April 3,
2023

References: [ GW1 ] Sections (4.1), (4.2); [ Mu ] II.6.

As we have discussed in Algebraic Geometry 1, to a scheme \(X\) we can attach its functor of \(T\)-valued points:

which on morphisms is just given by composition: \(f\colon T'\to T\) is mapped under \(h_X\) to the map \(X(T)\to X(T')\), \(\alpha \mapsto \alpha \circ f\).

Using the notion of a morphism of functors, we can speak of the category \(\widehat{{\mathscr C}} := \operatorname{Func}(({\rm Sch})^{{\rm opp}}, {\rm (Sets)})\) of all such functors, and we obtain a functor \(h\colon ({\rm Sch}) \to \widehat{{\mathscr C}}\), \(X\mapsto h_X\), which on morphisms is – once again – defined by composition: \(\alpha \colon X'\to X\) is mapped by \(h\) to the morphism \(h_{X'}\to h_X\) of functors given by \(X'(T)\to X(T)\), \(\beta \mapsto \beta \circ \alpha \).

Even though at first sight this may look complicated, this is an entirely “formal” (i.e., category-theoretic) procedure which has nothing to do with schemes. In fact, if \({\mathscr C}\) is any category, for an object \(X\) of \({\mathscr C}\) we can define the functor

and now setting \(\widehat{{\mathscr C}} := \operatorname{Func}({\mathscr C}^{{\rm opp}}, {\rm (Sets)})\), we obtain a functor \(h\colon {\mathscr C}\to \widehat{{\mathscr C}}\).

We will mostly apply the Yoneda Lemma to the category of schemes, or the category of \(S\)-schemes for some fixed scheme \(S\). Let us list some of its consequences (as before, these facts are not specific to the category of schemes):

1. Let \(X\), \(Y\) be schemes. The following are equivalent:

   1. \(X\cong Y\),

   2. \(h_X \cong h_Y\) (isomorphism of functors)

   3. there exists a family \(f_T\colon X(T)\to Y(T)\) of bijections of sets that is functorial in \(T\), i.e., for every scheme morphism \(T'\to T\), the diagram

      

      is commutative.

2. Let \(X\), \(Y\) be schemes. Giving a scheme morphism \(X\to Y\) is equivalent to giving a family of maps \(f_T\colon X(T)\to Y(T)\) of sets for each scheme \(T\), that is functorial in \(T\) (same condition as in (1) (iii)). Example. The determinant of a matrix is a scheme morphism \(\mathbb A^{n^2}\to \mathbb A^1\).

3. A diagram of scheme morphisms is commutative if and only if for every scheme \(T\) the diagram (in the category of sets) obtained by replacing each scheme by its set of \(T\)-valued points, and replacing the scheme morphisms by the induced maps of sets, is commutative.

April 5,
2023

References: [ GW1 ] Sections (4.4)–(4.6); [ H ] II.3; [ Mu ] II.2.

The universal property of a fiber product generalizes the universal property of a product (of two objects, in any category). It is defined as follows. (See the lecture notes for Algebraic Geometry 1 for a bit more background, e.g., an explanation of the term fiber product.)

We say that a commutative square

is cartesian, if it is a fiber product diagram, i.e., if \(A\) satisfies the universal property defining the fiber product of \(B\) and \(C\) over \(D\).

April 12,
2023

References: [ GW1 ] Chapter 4, in particular Sections (4.7)–(4.10).

Given scheme morphisms \(f\colon X\to S\) and \(g\colon S'\to S\), we call the projection \(X\times _SS'\to S'\) the morphism obtained from \(f\) by base change along \(g\). This defines a functor from the category of \(S\)-schemes to the category of \(S'\)-schemes.

A particularly simple example is the case where \(g\colon V\to S\) is an open immersion. In that case the base change of \(f\) is just the restriction of \(f\) to \(f^{-1}(V)\to V\).

Many properties of scheme morphisms are “stable under base change” in the following sense: A property \(\mathbf P\) of scheme morphisms is called stable under base change if for every morphism \(f\colon X\to Y\) of \(S\)-schemes that has property \(\mathbf P\) and every scheme morphism \(S'\to S\), the induced morphism \(X\times _SS'\to Y\times _SS'\) also has property \(\mathbf P\).

Given a property \(\mathbf P\), to check that it is stable under base change, it is enough to check that whenever \(f\colon X\to S\) has the property, and \(g\colon S'\to S\) is a scheme morphism, then \(X\times _SS'\to S'\) also has the property. In fact, this is clearly a special case of the above definition (namely the case where \(Y=S\)). On the other hand, suppose this special case is true and \(f\colon X\to Y\) is any morphism of \(S\)-schemes. Identifying \(X\times _SS' = X\times _Y(Y\times _SS')\) using the rules of “computations with fiber products” (Lemma 1.8), the base change \(X\times _SS'\to Y\times _SS'\) is identified with the projection \(X\times _Y(Y\times _SS')\to Y\times _SS'\). Applying the special case to \(X\to Y\) and the base change \(Y\times _SS'\to Y\), we obtain that \(X\times _SS' = X\times _Y(Y\times _SS')\to Y\times _SS'\) has property \(\mathbf P\).

A notable exception is the property of being injective: Can you find an example of an injective morphism \(X\to S\) of schemes and a morphism \(S'\to S\) such that the base change \(X\times _SS'\to S'\) is not injective?

All the properties in the above list, and also being injective, are stable under composition, i.e., if two composable morphisms both have the property, then so does the composition.

References: [ GW1 ] Sections (9.3), (9.4); [ H ] II.4.

Recall the following description of a topological space \(X\) being Hausdorff: \(X\) is Hausdorff if and only if the “diagonal” \(\{ (x,x)\in X\times X;\ x\in X\} \) is a closed subset of \(X\times X\) (with respect to the product topology).

As we have discussed, the underlying topological space of a scheme usually is not Hausdorff. On the other hand, the (fiber) product of schemes usually does not carry the product topology, and it turns out that using the above condition in the context of schemes gives rise to an interesting and useful notion for schemes (which in some sense is a good replacement for the Hausdorff property; see Proposition 1.17 for an example).

April 17,
2023

Proof

The key point is noting that for \(U, V\subseteq X\) open, \(U\times _XV\) can be identified with \(U\cap V\), and that as a consequence the diagram

is cartesian.

The following proposition illustrates (why?) that in some sense the property of being separated resembles the Hausdorff property of topological spaces. (See Algebraic Geometry 1, Problem 52, for an example which shows that the reducedness assumption cannot be dropped.)

References: [ Bou-TG ] Ch. I §10.

Most schemes that we have encountered so far (in particular, all affine schemes, projective space over any ring, subschemes \(V_+(I)\) of projective space over a ring, …) are quasi-compact. On the other hand, from a geometric point of view, e.g., the affine line (or higher-dimensional affine space) “should not be viewed” as a compact space. The notion of properness is a suitable replacement in algebraic geometry for the notion of compactness in topology/differential geometry.

Similarly as separatedness, we will define properness in terms of fiber products of schemes, starting from a characterization of quasi-compact topological spaces, given by the notion of proper map between continuous spaces, which we discuss below as a motivation for the definition of proper scheme morphisms. The purpose of motivation aside, the rest of this section plays no role in the course.

Note that fiber products in the category of topological spaces exist. In fact, for continous maps \(X\to S\), \(Y\to S\), the set-theoretic fiber product \(X\times _SY\), equipped with the subspace topology for the inclusion \(X\times _SY\subseteq X\times Y\) (where the right hand side carries the product topology) is easily seen to satisfy the required universal property.

Recall that a continous map \(f\colon X\times Y\) is closed, if for every closed subset \(C\subseteq X\), the image \(f(C)\subseteq Y\) is closed.

In the remainder of the section, we will discuss the proof of the proposition. It is easy to see that (i) implies (ii). In fact, if \(Z\) is a topological space, then \(X\times Z = X\times _Y(Y\times Z)\), so the map occurring in (ii) can be written as the base change of \(f\) along the projection \(Y\times Z\to Y\).

Conversely, if \(Y\) is Hausdorff, and \(g\colon Z\to Y\) is continuous, then \(X\times _YZ\) is closed in \(X\times Z\). Similarly the map \(Z\to Y\times Z\), \(z\mapsto (g(z), z)\), which up to switching the factors is the graph of \(g\), identifies \(Z\) with a closed subset of \(Y\times Z\). Now let \(C\subseteq X\times _YZ\) be closed, and write \(f_Z\colon X\times _YZ\to Z\) for the base change of \(f\). Then \(f_Z(C)\subseteq Z\), under the inclusion \(Z\hookrightarrow Y\times Z\), is identified with the image of \(C\) under the composition \(X\times _YZ\to X\times Z\to Y\times Z\). This shows that (ii) implies (i), if \(Y\) is Hausdorff.

Denote by \(P\) the topological space consisting of a single point.

In the following lemma, which in a sense is the most interesting part of the proposition, we consider the special case \(Y=P\). In this case, properties (i) and (ii) are equivalent for trivial reasons (since \(P\) is a terminal object in the category of topological spaces, fiber products over \(P\) are just products). Likewise, (iii) and (iv) are equivalent in this case. The lemma proves that, for \(Y=P\), conditions (i) and (ii) imply conditions (iii) and (iv).

It is easy to see that whenever \(f\colon X\to Y\) satisfies (ii) and \(Y'\subseteq Y\) is a subspace, then the restriction \(f^{-1}(Y')\to Y'\) of \(f\) satisfies (ii) as well. Applying this to singletons \(Y'\), from the lemma we obtain the implication (ii) \(\Rightarrow \) (iii) in the proposition.

For the implication (iii) \(\Rightarrow \) (iv), to ease the notation we first note that (replacing \(Y\) by \(K\) and \(X\) by \(f^{-1}(K)\) and observing that (iii) still holds) it is enough to handle the case that \(K=Y\) is quasi-compact. We then have to show that \(X\) is quasi-compact. So let \(X = \bigcup U_i\) be an open cover of \(X\). By quasi-compactness of the fibers of \(f\), for each \(y\) there exists an open \(U_y\subseteq X\) containing \(f^{-1}(y)\) which is a finite union of subsets of the form \(U_i\). Since \(f\) is closed, \(V_y:=Y\setminus f(X\setminus U_y)\) is open in \(Y\). It clearly contains \(y\), so all the \(V_y\) together cover \(Y\). By quasi-compactness of \(Y\), finitely many \(V_y\) suffice to cover \(Y\). But then the corresponding \(U_y\) cover \(X\), and we are done.

Let us show that (iii) implies (ii), so let \(f\colon X\to Y\) as in (iii) be given and let \(Z\) be any topological space. Let \(C\subseteq X\times Z\) be closed. We show that \(W:=(Y\times Z)\setminus (f\times \operatorname{id}_Z)(C)\) is open in \(Y\times Z\). For \((y,z)\in W\), we need to find open neighborhoods \(U\) of \(y\) in \(Y\) and \(V\) of \(z\) in \(Z\) such that \(U\times V \subseteq W\), or in other words: \((f^{-1}(U)\times V)\cap C=\emptyset \). If \(y\) does not lie in the image of \(f\), we can take \(U\) to be any neighborhood of \(y\) which is disjoint from the image of \(f\) (which exists since \(f\) is closed). Otherwise, for each \(x\in f^{-1}(y)\), let \(U_x\) be an open neighborhood of \(x\) in \(X\) and \(V_x\) an open neighborhood of \(z\) in \(Z\) auch that \((U_x\times V_x)\cap C = \emptyset \). Since \(f^{-1}(y)\) is quasi-compact, finitely many of the \(U_x\) suffice to cover this fiber. Let \(U'\) denote their union, and \(V\) the intersection of the corresponding sets \(V_x\). Since \(f\) is closed, \(U:= Y\setminus f(X\setminus U')\) is an open neighborhood of \(y\). One checks that \(U\) and \(V\) constructed in this way have the desired property.

Finally, let us discuss the implication (iv) \(\Rightarrow \) (iii). As we will see, we will have to assume that \(Y\) is Hausdorff and that every \(y\) admits a quasi-compact neighborhood, i.e., there exist an open \(U\) and a quasi-compact \(K\subseteq Y\) such that \(y\in U\subseteq K\). Clearly, conditions (iv) implies that all fibers of \(f\) are quasi-compact (without any assumption on \(Y\)). We need to show that \(f\) is closed.

Let \(C\subseteq X\) be a closed subset. To check that \(f(C)\) is closed, we may work locally on \(Y\). Since property (iv) still holds if we replace \(Y\) by a subspace (and \(X\) by the inverse image of that subspace), and since \(Y\) is locally compact, we may assume without loss of generality that \(Y\) is quasi-compact, and — using our assumption (iv) — that \(X\) is quasi-compact as well. Then \(C\) is also quasi-compact, being closed in a quasi-compact space, and thus the image \(f(C)\) is quasi-compact, as well. Since \(Y\) is Hausdorff and quasi-compact subspaces of a Hausdorff space are necessarily closed, we see that \(f(C)\) is closed in \(Y\), as desired.

References: [ GW1 ] Sections (12.13); [ H ] II.4; [ Mu ] II.7.

April 19,
2023

To define proper morphisms of schemes, we also need the following ingredients. We have already defined quasi-compact morphisms, and a special case of morphisms of finite type in Algebraic Geometry 1.

The difficult implication was Problem 46 in Algebraic Geometry 1. Note that in part (iii) of the lemma it is important to consider a cover by affine open subschemes.

Recall that an algebra \(B\) over a ring \(A\) is called of finite type (or equivalently, finitely generated) if there exists \(n\ge 0\) and a surjective \(A\)-algebra homomorphism \(A[X_1,\dots , X_n]\to B\).

Each of the properties of being locally of finite type, quasi-compact, and of finite type is stable under composition and under base change.

This definition of projective schemes differs slightly from the one in  [ GW1 ] (Definition 13.68, which requires only that the above property holds locally on \(S\)). If \(S\) is affine, they coincide, however, and the difference will not be of any concern for us in this course. See  [ GW1 ] , Summary 13.71 for a discussion. The definition given here is the one used in  [ H ] and in  [ Stacks ] .

Suppose that \(S= \operatorname{Spec}R\) is an affine scheme. For any homogeneous ideal \(I\subseteq R[X_0,\dots , X_N]\), \(V_+(I)\) is a closed subscheme of \(\mathbb {P}^N_S\), and hence in particular a projective \(S\)-scheme. One can show that for \(S\) affine every projective scheme is isomorphic to a scheme of this form.

We will study this notion in more detail later (see Chapter 5).

Before we come to the main theorem of this section (Theorem 1.32), recall that for a homogeneous ideal \(I\subseteq R[X_0,\dots , X_n]\) (where \(R\) is some ring) we have defined a closed subscheme \(V_+(I)\) of \(\mathbb {P}^n_R\). We need the following two results on closed subschemes of projective space.

In addition, we will use the following commutative algebra lemma which is easily proved using the definitions of the localizations appearing in the lemma. (We will later generalize the lemma when we prove that given an \({\mathscr O}_X\)-module \({\mathscr F}\) of finite type on a locally ringed space \(X\), the support of \({\mathscr F}\), i.e., the set of all points \(x\) such that the stalk \({\mathscr F}_x\) does not vanish, is closed. See Proposition 2.18.)

April 24,
2023