1. General Sheaf Theory Exercises 1.1

SHEAF THEORY EXERCISES

PETER J. HAINE

1. General Sheaf Theory Exercises 1.1. Exercise. Let 푋 be a space. Prove that the opposite of the poset Op(푋) of open subsets of 푋 is filtered.

1.2. Exercise. Let 푋 be a space and 푥 ∈ 푋. Write Op푥(푋) for the full subposet of Op(푋) op spanned by those open subsets containing the point 푥. Prove that Op푥(푋) is filtered. 1.3. Exercise. Prove that the subcategory of CRing of domains is stable under filtered col- imits, i.e., that if 퐹∶ 퐼 → CRing be a filtered diagram of commutative rings, where 퐹(푖) is a domain for all 푖 ∈ 퐼, then colim퐼 퐹 is a domain (where the colimit is taken in CRing). You’ll need to use the explicit description of filtered colimits in Set, which also describes the underlying set of the filtered colimit in CRing. You should think about what the ring structure should be. More generally, this description gives the underlying set of a filtered colimit in Mod푅 or Alg푅, where 푅 is a commutative ring (taking 푅 = 퐙, we recover the special cases of Ab = Mod퐙 and CRing = Alg퐙). 1.4. Exercise. Let 퐶 be a category and 푓∶ 푋 → 푌 be a morphism in 퐶. Prove 푓 is a monomorphism if and only if the square 푋 푋

푓

푋 푌 푓 is a pullback square. 1.5. Exercise. Suppose that 퐶 and 퐷 are categories and 퐹, 퐺∶ 퐶 → 퐷 functors. Prove that if 훼∶ 퐹 ⇒ 퐺 is a natural transformation whose components are monomorphisms in 퐷, then 훼 is a monomorphism in 퐷퐶. Conversely, if 퐶 is small, 퐷 has pullbacks, and 훼 is a monomorphism in 퐷퐶, prove that each component of 훼 is a monomorphism in 퐷. 1.6. Definition. Let F be a presheaf of sets (or commutative rings, or abelian groups) on a space 푋. The stalk of F at 푥 is the colimit F ≔ colim F 푥 op Op푥(푋) op (of the diagram F restricted to Op푥(푋) ). 1.7. Exercise. Let 휙∶ F ↣ G be a monomorphism of presheaves of sets on a space 푋. Prove that for all 푥 ∈ 푋, the induced map on stalks 휙푥 ∶ F푥 → G푥 is an injection.

Date: March 9, 2018. 1 2 PETER J. HAINE

1.8. Definition. Let F be a presheaf of sets (or commutative rings, or abelian groups) on a space 푋, 푓 ∈ F(푋), and 푥 ∈ 푋. Write 푓푥 for the image of 푓 in the stalk F푥 under the leg 휆푋 ∶ F(푋) → F푥 of the colimit cone. The element 푓푥 is called the germ of 푓 at 푥. 1.9. Definition. Let F be a presheaf on a space 푋, and 푈 ⊂ 푋 be open. The germ map

푔F (푈)∶ F(푈) → ∏ F푥 푥∈푈 is the map given by the assignment 푓 ↦ (푓푥)푥∈푈. 1.10. Exercise. Show that if F is a sheaf on a space 푋, then for every open subset 푈 ⊂ 푋 the germ map 푔F (푈)∶ F(푈) → ∏푥∈푈 F푥 is injective. Have you used the entirety of the sheaf condition here? The next exercise is an extension of Exercise 1.7 for sheaves. 1.11. Exercise. Suppose that 휙∶ F → G is a morphism of sheaves of sets on a toplogical space 푋. Show that the following are equivalent: (1.13.a) 휙 is a monomorphism in the category of sheaves.

(1.13.b) 휙 is injective on the level of stalks: 휙푥 ∶ F푥 → G푥 is injective for all 푥 ∈ 푋. (1.13.c) 휙 is injective on the level of open sets: 휙(푈)∶ F(푈) → G(푈) is injective for all open 푈 ⊂ 푋. 1.12. Exercise. Let 휙, 휓∶ F → G be morphism of sheaves of sets on a toplogical space 푋.

Show that if 휙푥 = 휓푥 as maps F푥 → G푥 for all 푥 ∈ 푋, then 휙 = 휓 as morphisms of sheaves. 1.13. Exercise. Let 휙∶ F → G be morphism of sheaves of sets on a toplogical space 푋. Show that 휙 is an isomorphism if and only if 휙푥 ∶ F푥 → G푥 is a bijection for all 푥 ∈ 푋. 1.14. Exercise. Let F be a presheaf of commutative rings on a space 푋, and 푓 ∈ F(푋). Prove that the set × { 푥 ∈ 푋 | 푓푥 ∈ F푥 } ⊂ 푋 × is open, where F푥 is the group of units of the stalk F푥. 1.15. Definition. Let 푋 be a space and 푥 ∈ 푋. Write 푥∶ {푥} ↪ 푋 for the inclusion of the point 푋. The skyscraper sheaf functor is the pushforward

푥⋆ ∶ Set ≅ Shv({푥}) → Shv(푋) .

1.16. Exercise. Let 푋 be a space, 푥 ∈ 푋, and 푆 a set. Describe the skyscraper sheaf 푥⋆(푆) explicitly. 1.17. Exercise. Let 푋 be a space and 푥 ∈ 푋. Show that the skyscraper sheaf functor

푥⋆ ∶ Set → Shv(푋) is right adjoint to the stalk functor (−)푥 ∶ Shv(푋) → Set. 1.18. Definition. Let 푋 be a space and 푆 a set. The constant presheaf on 푋 with value 푆 is the constant functor 푆∶ Op(푋)op → Set at 푆. 1.19. Exercise. Let 푋 be a space. For which sets 푆 is the constant presheaf at 푆 a sheaf? Hint: The axioms for a topology on a set 푋 guarantee that certain subsets of 푋 are open, and the value of the constant presheaf is completely determined by where it sends any open subset 푈 ⊂ 푋. SHEAF THEORY EXERCISES 3

1.20. Exercise. Let F and G be presheaves on a space 푋. In this exercise we define a presheaf Hom(F, G) on 푋. On objects, send an open set 푈 ⊂ 푋 to

Hom(F, G)(푈) ≔ Shv푈(F|푈, G|푈) . Now: (1.20.a) Extend this assignment on objects to morphisms (so that Hom(F, G) defines a presheaf on 푋). (1.20.b) Show that if G is a sheaf, then Hom(F, G) is a sheaf.

2. Exercises Related to Sheafification 2.1. Definition. The Godement construction God∶ PShv(푋) → Shv(푋) is the functor given by sending a presheaf F ∈ PShv(푋) to the sheaf God(F) defined by sending an open set 푈 to

God(F)(푈) ≔ ∏ F푥 , 푥∈푈 with the restriction morphisms given by product projections. The assignment of God on morphisms is given by sending a morphism of presheaves 휙∶ F → G to the morphism

God(휙)(푈) ≔ ∏ 휙푥 ∶ ∏ F푥 → ∏ G푥 . 푥∈푈 푥∈푈 푥∈푈 2.2. Exercise. Verify that if F is a presheaf on a space 푋, then God(F) is a sheaf (not merely a presheaf). 2.3. Exercise. Let F be a presheaf on a space 푋. Show that the germ maps

푔F (푈)∶ F(푈) → ∏ F푥 = God(F)(푈) , 푥∈푈 for 푈 ⊂ 푋 open, define a morphism of presheaves 푔F ∶ F → God(F).

2.4. Definition. Let F be a presheaf on a space 푋 and 푈 ⊂ 푋 open. An element (푓푥)푥∈푈 ∈ ∏푥∈푈 F푥 consists of compatible germs over 푈 if for all 푥 ∈ 푈 there is an open cover {푈훼}훼∈퐴 of 푈 and sections 푠훼 ∈ F(푈훼) for each 훼 ∈ 퐴 so that if 푥 ∈ 푈훼, then (푠훼)푥 = 푓푥. Note that any section 푓 ∈ F(푈) clearly gives a choice of compatible germs over 푈.

2.5. Exercise. Let F be a sheaf on a space 푋, and 푈 ⊂ 푋 open. Prove that any choice (푓푥)푥∈푈 of compatible germs for F over 푈 is the image of a section 푓 ∈ F(푈) under the germ map 푔F (푈)∶ F(푈) → ∏푥∈푈 F푥. Have you used the entirety of the sheaf condition here? 2.6. Exercise. Let F be a sheaf on a space 푋, and 푈 ⊂ 푋 open. Identify the image of the germ map 푔F (푈)∶ F(푈) → ∏푥∈푈 F푥. Use this to identify F with a subsheaf of the Godement sheaf God(F). 2.7. Exercise. Give an example of a space 푋 and a sheaf F on 푋 so that F is not isomorphic to the Godement sheaf God(F).

2.8. Exercise. Let 푋 be a space. Define a subfunctor 퐿푋 of God by setting

퐿푋(F)(푈) ≔ {compatible germs over 푈} ⊂ God(F)(푈) , for F ∈ PShv(푋) and 푈 ⊂ 푋 open.

(2.8.a) Verify that 퐿푋(F) is a well-defined presheaf for any F ∈ PShv(푋). (2.8.b) Show that 퐿푋(F) is actually a sheaf for any F ∈ PShv(푋). 4 PETER J. HAINE

(2.8.c) Show that the germ morphism 푔F ∶ F → God(F) factors through 퐿푋(F) ⊂ God(F). (2.8.d) Show that the germ morphism 푔F ∶ F → 퐿푋(F) exhibits 퐿푋(F) as a sheafification of F, hence 퐿푋 defines a left adjoint to the inclusion Shv(푋) ↪ PShv(푋). 2.9. Exercise. Let 푋 be a space. Show explicitly from the construction of the sheafification in Exercise 2.8 that the germ map 푔F ∶ F → 퐿푋(F) induces an isomorphism on stalks for any presheaf F on 푋.

2.10. Exercise. Let 푋 be a space. The constant sheaf functor (−)푋 ∶ Set → Shv(푋) is the composite of the constant presheaf functor Set → PShv(푋) with the sheafification func- tor 퐿푋 ∶ PShv(푋) → Shv(푋). The constant sheaf functor has a right adjoint. Describe this adjoint explicitly.

3. Exercises Related to Inverse Image Sheaves 3.1. Exercise. Let 푋 be a space and 푗∶ 푈 ↪ 푋 the inclusion of an open subspace.

(3.1.a) Show that the functor 푗⋆ ∶ PShv(푈) → PShv(푋) is fully faithful and note that the same is true if we restrict to categories of sheaves. (3.1.b) Define a functor 푗−1 ∶ PShv(푋) → PShv(푈) given by setting 푗−1(F)(푉) ≔ F(푉) for 푉 ⊂ 푈 open, and with the obvious restriction maps and assignment on mor- −1 phisms of presheaves. Show that 푗 is left adjoint to 푗⋆ ∶ PShv(푈) → PShv(푋). (3.1.c) Show that 푗−1 preserves sheaves, and deduce that the functors −1 푗 ∶ Shv(푋) ⇄ Shv(푈) ∶ 푗⋆ are adjoint. (3.1.d) Prove that the functor 푗−1 ∶ PShv(푋) → PShv(푈) preserves all limits. 3.2. Definition. Let 푓∶ 푋 → 푌 be a map of spaces. The presheaf inverse image functor −1 푓푝 ∶ PShv(푌) → PShv(푋) is defined as follows: (3.2.a) For a presheaf F on 푌 and open set 푉 ⊂ 푋, 푓−1(F)(푉) ≔ colim F(푈) , 푝 op 푈∈Op푓(푉)(푋) −1 −1 ′ ′ and the restriction morphism 푓푝 (F)(푉) → 푓푝 (F)(푉 ) for 푉 ⊂ 푉 open is in- op duced via the universal property of the colimit by the inclusion Op푓(푉)(푋) ⊂ op Op푓(푉′)(푋) . (3.2.b) Given a morphism of presheaves 휙∶ F → G and 푉 ⊂ 푋 open, the morphism −1 −1 −1 푓푝 (휙)∶ 푓푝 (F)(푉) → 푓푝 (G)(푉)

is the unique morphism induced by the morphism of diagrams 휙|Op (푋)op in the −1 −1 푓(푉) definining colimits of 푓푝 (F)(푉) and 푓푝 (G)(푉) from (3.2.a). 3.3. Definition. Let 퐶 and 퐷 be categories with finite limits. A functor 퐹∶ 퐶 → 퐷 is left exact if 퐹 preserves finite limits. −1 3.4. Exercise. Let 푓∶ 푋 → 푌 be a map of spaces. Show that 푓푝 ∶ PShv(푌) → PShv(푋) is −1 left adjoint to 푓⋆ ∶ PShv(푋) → PShv(푌) as well as that 푓푝 is left exact. SHEAF THEORY EXERCISES 5

3.5. Exercise. Let 푋 and 푌 be spaces and 퐺∶ PShv(푋) → PShv(푌) be a functor. Show that if 퐺 preserves limits, then 퐺 preserves sheaves. In particular, right adjoints preserves sheaves. 3.6. Exercise. Let 퐶′ ⊂ 퐶 be a full subcategory and 퐷′ ⊂ 퐷 be reflective subcategory with ′ localization functor 퐿퐷 ∶ 퐷 → 퐷 . Let 퐹∶ 퐶 ⇄ 퐷 ∶ 퐺 be an adjunction (with left adjoint ′ ′ ′ ′ 퐹 and right adjoint 퐺). Show that if 퐺(퐷 ) ⊂ 퐶 , then 퐿퐷퐹∶ 퐶 → 퐷 is left adjoint to 퐺∶ 퐷′ → 퐶′.

3.7. Exercise. Let 푓∶ 푋 → 푌 be a map of spaces. Deduce that 푓⋆ ∶ Shv(푋) → Shv(푌) admits a left exact left adjoint 푓−1 ∶ Shv(푌) → Shv(푋). 3.8. Exercise. This exercise is a continuation of Exercise 3.1. Let 푋 be a space and 푗∶ 푈 ↪ 푋 the inclusion of an open subspace. (3.1.a) Show that the functor 푗−1 ∶ PShv(푋) → PShv(푈) admits a left adjoint 푝 푗! ∶ PShv(푈) → PShv(푋) . Hint: Categories of presheaves are presentable. −1 (3.1.b) Prove that the functor 푗 ∶ PShv(푋) → PShv(푈) admits a left adjoint 푗!. 푝 (3.1.c) Can you describe 푗! and 푗! explicitly? The results of this exercise and Exercise 3.1 show that we have a chain of adjunctions

푗!

⊥ Shv(푋) 푗−1 Shv(푈) ⊥

푗⋆

Conclude that, in addition, 푗! is fully faithful. 3.9. Exercise. Let 푋 be a space and 푥 ∈ 푋. Write 푥∶ {푥} ↪ 푋 for the inclusion of the point 푥 in 푋. Deduce that the stalk functor (−)푥 ∶ Shv(푋) → Set ≅ Shv({푥}) is naturally isomorphic to the inverse image functor 푥−1 ∶ Shv(푋) → Shv({푥}). 3.10. Exercise. Let 푋 be a space and F a presheaf on 푋. Give a better proof (than the one you gave in Exercise 2.9) that the unit of the sheafification adjunction 휂F ∶ F → 퐿푋(F) is an isomorphism on stalks. 3.11. Exercise. Let 푋 be a topological space and write 푋훿 for the space whose underlying set 훿 훿 is 푋 and the topology on 푋 is the discrete topology. The identity map 휀푋 ∶ 푋 → 푋 is con- tinuous. Show that for a sheaf F on 푋, the Godement sheaf is isomorphic to the composite −1 휀푋,⋆휀푋 (F), where 휀푋,⋆ ≔ (휀푋)⋆.