Last update: 3 June 2013

The subject-matter of algebraic geometry, from the time of Descartes onwards, has been the study of the solutions of systems of polynomial equations in several variables:

$$\begin{array}{cc}{f}_{\alpha}({x}_{1},\dots ,{x}_{n})=0\text{.}& \text{(1)}\end{array}$$
Originally the ${f}_{\alpha}$ were taken to have real coefficients, and one looked for real solutions. However, fairly soon it was
realised that it made better sense to include *complex* solutions, since there was then a better chance of their existence (e.g.,
${x}^{2}+{y}^{2}+1=0$ has no real solutions, but plenty of
complex ones).

Equally, one of the main preoccupations of number theory has been Diophantine problems, i.e., the solutions (if any) of a system of equations (1) in rational integers, the ${f}_{\alpha}$ now being supposed to have integer coefficients: for example, 'Fermat's last theorem' , the equation ${x}^{n}+{y}^{n}={z}^{n}\text{.}$ As this example indicates, the problem thus set was often too hard, so it was natural to modify it by asking either for rational solutions or for solutions mod. $p$ $\text{(}p$ a prime number), i.e., to regard the equations (1) as having their coefficients in the rational field $\mathbb{Q}$ or the finite field ${\mathbb{F}}_{p}$ and to ask for solutions in that field. More generally, we may reduce the equations (1) mod. ${p}^{n},$ thereby replacing the coefficient domain by the Artin local ring $\mathbb{Z}/\left({p}^{n}\right),$ and we may then pass to the ring of $p\text{-adic}$ integers ${\mathbb{Z}}_{p}=\underleftarrow{\text{lim}}\hspace{0.17em}\mathbb{Z}/\left({p}^{n}\right),$ or its field of fractions ${\mathbb{Q}}_{p}\text{.}$

Thus it is natural to consider systems of equations (1) with coefficient domains other than the fields of real or complex numbers, and these coefficient domains may not always, be fields. However, if we stick to a coefficient field, we had better let it be quite arbitrary if we want a theory which is of sufficient generality for its applications. In particular, our field should be allowed to have positive characteristic (e.g., the finite fields). So we are led to study the solutions of (1), where the ${f}_{\alpha}$ are now polynomials over an a arbitrary field $k\text{.}$ As already observed, it is not enough to consider only the solutions in $k,$ because there may not be any, or at any rate not enough: we should therefore take an algebraically closed field $K\supseteq k,$ and consider the solutions of (1) in $K\text{.}$ This is roughly the point of view of Weil (Foundations of Algebraic Geometry). If we agree to ignore questions of rationality, we can jettison $k$ and use only $K\text{.}$ But this is inadequate for many purposes, e.g., Weil's conjectures on the number of points of an algebraic variety over a finite field.

Let $k$ be a field, $K$ an algebraically closed field containing $k,$ and let
$S$ be a subset of the polynomial ring $k[{t}_{1},\dots ,{t}_{n}]$
(which we shall abbreviate to $k\left[t\right]\text{).}$ The *variety*
$V\left(X\right)$ defined by $S$ is the set of all
$x=({x}_{1},\dots ,{x}_{n})\in {K}^{n}$
such that $f\left(x\right)=0$ for all $f\in S\text{.}$
If $a$ is the ideal generated by $S$ in $k\left[t\right],$
then clearly $V\left(S\right)=V\left(a\right)\text{.}$
Now let ${a}^{*}$ be the ideal consisting of all $f\in k\left[t\right]$
which vanish at every point of $V\text{.}$ Clearly ${a}^{*}\supseteq a,$
and the inclusion may be strict (for example, $a=\left({t}_{1}^{2}\right),$
${a}^{*}=\left({t}_{1}\right)\text{).}$ The
relationship between $a$ and ${a}^{*}$ is given by a theorem of Hilbert (the Nullstellensatz) which
asserts that ${a}^{*}$ is the *radical* of $a,$ that is to say it is the set
of all polynomials $f$ some power of which lies in $a\text{.}$
$V=V\left(S\right)$ is an *affine*
$(k,K)\text{-variety.}$

Each polynomial $f$ in $k\left[t\right]$ determines a function
$x\mapsto f\left(x\right)$ on ${K}^{n}$ with values
in $K,$ and the restriction of this function to $V$ is called a *regular function*
$V\text{.}$ The regular functions form a ring $A,$ clearly isomorphic to
$k\left[t\right]/{a}^{*}\text{;}$ this ring is called the
*coordinate ring* (or affine algebra) of $V\text{.}$ Obviously $A$ is finitely generated as a
$k\text{-algebra,}$ and from Hilbert's theorem it follows immediately that $A$ has no non-zero nilpotent
elements. Conversely, every finitely generated $k\text{-algebra}$ $A$ with no nilpotent elements
$\ne 0$ arises as the coordinate ring of some $(k,K)\text{-variety}$
$V$ in ${K}^{n}$ (for some $n\text{):}$ we have only to take a set
of generators ${u}_{1},\dots ,{u}_{n}$ of
$A,$ which defines a $k\text{-algebra}$ homomorphism of
$k[{t}_{1},\dots ,{t}_{n}]$
onto $A\text{;}$ the kernel $a$ of this homomorphism is an ideal which is equal to its own radical, and
$V\left(a\right)$ is the variety sought. But there is a more intrinsic way of getting $V$
from $A\text{:}$ namely, the points of $V$ are in one-to-one correspondence with the
$k\text{-homomorphisms}$ of $A$ into $K\text{.}$ For if
$x\in V,$ then $f\mapsto f\left(x\right)$
is a $k\text{-homomorphism}$ $A\mapsto K\text{;}$ and conversely,
if $\phi :A\to K$ is a $k\text{-homomorphism,}$ let
${x}_{i}=\phi \left({u}_{i}\right),$ then
$x=({x}_{1},\dots ,{x}_{n})$ is a
point of $V\text{.}$ Thus an affine algebraic variety is determined by its coordinate ring.

If $U,V$ are affine $(k,K)\text{-varieties,}$
say $U\subseteq {K}^{m},$
$V\subseteq {K}^{n},$ a mapping
$f:U\to V$ is $(k,K)\text{-regular}$
if it is induced by a $k\text{-polynomial}$ mapping of ${K}^{m}$ into
${K}^{n}\text{.}$ We how have a *category* of affine varieties and regular maps (we shall drop the
prefix $(k,K)$ from now on). If $A,B$
are the coordinate rings of $U,V$ respectively, then the regular maps
$f:U\to V$ correspond one-to-one to the $k\text{-algebra}$
homomorphisms $\phi :B\to A\text{:}$ if
$u\in B$ (i.e., $u:V\to K$ is regular) then
$u\circ f:U\to K$ is regular and thus we have a mapping
$u\mapsto u\circ f$ of $B$ into $A,$
which of course is a homomorphism. Moreover, this correspondence is *functorial*: if $g:V\to W$
corresponds to $\psi :C\to B$ (where $C$ is the coordinate ring of
$W\text{)}$ then $g\circ f$ corresponds to
$\phi \circ \psi \text{.}$ In this way it appears that the category of affine
$k\text{-varieties}$ is equivalent to the dual of the category of finitely-generated
$k\text{-algebras}$ with no nilpotent elements. In other words, the theory of affine algebraic varieties over $k$
is equivalent to the theory of a rather special class of commutative rings, and one can compile a dictionary for translating statements about affine vatieties
into statements of commutative algebra. Thus, in the hands of the German school of the 1920's and 1930's, algebraic geometry became the study of ideals in
polynomial rings.

Let $V$ be an affine $k\text{-vartety,}$ $A$ its coordinate ring. The elements of
$A$ are functions from $V$ to $K\text{.}$ If $S$ is any subset
of $A,$ let $V\left(S\right)$ denote the set of common zeros of
the functions in $S\text{;}$ then it is easily verified that by taking the $V\left(S\right)$
as closed sets we have a *topology* on $V,$ called the *Zariski topology* (strictly, the
$k\text{-topology).}$ From the topologist's point of view, this is a very bad topology: in general it is not even
${T}_{0}$ (unless $k=K,$ when it is
${T}_{1}$ (but not ${T}_{2}\text{)).}$ If
$x\in V,$ the closure of the set $\left\{x\right\}$ in the
Zariski topology is the intersection of all the closed sets $V\left(S\right)$ which contain
$x\text{:}$ it is what Weil calls the *locus* of $x,$ and its points are the
*specializations* of $x\text{.}$ Thus $y$ is a specialization of $x$ if and
only if $y\in \left\{\stackrel{\u203e}{x}\right\}\text{.}$

If $U\subseteq {K}^{m},$
$V\subseteq {K}^{n}$ are two affine varieties then
$U\times V\subseteq {K}^{m+n}$ is an affine variety,
the *product* of $U$ and $V$ (it is the product of $U$ and $V$ in the category of all
affine $k\text{-varieties,}$ that is to say it satisfies the usual universal mapping property in this category). If
$A,B$ are the coordinate rings of $U,V$ respectively then one
might hope that the coordinate ring of $U\times V$ would be the tensor product
$A{\otimes}_{k}B\text{.}$ Unfortunately it isn't, in general, because
$A{\otimes}_{k}B$ may well have nilpotent elements (unless $k$ is perfect),
and to get the coordinate ring of $U\times V$ one has to factor out the ideal of nilpotent elements in the tensor
product. This is one example where the exclusion of nilpotent elements leads to an unsatisfactory situation.

It should also be remarked that the Zariski topology on $U\times V$ is *not* (in general) the product
topology: generally it is strictly finer than the product topology, i.e. it has more open sets. The standard example is the affine plane
$K\times K\text{.}$

It was realised early on that affine geometry is in many respects unsatisfactory. For example, two subvarieties of an affine variety may have empty intersection even
if their dimensions are right, and Bézout's theorem does not hold without qualification; or a point or subvariety may escape 'to infinity'. This was rectified
by 'completing' affine space by sticking on suitable 'points at infinity', as everyone knows, and the result is projective space
${P}_{n}\left(K\right)\text{.}$ From a geometrical point of view, projective
space and projective varieties are much more satisfactory to deal with. The process outlined above of constructing coordinate rings etc. can be imitated in
the projective case, but it doesn't work nearly as well. A projective variety $V$ in
${P}_{n}\left(K\right)$ is given by a set of homogeneous polynomial equations
${f}_{\alpha}({x}_{0},{x}_{1},\dots ,{x}_{n})=0$
(with coefficients in $k\text{);}$ these generate a *homogeneous* ideal $a$ in the *graded*
polynomial ring $k[{t}_{0},\dots ,{t}_{n}]\text{.}$
The radical ${a}^{*}$ of $a$ is again a homogeneous ideal, so we can form
$A=k\left[t\right]/{a}^{*}$ which is a
*graded* $k\text{-algebra.}$ But: (i) the elements of $A$ do not correspond to regular functions on
$V,$ because the only everywhere-defined regular functions on $V$ are in fact constants; and (ii)
there is no longer a one-to-one correspondence (as in the affine case) between graded coordinate rings and projective varieties: non-isomorphic rings can give rise to
isomorphic varieties. For example, the coordinate ring of ${P}_{1}\left(K\right)$ and of a
conic in ${P}_{2}\left(K\right)$ are not isomorphic.

A different approach is the following. ${P}_{n}\left(K\right)$ can be regarded as the
union of a finite number of overlapping affine spaces - for example, the complements of $n+1$ hyperplanes with no common point
- which are open sets in the Zariski topology, and hence any projective variety $V$ is the union of a finite number of overlapping affine varieties
${U}_{i},$ which are open sets in $V\text{:}$ thus
$V$ is 'locally affine'. The situation is analogous to that for a manifold, which is 'locally Euclidean', i,e., is obtained by sticking together
overlapping Euclidean spaces in a suitable way. Thus it is natural to go further, as Weil did, and define an 'abstract variety' as one which is obtained by pasting
together overlapping affine varieties. The resulting object mayor may not be projective (i.e., embeddable in a projective space). The characteristic 'good'
property of projective varieties, that they are in some sense 'compact' or that they don't have bits missing at infinity, is then replaced by the property of
*completeness*, which can be formulated in various ways. Probably the simplest of these is the following: an (abstract) variety $V$ is
*complete* if, for every variety $W,$ the projection
$V\times W\to W$ is a closed map (with respect to the Zariski topology).

To give meaning to the definition of an abstract variety, it is necessary to specify how the affine varieties which make it up are to be stuck together. There are
various ways of doing this: one is the following. If $V$ is an affine variety, say
$V\subseteq {K}^{n},$ we associate with $V$ a
*structure sheaf* ${\mathcal{O}}_{V},$ which may be defined as follows. A *rational* function
$u\in k({t}_{1},\dots ,{t}_{n})$
said to be *regular* at $x\in {K}^{n},$ or *defined* at
$x,$ if u can be put in the form $f/g,$ where
$f,g$ are polynomials and $g\left(x\right)\ne 0$
(so that $u\left(x\right)=f\left(x\right)/g\left(x\right)$
is well-defined). The domain of definition of a rational function is an open set in ${K}^{n}\text{.}$ A
*rational function* $u\prime $ *on* $V$ is by definition the restriction to
$V$ of a rational function $u$ on ${K}^{n}$ (so the domain of
$u\prime $ is an open set in $V\text{).}$ If $U$
is any open set in $V,$ the rational functions on $V$ which are defined at every point of $U$
form a ring $A\left(U\right),$ and the assignment
$U\mapsto A\left(U\right)$ is a presheaf of rings on $V$ which is
immediately verified to be a *sheaf*. This is the structure sheaf ${\mathcal{O}}_{V},$ and it is,
intrinsically related to $V,$ i.e., it does not depend on the embedding of $V$ in an affine space.
One then defines a *prealgebraic variety* to be a topological space $X$ together with a sheaf of rings
${\mathcal{O}}_{X},$ this sheaf being a sheaf of germs of functions on $X$ with values in
$K,$ with the following property: there exists a finite open covering
${\left({V}_{i}\right)}_{1\le i\le n}$
of $X$ such that each ${V}_{i},$ together with the restriction of
${\mathcal{O}}_{X}$ to ${V}_{i},$ is isomorphic, sheaf and all, to an affine
algebraic variety. $X$ is an (abstract) *algebraic variety* if in addition it satisfies a 'separation axiom' which is the formal analogue
of Hausdorff's axiom for topological spaces, namely that the diagonal should be a closed subset of the product $X\times X$
(only here, as we have already seen, the topology on $X\times X$ is not the product topology).

This definition is due to Serre (Faisceaux algébriques cohérents). Thus the philosophy is this: an affine variety is equivalent to a commutative ring (of a rather restricted type) and an abstract variety is obtained by sticking a number of these together by means of their structure sheaves.

We have now more or less set the stage. Going back for a moment to the affine case, we have remarked that any situation or theorem relating to affine varieties can
be transcribed into one relating to their coordinate rings, and it has been recognised for a long time that in this way one gets more general statements, for
generally the theorems of commutative algebra that arise are valid under much less restrictive hypotheses on the rings in question: often it is enough that
they should be Noetherian. So, to obtain a satisfactorily general theory, one should start with a quite arbitrary commutative ring and construct something like an
'affine variety' from it, and then stick these objects together by means of structure sheaves to obtain generalised abstract varieties or *preschemes*.

This is a typed excerpt of the book "Algebraic Geometry: Introduction to Schemes - I.G. Macdonald".