Hi! My name is Matt DeLand, I’m a graduate student at Columbia and I’m responding to Charles’ call for cobloggers. I also study Algebraic Geometry, and have been enjoying Charles’ posts; hopefully I can help out and make some positive contributions. I apologize in advance for the quality of my first post….

Introductions aside, it’s time for some math! In previous posts, the importance of moduli spaces in algebraic geometry has already been underlined (Representable Functors, Grassmannians, Hilbert Schemes, Hom Schemes, Families of Cartier Divisors, etc.). There’s even already a post-series outlining how to take quotients of varieties by finite (in fact reductive) algebraic groups. I’ll start out slowly here toward the goal of explaining what this means.

First, a group variety has already been defined in this series. Since the first applications we have in mind are geometric, we’ll work over a fixed algebraically closed field \text{Spec } k. All schemes, varieties, and morphisms should be assumed to be over \text{Spec } k. At some point we reserve the right to assume that the characteristic is 0. Recall that a group scheme is a scheme G with morphisms m: G \times G \rightarrow G, i: G \rightarrow G, and e: \text{Spec } k \rightarrow G which satisfy the usual relationships for groups. By an Affine Group Scheme, we will mean that G is isomorphic to an affine scheme \text{Spec } A. This will imply that A has a Hopf Algebra structure, but we won’t focus on that for now. For the purposes of this post, all group schemes will be affine. The theory of complete group varieties (also known as Abelian Varieties) is a topic for another day(s).

Examples: The following are the classic examples of Group Schemes that we should all be familiar with.

1. G = \text{Spec } A where A = k[t] and the defining morphism are given by: m : G \times G \rightarrow G corresponds to t \rightarrow t \otimes 1 + 1 \otimes t, i: G \rightarrow G corresponds to t \rightarrow -t, and e: \text{Spec } k \rightarrow G corresponds to t \rightarrow 0. This group scheme (in fact variety) is denoted \mathbb{G}_a(k) but we’ll leave out the field notation from here on out. Notice that the functor of points of G defined by h_G : \text{Affine k-Schemes} \rightarrow \text{Groups} (note: for any group scheme G, the functor h_G actually takes values in the category of Groups rather than Sets because of the axioms) defined by Y \rightarrow Hom(Y,G) sends \text{Spec } R to the ring R as an additive group.

2. G = Spec A where A = k[t, t^{-1}] and the morphism are given by: m : G \times G \rightarrow G corresponds to t \rightarrow t \otimes t, i: G \rightarrow G corresponds to t \rightarrow t^{-1} and e: \text{Spec } k \rightarrow G corresponds to t \rightarrow 1. This group variety is denoted \mathbb{G}_m. With the notation as above, h_G(\text{Spec } R) = R^\times, the group of invertible elements of R, as a group under multiplication.

3. G = Spec A where A = k[t]/(t^n - 1). The morphisms are the same as in Example 2 above, except the map i corresponds to t \rightarrow t^{n-1}. For obvious reasons, this scheme is called the group of n-th roots of unity, and is denoted \mu_n. Notice that if n is a multiple of the characteristic of the field, then we have encountered our first group scheme which is not a group variety.

4. We leave it as an exercise to work out other standard group schemes GL_n, SL_n, O_n, SO_n, SP_{2n} and any others that motivate you.

Definition A morphism of group schemes f : G \rightarrow H is a morphism of schemes which is also a homomorphism of groups.

As an example, there is an exact sequence of groups schemes 0 \rightarrow \mu_n \rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m \rightarrow 0, where the second arrow is given by t \rightarrow t^n.

Suppose that V is a finite type scheme, then we define what it means for a group scheme G to act on V.

Definition Suppose G is a group scheme and V is a scheme. An action of G on V is given by a map \rho: G \times V \rightarrow V such that \rho \circ (Id \otimes e) : V \cong V \times pt \rightarrow V \times G \rightarrow V is the identiy and such that the two maps \rho \circ (Id \times m), \rho \circ (\rho \times Id) : V \times G \times G \rightarrow V are equal. This definition simply encodes the usual definition for a group acting on a set.

At the level of rings, there is a dual notion:

Definition: A representation of a group scheme G = Spec A is a k-vector space (not necessarily finite dimensional) V along with a linear map \mu : V \rightarrow V \otimes A which satisfy the dual relations for those of an action. A vector v \in V is called invariant if \mu(v) = v \otimes 1 and a subspace U \subset V is called a subrepresentation if \mu(U) \subset U \otimes A.

Here are some Examples of group actions on V = \mathbb{A}^2: (In each case we leave it to the reader to check that we actually have a group action). If I could draw pictures here I would…

1. G = \mathbb{G}_m acts on V by the map (t, (x,y)) \rightarrow (tx, ty).

2. G= \mathbb{G}_m acts on V in another way by the map (t, (x,y)) \rightarrow (tx, t^{-1}y).

3. G = \mathbb{G}_a acts on V by the map (s, (x,y)) \rightarrow (x + sy, y).

4. G = \mathbb{G}_m \times \mathbb{G}_a acts on V by the map ((t,s), (x,y)) \rightarrow (x + sy, ty).

5. G = \mathbb{G}_m \times \mathbb{G}+a acts on V in another way by the map ((t,s), (x,y)) \rightarrow (tx + uy, t^{-1}y).

6. Suppose for simplicity that k is algebraically closed and that the characteristic is prime to n. Fix a primitive n-th root of unity, z. Then G = \mu_n acts on V by sending (a, (x,y)) \rightarrow (z^ax, z^ay).

Now we can ask what it should mean to take the quotient of V by a group action G. The original scheme should certainly map to the quotient, which we will call Y. In the best case scenario, the points of Y will correspond to orbits of the action. However, if f: V \rightarrow Y is a quotient, then fibers over closed points are closed in V. If there are non-closed orbits then, it can’t be the case that points of Y correspond to orbits uniquely.

Let’s analyze the oribts in the above examples. In Example 1, the origin is an orbit, and all lines through the origin (not including the origin) are also orbits. In Example 2, the origin is an oribt, as are all hyperbolas xy = a, as are the x-axis and the y-axis if we leave out the origin. In Example 3, the orbits are all closed, they are “horizontal lines” andall (isolated) points on the x-axis. In Example 4, there is an open orbit which is the complement of the x-axis, and then isolated points on the x-axis. In Example 5, there is the origin, the x-axis minus the origin, and the plane minus the x-axis. Finally in Example 6, the orbits are collections of n points except for the origin which is its own orbit. Even with relatively simple group actions we’ve run into non-closed orbits.

The ideas involved in taking a quotient are made clear by looking at the affine case. Suppose then that G = \text{Spec } A and V = \text{Spec} R. Let R^G = \{ f \in R | \mu(f) = f \otimes 1 \} is the set (actually subalgebra) of G invariants for the action. Consider the map V \rightarrow \mathbb{A}^n given by some G invariant functions f_1, \ldots, f_n \in R^G. We have an induced (surjective) map V \rightarrow W = \text{Spec } R^G corresponding to the inclusion. Since each f_i is an invariant, this map is constant on G orbits of the action, that is, it sends each orbit to a point. We can ask then when do distinct orbits map to distinct points? When the algebra R^G is finitely generated, we can take a set of generators to define the map, and hope that the image is a variety and is the quotient we want. We’ll see in the future that this is always the case when the action is nice (see below).

For the technical definition , we’ll follow Mumford:

Definition Suppose a group scheme G acts on a k-scheme V. We say that f: V \rightarrow Y is a geometric quotient if:
i) f \circ \rho = f \circ p_2 (p_2 is the second projection on G \times V).
ii) The map f is surjective and the map (f, p_2) : G \times V \rightarrow V \times V has image V \times_Y V.
iii) The map f is submersive.
iv) The sheaf \mathcal{O}_Y is the subsheaf of f_*(\mathcal{O}_V) consisting of invariant funtions. Said another way, if h \in f_* (\mathcal{O}_V)(U)) = \mathcal{O}_V(F^{-1}(U) then h \in \mathcal{O}_Y(U) if and only if the two maps H \circ \rho, H \circ p_2 : X \times f^{-1}(U) \rightarrow \mathbb{A}^1 are equal. Here H is the map determined by h.

It’s a mouthful, and vaguely translated, condition i) is the property the morphism contracts orbits, condition ii) is the property that fibers over closed points correspond to orbits (see the discussion below), and all the conditions together¬†assure that it is the “smallest” variety (that is, satisfying a universal property) that has properties i) and iv).

Now of course, the question becomes, when do geometric quotients exist? The answer will be when the group is reductive (a notion that we won’t define until next time). In fact, when V = \text{Spec } R is an affine variety and the action is nice, the coordinate ring of the quotient will be exactly R^G. Let’s analyze the above examples.

In Example 1, the affine quotient is a single point (the only invariant functions are constant)! This will be fixed in the future when we remove the origin and we’ll see that the quotient is \mathbb{P}^1 as expected. In Example 2, the quotient is \mathbb{A}^1 = \text{Spec } (k[xy]). Notice that the non-closed orbits fail to be separated by the quotient map. In Example 3, the quotient is \mathbb{A}^1 = \text{Spec } (k[x]). Here, even closed orbits (the points) are not separated, indeed the group \mathbb{G}_a is not reductive. We’ll leave the rest as exercises, it should be similar.

Since we haven’t covered any theory at all, and since actions of \mathbb{G}_m and \mathbb{G}_a are ubiquitous (though mostly the former), we’ll discuss such actions slightly more here.

In the special case when G = \mathbb{G}_m, the representations are particularly simple. Given V and an integer a, consider the map V \rightarrow V \otimes k[t,t^{-1}] which sends v \mapsto v \otimes t^a. This is called a representation of weight a.

Proposition: For each representation V of \mathbb{G}_m, there is a direct sum decomposition V = \bigoplus V_m where each V_m is a subrepresentation of weight m.

Proof: Define V_m = \{ v \in V | \mu(v) = v \otimes t^m \}. This is a subrepresentation of weight m. To verify the direct sum decomposition, for an arbitrary vectory write \mu (v) = \Sigma v_m \otimes t^m \in V \otimes k[t,t^{-1}] (this sum will be finite). By property i) of an action, we’ll have v = \Sigma v_m so we just must verify that each v_m \in V_m. By property ii) of an action though, we have that \Sigma \mu (v_m) t^m = \Sigma v_m \otimes t^m \otimes t^m \in V \otimes k[t,t^{-1}] \otimes k[t,t^{-1}]. By linear independence of the t^m, we must have then that each v_m \in V_m.

From this we see that to give a \mathbb{G}_m action on \text{Spec } R is equivalent to specifying a grading decomposition R = \bigoplus R_m. The invariants of the action correspond to elements of weight 0.

In characteristic 0, something similar for an action of \mathbb{G}_a is true:

Proposition: Every representation V of \mathbb{G}_a is given by \mu (v) = \Sigma f^n (v) \otimes \frac{ t^n}{n!} (sum taken over non-negative integers) for some f \in End(V) which is locally nilpotent (that is, every vector is eventually killed).

We’ll leave this proof as an exercise, it’s not much harder than the previous one and isn’t used as often. Hint : Consider the linear maps h_n defined by \mu(v) = \Sigma h_n(v) \otimes s^n \in V \otimes k[s].

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