Let
*L* = *V*(*F*_{1},..., *F*_{n};*G*_{1},..., *G*_{m}) be a quasi-projective scheme in
^{p} and
*K* = *V*(*D*_{1},..., *D*_{k};*E*_{1},..., *E*_{l}) be a quasi-projective
scheme in
^{q}. As before we can and do assume that the
collections {*D*_{i}}, {*E*_{i}}, {*F*_{i}} and {*G*_{i}} have
constant degrees. Let *X*_{i}'s be the *p* + 1 variables for
^{p} and
*Y*_{j} be the *q* + 1 variables for
^{q}. If *d*_{1} is the degree of
the *D*_{t}'s then the bi-homogeneous polynomials of the form
*D*_{t}^{ . }*M* where *M* is a monomial of degree *d*_{1} in the variables *X*_{i} can
be written as polynomials in the variables
*Z*_{ij} = *X*_{i}*Y*_{j} (by
choosing some arbitrary pairing of *X*'s with *Y*'s for each term).
Let
{} denote the resulting collection of polynomials in
*Z*_{ij} as *M* varies over all possible monomials in the *X*'s and
*F*_{t}'s vary. We have similar collections
{},
{} and
{}. One then checks quite
easily that
*L*(*A*)×*K*(*A*) is the subset of
^{pq + p + q}(*A*)
defined by the conditions:

- The equations
*Z*_{ij}*Z*_{kl}-*Z*_{il}*Z*_{kj}= 0 hold. - All the 's and the 's vanish.
- The evaluation of the collection
{} results in a tuple that generates
the ring
*A*.

Thus, when *X* and *Y* are quasi-projective schemes, then so is
*X*×*Y*. Hence, for a sub-functor *Z* of *X*×*Y* it makes sense
say that it is a subscheme; or more specifically a closed or open
subscheme. In particular, if *W* is a subscheme (resp. closed or open
subscheme) of *Y*, we see that *X*×*W* is a subscheme (resp.
closed or open subscheme) of *X*×*Y*. Similarly, for subschemes of
*X*. Another useful closed subscheme is
*X*×*X*,
the diagonal subscheme, which is defined by intersecting *X*×*X*
with the diagonal subscheme of
^{q}×^{q} when *X* is given
a a subscheme of
^{q}.

A *correspondence* from *X* to *Y* is a closed subscheme of
*X*×*Y*. For any natural transformation *f* : *X**Y* the graph
is the subfunctor of *X*×*Y* which gives for each
finite ring *A* the graph of
*f* (*A*) : *X*(*A*)*Y*(*A*). We say that *f* is a
*morphism* if is a closed subscheme of *X*×*Y*. In
other words, a morphism is a natural transformation which is also a
correspondence. Alternatively, if
*Z* *X*×*Y* is a
correspondence so that the projection
*Z*(*A*)*X*(*A*) is a bijection
for all finite rings *A*, then *Z* is the graph of a morphism.

Now it follows easily that the identity natural transformation *X**X* is a morphism with the diagonal as the associated correspondence.
Moreover, each of the projections
*X*×*Y**X* and
*X*×*Y**Y* is a morphism. It is also clear that if
*W* *X* is a subscheme
then the intersection of *W*×*Y* with gives the graph
of the restriction of *f* : *X**Y* to *W*; as a result this restriction
is also a morphism. If
*Z* *X*×*Y* is the graph of a morphism
then the projection *Z**X* is a morphism; its graph in
*Z*×*X* *X*×*Y*×*X* is the intersection of the diagonal of the
extreme terms (consisting of (*x*, *y*, *x*)) with *Z*×*X*. The map
*Z*(*A*)*X*(*A*) is a bijection; let *g* : *X**Z* be the inverse natural
transformation. The graph of *g* in
*X*×*Z* *X*×*X*×*Y* is the intersection of
×*Y* with *X*×*Z*. Thus
*g* is also a morphism. In other words, there are morphisms *Z**X*
and *X**Z* with composition either way being identity. Thus *Z**X*
is an *isomorphism*.

Now, let *f* : *X**Y* be a morphism and *g* : *Y**Z* be another
morphism. Let *W* be the intersection of
×*Z* with
*X*× in
*X*×*Y*×*Z*. Under the above
isomorphism
*X*, we can identify *W* as a subscheme of
*X*×*Z*. It clear that *W*(*A*) is the graph of the composite
natural transformation *g*`o`*f*. Thus, morphisms can be *composed*.

Let *f* : *X**Y* be a morphism and
*W* *Y* be a subscheme. Then, we
have a subscheme of given by its intersection with *X*×*W*. Since
*X* is an isomorphism, we obtain a subscheme of
*X* as well; this scheme is usually denoted *f*^{-1}(*W*) and called the
inverse image of *W* under *f*. In some cases it may happen that
is *contained* in *X*×*W* so that
*f*^{-1}(*W*) = *X*. In
this case we say that the morphism *f* factors through or lands inside
*W*.

The theorem of Chevalley asserts that there is a smallest subscheme
*W* of *Y* so that *f* factors through *W*; we can refer to *W* as the
*categorical image* of *f*. Note that it may not be true that
*W*(*A*) is the image of *X*(*A*) in *Y*(*A*) even for *one* non-zero
finite ring *A*.

Given morphisms *X**W* and *X**Z* we easily check that the natural
transformation
*X**W*×*Z* is a morphism. Given morphisms *X**S* and *Y**S*, we obtain the compositions
*a* : *X*×*Y**X**S* and
*b* : *X*×*Y**Y**S*. Thus we a morphism
*X*×*Y**S*×*S*.
The inverse image of the diagonal is denoted
*X*×_{S}*Y* and is
called the *fibre product* of *X* and *Y* over *S*. For any
morphisms
*Z**X*×*Y* such that the resulting composites with *a*
and *b* are equal, we see that the morphism actually lands in the
subscheme
*X*×_{S}*Y*.

One important example of a correspondence is the subscheme *Z* of
^{p + q}×^{q} defined by the conditions
*X*_{i}*Y*_{j} = *X*_{j}*Y*_{i} for
0 *i*, *j* *q*. Let *U* be the open subscheme of
^{p + q} given
by
*U* = *V*(0;*X*_{0}, *X*_{1},..., *X*_{q}). For
(*a*_{0} : ^{ ... } : *a*_{p + q}) in *U*(*A*),
the tuple
(*a*_{0},..., *a*_{q}) generates the ring *A*, thus we see that
we see that
((*a*_{0} : ^{ ... } : *a*_{p + q}),(*a*_{0} : ^{ ... } : *a*_{q})) gives an element
of
^{p + q}(*A*)×^{q}(*A*) which clearly lies in *Z*(*A*).
Conversely, if
((*a*_{0} : ^{ ... } : *a*_{p + q}),(*b*_{0} : ^{ ... } : *b*_{q})) lies in
*Z*(*A*) *and*
(*a*_{0},..., *a*_{q}) generate the ring *A*, then the
above equations show that there is a unit *u* in *A* so that
*b*_{i} = *ua*_{i} (apply the Chinese Remainder theorem for finite rings!).
Thus, the projection
*Z*(*A*)^{p + q}(*A*) is a bijection over *U*(*A*)
and gives a morphism
*U*^{q}. This morphism is called the * projection* on
^{p + q} *away* from the linear subscheme (or
subspace!)
*V*(*X*_{0},..., *X*_{q}); more generally, we can refer to the
above correspondence as the projection correspondence.

A natural generalisation of this is to consider a collection
*F*_{0},...,*F*_{q} of homogeneous polynomials of the same degree in
variables *X*_{0},...,*X*_{p}; we can then take the subscheme *Z* of
^{p}×^{q} defined by the equations

For any functor *F* on the category of finite rings we can introduce a
new functor *T*_{F} which associates to a finite ring *A* the set
*F*(*A*[]) where
*A*[] denotes the finite ring
*A*[*T*]/(*T*^{2}). The morphism
*A*[]*A* that sends to
induces a natural transformation of functors *T*_{F}*F*. Now, if
*F* = ^{p} is the projective space then
*T*_{p}(*A*) consists of
equivalence classes of *p* + 1-tuples

(*a*_{0} + *b*_{0},..., *a*_{p} + *b*_{p}) (*ua*_{0} + (*a*_{0}*b* + *ub*_{0}),..., *ua*_{p} + (*a*_{p}*b* + *ub*_{0})

where s_{ij}s_{kl} - s_{ik}s_{jl} |
= | 0 | (1) |

t_{ij}s_{kl} + t_{jk}s_{il} + t_{ki}s_{jl} |
= | 0 | (2) |

Moreover, the entries