1 Flashcards by Andrew Frohmader

Define: presheaf, sheaf, stalk, germ, support, morphisms

pg14-15

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When are two sheaves isomorphic? Proof?
When do we have a s.e.s. of sheaves? Injective, surjective?

iff exists a morphism inducing isomorphisms of all stalks.

pg15, pg17

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Discuss sheafification

left adjoint to forgetful functor
pg 16

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What is the support of an object in derived cat of sheaves?

closure of support of all cohomologies -pg17

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Does Sh(X) have enough injectives? Projectives? Proof?

yes–use that mod has enough inj
no pg 17

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Discuss hypercohomology, relation to singular cohomology?

From global sections functor, in nice situations is equal to singular pg18-19

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Define: locally compact space, proper map

A space is locally compact if it is Hausdorff an if each point is contained in a pair of subsets U < K < X with U open and K compact.

A continuous map is proper if it is universally closed - for any other space f x id_Z : X x Z –> Y x Z is closed

If X,Y locally compactTFAE
1. Proper
2. For every compact set f^-1{K} is compact
3. The map f is closed and every point y has compact fiber.

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Discuss 4 sheaf functors coming from a continuous map

pullback
push-forward
proper push-forward
proper pullback <— tricky one

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Discuss exactness of f, f_, f_!

Derived functors?

f* exact, other two left exact
pg 21

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DIscuss examples of pushforwards, pullbacks in case of one point space

pg 22

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Discuss f* and f_* adjunction. What is the strategy for proving?

Show results of adjunction in case of 1-pt spaces

First do for presheaves using zig-zag equations.

Then sheafify

Finally, since enough injectives, descends to derived functor

pg 22

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What is the general strategy for proving natural isomorphisms of compositions of functors?

First prove at level of abelian categories
Second exhibit an adapted class
pg 23

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Define: flabby and c-soft sheaves,

Purpose?

restriction to open set is surjective

restriction to compact set is surjective

pg 23

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Prove (g o f)* = f* o g* and similarly for pushforwards

Implications? Consider again one of the spaces being 1-pt space

pg 23-24

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Induced map in hypercohomology?

pg 24

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Discuss/prove different commutative diagrams coming from cartesian squares

When are the different morphisms isomorphisms? state without proof

pg 25-26

1.9

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Prove proper base change theorems

1.2.13 and 1.2.15

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State and prove open base change

pg 28

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Define: locally close subset

examples?

pg 29

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Prove properties of the extension by zero functor

pg 29-30

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What is the right adjoint of extension by zero functor? Properties?

restriction with supports in Y
pg 30

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Prove: If j is an open embedding, j^! F = j* F (Lemma 1.3.3)

pg 30

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h^! is left exact (Lemma 1.3.4)

pg 30

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prove h_! h^! adjunction

pg 30

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Induced maps in hypercohomology with compact support?

pg 31

Prove isomorphims among different pushforward/pullback functors h*h_* etc as in Prop 1.3.9

pg 31-32

Discuss/prove distiguished triangles coming from closed embeddings (Key Theorem 1.3.10)

pg 32-33

Discuss/prove Meyer-Vietoris distinguished triangles

pg 33-34

Discuss tensor product and sheaf hom, exactness, derived versions, adjunction?

tensor right exact, not enough projectives - use flat resolutions sheaf home left exact, enough injectives pg 36, pg 38 Thm 1.4.8

How do tensor and sheaf hom interact with constant sheaves? Proof in derived case?

pg 37 1.4.4

How does derived tensor interact with f*?

commutes nicely pg 37

What is projection formula? Proof idea?

pg 38

Discuss change of scalars functors, exactness, derived versions

pg 39-40

Projection formula in context of extension of scalars? Proof? Universal Coefficient theorem?

pg 40 1.4.17, 1.4.18

Define external tensor product and discuss how it behaves w.r.t pullback, proper pushforward, tensor product, Kunneth formula?

pg 41-42

Define: finite cohomological dimension, how can we check this condition for f_! ? Why important? Define: c-soft dimension <= n

In order for right adjoint to f_! to exist, we must assume the sheaf functor o^f_! has finite cohomological dimension One way to check: 1.5.1: Every sheaf in Sh(X, k) admits a relativelt c-soft resolution of length at most n. X has c-soft dim <= n <=> every sheaf admits c-soft resolution of length <=n <=> cohomological dim <= n A nice case where all the above hold is for locally closed subsets of real n-dimensonal manifolds pg 43

Discuss the construction of the right adjoint to f_! in general

pg 43-44

Prove: f^!g^! F = (g o f)^! F

pg 45

Discuss base change with f^!

pg 45 Prop 1.5.7

How does f^! behave w.r.t. Hom and tensor?

pg 45 Prop 1.5.8 and 1.5.9

Define: topological submersion of rel dim n, relative orientation sheaf Why relevant to study of f^!? Show details

In the case of a top submersion of rel dim n, we can find f^! F by computing or_f tensor^L f* F[n] pg 46

What can be said about f^! in case f is a local homeomorphism?

Prop 1.5.12 natural iso with f*, so f* right adjoint to f_!

Define: dualizing complex, duality functor, pairing map

pg 48

Define: Local system, finite type, locally free, rank, grade

pg 54

Show f* takes local systems to local systems

pg 54

Discuss constant sheaf functor, prove fully faithful

Lemma 1.7.3 pg 54-55

Prove if X locally connected, Loc(X,k) abelian subcategory of Sh(X,K)

1.7.4 pg 55

Discuss criterion for showing a sheaf is locally constant

1.7.5 pg 55

Show that the pushforward of a constant sheaf on a connected open dense subset is a constant sheaf on the whole space

pg 55

Define: Monodromy representation and discuss construction

Defines an equivalence of categories between local systems and k[pi_1]-mod pg 56 - 57

How does Monodromy rep behave w.r..t f*?

Prop 1.7.10 pg 58

How does Monodromy behave w.r.t. tensor and Hom? Derived verions?

1.7.11, 1.7.12 pg 58

Define: dual local system Monodromy?

pg 59

Discuss pushforward and proper pushforward for local systems proof?

1.7.4 pg 59

How does Monodromy behave w.r.t. f_* and f_! ?

pg 60

Discuss pullbacks along deck transformations w.r.t Monodromy pushforwards?

pg 61 1.7.18 pg 62

What are the internal monodromy actions?

pg 62 - 63

Define derived categories of local systems

pg 64

Discuss how Loc(X, k) and Loc(X x [0,1], k) are related. Proof?

p65

What be said about induced maps on cohomology coming from two homotopic maps? Proof? Implications

pg 65-66

Prove Loc(X, k) closed under extensions

pg 66

What can be said about derived categories of local systems on homotopic spaces? Proof strategy?

The are equivalent pg 67-68