8.4 Relativisation and categorical constructions

Now that we have constructed morphisms it follows that quasi-projective schemes form a category. One standard construction is that of the slash category associated with an object S which is denoted by /S. The objects in this category are morphisms X$\to$S. The morphisms are commutative diagrams

$$\arraycolsep$$=0pt$$\begin{array}{ccc} X\hphantom{\to\to} & {\to} &\hphantom{\to\to} Y \\ \searrow&&\swarrow\\ & S \end{array}$$

The products in this category are provided by fibre products. Geometrically, we conceptualise the objects X$\to$S as families of spaces parametrised by S. Note that there is a natural and unique morphism from any scheme X to the scheme $\mathbb {1}$ (which we have called a point or Spec($\mathbb {Z}$) or $\mathbb {A}$⁰ or $\mathbb {P}$⁰ above in different places!). Thus schemes are in fact naturally parametrised by Spec($\mathbb {Z}$).

For any morphism T$\to$S we can ``re-parametrise'' or perform base change by associating X×_ST$\to$T with X$\to$S. One checks that this gives a functor from the slash category /S to the slash category /T.

For example, let N be any integer and consider the rings $\mathbb {Z}$/N$\mathbb {Z}$ and $\mathbb {Z}$[1/N]. The schemes over Spec($\mathbb {Z}$/N$\mathbb {Z}$) are the schemes ``modulo N''. The schemes over Spec($\mathbb {Z}$[1/N]) are schemes ``outside N''. In particular, we can take N = p a prime to get schemes over Spec($\mathbb {F}$_p) or schemes of characteristic p. We occasionally see statements like ``the following is true outside characteristic 2 and 3''; this can be interpreted as a statement about schemes over Spec($\mathbb {Z}$[1/6]).

For many algebraic object that can be defined diagram-theoretically, there are associated types of objects in the category /S. For example we can define a group as a set G with maps $\mu$ : G×G$\to$G for multiplication, $\iota$ : G$\to$G for inverse and e : 1$\to$G which maps the singleton set to the identity element of G. These satisfy various commutative diagrams which ensure that multiplication is associative, the product of an element and its inverse is identity and the identity multiplied with anything is identity.

$$\arraycolsep$$=0pt$$\begin{array}{ccc} G\times G\times G & \xrightarrow{1\times\mu} ... ...\mu} \\ G\times G\times G & \xrightarrow{1\times\mu} & G\times G \end{array}$$        $$\begin{array}{ccccc} G & \xrightarrow{\Delta} & G\times G & \xri... ...e\mu}\downarrow{\scriptstyle\mu} \\ & & 1 & \xrightarrow{e} & G \end{array}$$        $$\begin{array}{ccc} G & \xrightarrow{1\times e} & G\times G \\ ... ...& \hphantom{\scriptstyle\mu}\downarrow{\scriptstyle\mu}\\ & & G \end{array}$$

Thus we can define a group scheme over S as a morphism G$\to$S with morphisms in /S; $\mu$ : G×_SG$\to$G and $\iota$ : G$\to$G and e : S$\to$G which satisfy the same commutative diagrams. One example is the scheme $\mathbb {G}$_m = Spec($\mathbb {Z}$[X, Y]/(XY - 1)) which is called the multiplicative group of units since it associates to every finite ring A the group of units in A.

Similarly a ring R is a set with maps $\mu$ : R×R$\to$R for multiplication, $\alpha$ : R×R$\to$R for addition, - : R$\to$R for negation, 0 : 1$\to$R for the zero element and 1 : 1$\to$R for the multiplicative identity. The various laws of associativity, distributivity, commutativity (of addition) and additive and multiplicative identity can again be formulated in terms of commutative diagrams. We can use such diagrams to define the notion of a ring scheme. One important example is that of $\mathbb {G}$_a = Spec($\mathbb {Z}$[X]) called the additive group or the structure ring, since it associates to each finite ring A the ring A itself with its natural structure.

We can similarly define the notion of group scheme actions on a scheme and modules schemes over a ring scheme. One important example is that of vector space schemes, which are group schemes that are also modules over the ring scheme $\mathbb {G}$_a. These are so called because, if V$\to$S is a vector space scheme over S and k is a finite field, then the collection of all elements of V(k) that map to a fixed element in S(k) acquire the natural structure of a vector space over k. We can form a natural vector space scheme out of $\mathbb {A}$^q; we denote this scheme by $\mathbb {V}$^q. Clearly, $\mathbb {V}$^q×S$\to$S is a vector space scheme over S for any S. Another example of a vector space scheme the scheme T_S considered above. This is called the (Zariski) Tangent scheme of S.

Some other important examples of vector space schemes are as follows. Let H = V(0;X₀, X₁,..., X_p) be the complement of the point (0 : ^... : 0 : 1) in $\mathbb {P}$^{p + 1}. The projection way from this point gives a morphism H$\to$$\mathbb {P}$^p. This is a vector space scheme with ``zero section'' given by $\mathbb {P}$^p$\to$H which maps (a₀ : ^... : a_p) to (a₀ : ^... : a_p : 0). For any i between 0 and p we have sections $\mathbb {P}$^p$\to$H given by sending a₀ : ^... : a_q) to (a₀ : ^... : a_q : a_i). Considering the set $\mathbb {P}$^p(A) as equivalence classes of surjective A-module homomorphisms A^{p + 1}$\to$A, it is clear that the kernel of this homomorphism is independent of the chosen representative of the equivalence class. This defines a sub-vector space scheme of $\mathbb {V}$^{p + 1}×$\mathbb {P}$^p$\to$$\mathbb {P}$^p. Another vector space scheme over $\mathbb {P}$^p consists of the subscheme of $\mathbb {V}$^{p + 1}×$\mathbb {P}$^p which is defined by V_iX_j = V_jX_i; this vector space scheme is denoted L.

If V$\to$S is a vector space scheme then for any morphism T$\to$S it is clear that V×_ST$\to$T is one as well. In particular, vector space schemes can be restricted to subschemes. The restriction of the vector group scheme denoted H over $\mathbb {P}$^{$\binom{p+d}{d}$ - 1} to the Veronese embedding of $\mathbb {P}$^p is denoted H_d$\to$$\mathbb {P}$^p.