Direct Limits. Mathematics 683, Fall 2013


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1 Direct Limits Mathematics 683, Fall 2013 In this note we define direct limits and prove their basic properties. This notion is important in various places in algebra. In particular, in algebraic geometry and complex analysis, the fundamental notion of a stalk of a sheaf uses direct limits. We recall the definition of a direct limit, which we state for Abelian groups. Let I be a set with a partial order satisfying the property that for any i, j I, there is a k I with i k and j k. Such a set is called a directed set. Suppose we have the following data: an Abelian group A i for each i, and for each pair i j a map φ ij : A i A j with φ ii = id Ai for each i, and such that whenever i j k, we have φ ik = φ jk φ ij. Then {A i, φ ij } is called a directed system of groups. The direct limit lim A i is the unique up to isomorphism group L satisfying the following universal mapping property: there are maps φ i : A i L such that φ i = φ j φ ij for every pair i j, and if there is an Abelian group C together with maps τ i : A i C such that τ i = τ j φ ij for each i j, then there is a unique group homomorphism τ : L C with σ = τ φ i. τ i A i C φ i τ 7 L A routine exercise involving universal mapping properties shows that the direct limit of a group, if it exists, is unique up to isomorphism. Direct limits of Abelian groups do exist; here is a construction: Let M be the direct sum of the A i, and let N be the subgroup generated by all elements of the form a φ ij (a) for all i j and all a A i. Then M/N, together with φ i the compositions of the natural maps A i M M/N, satisfy the mapping property for the direct limit. In order to prove the basic properties of direct limits in the lemma below, we give an alternative description of them. This description is often how you may see direct limits used in complex analysis and algebraic geometry. Let {A i, φ ij } be a directed system of groups. Consider pairs (A i, a i ) with a i A i. Define a relation on such pairs by (A i, a i ) (A j, a j ) if there is a k i, j with φ ik (a i ) = φ jk (a j ). A short calculation shows that is an equivalence relation. We will write A i, a i for the equivalence class of a pair (A i, a i ). If G is the set of equivalence classes, then we can define an operation on G by A i, a i + A j, a j = A k, φ ik (a i ) + φ jk (a j ), 1
2 where k is any index with k i, j. Another short calculation shows that this operation is well defined, and that G is an Abelian group under this operation. The map σ i : A i G given by σ i (a) = A i, a is a group homomorphism. Furthermore, σ i = σ j φ ij for any pair i j since σ j (φ ij (a)) = A j, φ ij (a) = A i, a by the definition of the equivalence relation. We will prove that G = lim A i by proving that G has the same mapping property as does the direct limit. To do this, suppose that B is an Abelian group and that for each i there is a homomorphism τ i : A i B with τ i = τ j φ ij for each i j. Define τ : G B by τ ( A i, a ) = τ i (a). This is well defined since if A i, a i = A j, a j, then there is a k with i, j k and φ ik (a i ) = φ jk (a j ). Therefore, τ i (a i ) = τ k (φ ik (a i )) = τ k (φ jk (a j )) = τ j (a j ). The map τ is clearly a group homomorphism. Furthermore, τ i = τ σ i is clear from the definition of τ. Finally, if τ : G B satisfies τ i = τ σ i for each i, then τ ( A i, a i ) = τ (σ i (a i )) = τ i (a i ) = τ ( A i, a i ). Thus, τ = τ. This shows that G satisfies the mapping property of direct limits (along with the maps σ i ). A standard application of mapping properties will then show G = lim A i, as we now give. The maps σ i induce a unique homomorphism σ : lima i G with σ i = σ φ i for each i. Similarly, the mapping property applied to G yields a map τ : G lim A i satisfying φ i = τ σ i. Therefore, σ τ : G G satisfies σ i = (σ τ) φ i. However, id G : G G also satisfies σ i = id G φ i. By the uniqueness part of the mapping property, we conclude that σ τ = id G. Similarly, τ σ = id lim A i, so σ (and τ) is an isomorphism. We now prove the two most basic computational properties of direct limits. Lemma 1. Let lim A i be the direct limit of a directed system of groups. 1. Every element of lim A i can be written in the form φ i (a) for some a A i. 2. If a A i satisfies φ i (a) = 0, then there is a j i with φ i,j (a) = 0. Proof. These properties are easy to see from the definition of the group G defined above, which is isomorphic to lim A i. Every element of G is of the form A i, a i = σ i (a i ). Also, if A i, a i = 0, then (A i, a i ) (A i, 0), so by definition of the relation, there is a j i with φ ij (a i ) = φ ij (0) = 0. Finally, following these properties for G by the isomorphism τ : G lim A i yields the corresponding properties for lima i. Example 2. Let X be an algebraic variety and let P X. The collection I of open neighborhoods of P is a directed set with respect to reverse inclusion. For each U I we have the ring O(U) of regular functions on U. If U V, then the restriction map O(V ) O(U) is a ring homomorphism. The collection of the O(U), as U ranges over I, is then a direct system of commutative rings. An element of the direct limit has the form O(U), f for some U and some f O(U). Equality is given by O(U), f = O(V ), g 2
3 if and only if there is W I with W U V and f W = g W. Since regular functions are determined by their restriction to a dense open set, when this occurs, f U V = g U V. Thus, this direct limit is isomorphic to O P via the map U, f O(U), f. Example 3. Let X be an algebraic variety. We modify the previous example a little by setting I to be the set of all nonempty open subsets of X. Then I is a directed set; we are using that, since X is irreducible, the intersection of two nonempty open sets is again nonempty. By the same arguments as in the previous example, we see that the direct limit of O(U) as U ranges over all nonempty open subsets of X is isomorphic to the function field K(X) of X. Example 4. Let A be an Abelian group. Let {A i } i I be the set of finitely generated subgroups of A. Then, by ordering I by i j if A i A j, the set I is a directed set, since for any pair i, j, the group A i + A j is both finitely generated and contains A i and A j. If we let φ i,j : A i A j be the inclusion map whenever i j, we have a directed system {A i, φ i,j }. Thus, the direct limit lima i exists. We claim that A = lim A i. We write H = lima i for convenience. To prove this, we have inclusion maps σ i : A i A. We also have the canonical maps φ i : A i H for each i. Since σ i = σ j φ i,j, as both sides are the inclusion maps A i A, the universal mapping property gives a unique homomorphism σ : H A with σ φ i = σ i for each i. The map σ is surjective, since if g A, then g A i for some i; the cyclic group g is a finitely generated subgroup of A, so it is equal to A i for some i. Thus, g = σ i (g) = σ(φ i (g)), proving that σ is surjective. Finally, if h ker(σ), then let h = φ i (g) for some g A i. Then 0 = σ(h) = σ(φ i (g)) = σ i (g) = g since σ i is the inclusion map. Thus, g = 0, so h = φ i (g) = 0. We have thus proven that σ is bijective, so A = H. Example 5. Let I be a directed set that has a maximum element k. That is, i k for every i I. We claim thatunderrightarrowlima i = A k for any directed system. Write A = lim A i. To prove this claim, for each i we have the canonical maps φ i,k : A i A k, and since φ i,k = φ j,k φ i,j for any j with i j, the universal mapping property gives a uniquely determined map σ : A A k with σ φ i = φ i,k for every i. In particular, for i = k, we have σ φ k = φ k,k = id Ak. Thus, σ is surjective. For injectivity, take g A with σ(g) = 0. Write g = φ i (g i ) for some g i A i. Then 0 = σ(g) = σ(φ i (g i )) = φ i,k (g i ). By definition of directed systems, we then have 0 = φ k (φ i,k (g i )) = φ i (g i ) = g. Therefore, σ is also injective, so A = A k. Example 6. Let F be a sheaf on a topological space X. Then the stalk F x is defined as F x = lim F(U), where the direct limit is over all open neighborhoods of x. This set is a directed set by ordering it with reverse inclusion: if U and V are neighborhoods of x, then U V is a neighborhood of x contained in both U and V. If V U, then the canonical map F(U) F(V ) is the restriction map res U,V that comes along with the sheaf F. Since res V,W res U,V = res U,W whenever W V U, and res U,U = id F(U), these maps satisfy the axioms to have a directed system of groups. Therefore, the direct limit limf(u) does exist. We write f x for the image of f F(U) in F x for x U. The properties of the lemma 3
4 translate to the following two: (i) if α F x, then α = f x for some open neighborhood U of x and some f F(U), and (ii) if f F(U) with f x = 0, then there is some V U with res U,V (f) := f V = 0. Example 7. Here is another example that arises in the theory of sheaves. Let f : X Y be a continuous map of topological spaces. If G is a sheaf on Y, then we can define the inverse image presheaf F by F(U) = limg(v ), as the limit runs over all open sets V of Y with f(u) V. This set of open sets is a directed set by ordering it with reverse inclusion, since if V and W are open and containing f(u), then so is V W. Example 5 shows that if U = f 1 (V ), then this set of open sets has f 1 (V ) as a maximum element, sof(f 1 (V )) = G(V ). Example 8. To continue the previous example further, suppose that X is a topological space and x X. The unique map f : X {x} is continuous. An Abelian group A gives rise to a sheaf on {x} which we will also denote by A, since the only nonempty open set of {x} is {x} itself. We then have the inverse image presheaf F on X, defined by F(U) = lima(v ), as V runs over open sets of {x} containing f(u). However, the only choice for V is {x}. Therefore, F(U) = A({x}) = A. Therefore, F is the constant presheaf that sends every open set to the same Abelian group. Example 9. To give another version of the inverse image presheaf example, let X be a topological space and let x X. If j : {x} X is the inclusion map, then j is continuous. Let G be a sheaf on X. Then the inverse image presheaf F is a presheaf on {x}. Since {x} is a one point space, this sheaf is nothing more than the Abelian group F ({x}) = lim G(V ), as the limit ranges over open sets V of X containing j (x) = x. Thus, this group is just the stalk G x. Example 10. For a final example, we look at pushouts. groups A f B Given the diagram of Abelian g C the pushout is a group P with maps α and β such that the following diagram is commutative A f B g C β P α and such that for any Abelian group D and maps α : C D and β : B D, there is a unique map θ : L D with α = θ αand β = θ β. We see that L exists and is just a direct limit. Let I be the directed set {1, 2, 3} ordered by divisibility. That is, 1 2 and 4
5 1 3, but 2 3. Set A = A 1, B = A 2, and C = A 3. Then with the maps f and g, we have a directed set. Thus, set L = lima i. Then the universal mapping property for direct limits is exactly that described above, where we set the map A L to be β f = α g. In the remainder of this note we look at direct limits from a categorical point of view. Let I be a directed set. From I we have a category I whose objects are the elements of I, and whose morphisms are arrows i j for each pair i j in I. In other words, hom I (i, j) contains one map i j if i j, and is empty otherwise. This mimics the category Top(X) of open sets of a topological space X. A directed system of Abelian groups is then nothing but a functor from I to Ab. Thus, the functor category Ab I is then the category of directed systems of Abelian groups on I. We claim that the direct limit gives a functor from Ab I to Ab. We already know how it acts on objects, it sends a directed system {A i } to the direct limit lima i. To see how it acts on morphisms, let f : {A i, φ ij } {B i, ϕ ij } be a morphism of directed systems. Recall from the study of functor categories that f is a natural transformation of functors. In other words, for each i there is a group homomorphism f i : A i B i, and if i j, then the following diagram commutes. f i A i B i φ ij A j f j B j ϕ ij If φ i : A i lim A i and ϕ i : B i lim B i are the canonical maps, then ϕ i f i : A i lim B i is a homomorphism for each i that has the correct properties to yield a unique group homomorphism f : lim A i lim B i with φ i f = f i for all i. The association f f is then how the direct limit functor operates on maps. It is an easy exercise to show that composition of morphisms is preserved, and that id {A i } = id lim A i. In other words, direct limit does give a functor Ab I Ab. We have a simple functor C in the opposite direction. For B an Abelian group, let C(B) = {B}. This is the direct system with B i = B for all i, and the map B i B j for i j is the identity map. It is easy to see that C is an exact functor. Furthermore, the universal mapping property for direct limits yields immediately that hom(lim A i, B) = hom Ab I({A i }, C(B)). In other words, lim is a left adjoint to C. Therefore, by Proposition of Weibel s book, An Introduction to Homological Algebra, the direct limit functor lim is right exact. In fact, a straightforward calculation shows that lim is actually an exact functor. One can dualize definitions to define inverse systems of Abelian groups: such a system {A i } has, for each i j, a homomorphism φ ij : A j A i. One can define the inverse limit lim A i, together with homomorphisms lima i A i, via the following universal mapping property: if B is an Abelian group and if f i : B A i are homomorphisms with f j = φ ij f i for each i j, then there is a unique homomorphism f : B lim A i with f i = φ i f for 5
6 all i. As we did above, we obtain a category of inverse systems over I, and lim is a functor from this category to Ab. The constant functor C that sends B to {B} i I is a functor in the opposite direction, and we have hom(b, lim A i ) = hom(c(b), {A i }) from the universal mapping property of inverse limits. This means that lim is a right adjoint to C. Since C is clearly exact, lim is then left exact. Unlike direct limits, however, lim is not exact. Therefore, one can study the derived functors of lim, whereas the derived functors of the direct limit lim are trivial since this functor is exact. 6
which is a group homomorphism, such that if W V U, then
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