general ring homomorphism
is not necessarily a maximal ideal if
is. It *is* a prime ideal though. The idea to then
keep track of prime ideals arises, in order to give an analogy to the
notion of a variety morphism. One can think of this as retaining the
information of irreducible subvarieties, since this is what prime ideals
correspond to classicaly. This gives rise to the notion of
. Confusingly, points
don’t have to be closed, even though they correspond to
closed sets, i.e.
. Another way to think of these points is as generic points
of the subvariety corresponding to
. Note that in general,
needn’t be prime. This is already true in the classical
notion if we move to a non-algebraically closed field
( gives us an example). We keep track of this information
about ’irreducibility’ through the generic points
(.

For all this and more, join us next time, hopefully soon.