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Summer School > 1 > Flashcards

1 Flashcards

1
Q

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

A

pg14-15

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2
Q

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

A

iff exists a morphism inducing isomorphisms of all stalks.

pg15, pg17

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3
Q

Discuss sheafification

A

left adjoint to forgetful functor
pg 16

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4
Q

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

A

closure of support of all cohomologies -pg17

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5
Q

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

A

yes–use that mod has enough inj
no pg 17

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6
Q

Discuss hypercohomology, relation to singular cohomology?

A

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

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7
Q

Define: locally compact space, proper map

A

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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8
Q

Discuss 4 sheaf functors coming from a continuous map

A

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

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9
Q

Discuss exactness of f, f_, f_!

Derived functors?

A

f* exact, other two left exact
pg 21

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10
Q

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

A

pg 22

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11
Q

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

A

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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12
Q

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

A

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

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13
Q

Define: flabby and c-soft sheaves,

Purpose?

A

restriction to open set is surjective

restriction to compact set is surjective

pg 23

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14
Q

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

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

A

pg 23-24

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15
Q

Induced map in hypercohomology?

A

pg 24

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16
Q

Discuss/prove different commutative diagrams coming from cartesian squares

When are the different morphisms isomorphisms? state without proof

A

pg 25-26

1.9

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17
Q

Prove proper base change theorems

A

1.2.13 and 1.2.15

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18
Q

State and prove open base change

A

pg 28

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19
Q

Define: locally close subset

examples?

A

pg 29

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20
Q

Prove properties of the extension by zero functor

A

pg 29-30

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21
Q

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

A

restriction with supports in Y
pg 30

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22
Q

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

A

pg 30

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23
Q

h^! is left exact (Lemma 1.3.4)

A

pg 30

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24
Q

prove h_! h^! adjunction

A

pg 30

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25
Q

Induced maps in hypercohomology with compact support?

A

pg 31

26
Q

Prove isomorphims among different pushforward/pullback functors hh_ etc as in Prop 1.3.9

A

pg 31-32

27
Q

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

A

pg 32-33

28
Q

Discuss/prove Meyer-Vietoris distinguished triangles

A

pg 33-34

29
Q

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

A

tensor right exact, not enough projectives - use flat resolutions

sheaf home left exact, enough injectives

pg 36, pg 38 Thm 1.4.8

30
Q

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

A

pg 37 1.4.4

31
Q

How does derived tensor interact with f*?

A

commutes nicely pg 37

32
Q

What is projection formula? Proof idea?

A

pg 38

33
Q

Discuss change of scalars functors, exactness, derived versions

A

pg 39-40

34
Q

Projection formula in context of extension of scalars? Proof?

Universal Coefficient theorem?

A

pg 40 1.4.17, 1.4.18

35
Q

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

A

pg 41-42

36
Q

Define: finite cohomological dimension, how can we check this condition for f_! ? Why important?

Define: c-soft dimension <= n

A

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

37
Q

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

A

pg 43-44

38
Q

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

A

pg 45

39
Q

Discuss base change with f^!

A

pg 45 Prop 1.5.7

40
Q

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

A

pg 45 Prop 1.5.8 and 1.5.9

41
Q

Define: topological submersion of rel dim n, relative orientation sheaf

Why relevant to study of f^!? Show details

A

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

42
Q

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

A

Prop 1.5.12

natural iso with f, so f right adjoint to f_!

43
Q

Define: dualizing complex, duality functor, pairing map

A

pg 48

44
Q

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

A

pg 54

45
Q

Show f* takes local systems to local systems

A

pg 54

46
Q

Discuss constant sheaf functor, prove fully faithful

A

Lemma 1.7.3

pg 54-55

47
Q

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

A

1.7.4 pg 55

48
Q

Discuss criterion for showing a sheaf is locally constant

A

1.7.5 pg 55

49
Q

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

A

pg 55

50
Q

Define: Monodromy representation and discuss construction

A

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

51
Q

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

A

Prop 1.7.10

pg 58

52
Q

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

Derived verions?

A

1.7.11, 1.7.12

pg 58

53
Q

Define: dual local system

Monodromy?

A

pg 59

54
Q

Discuss pushforward and proper pushforward for local systems

proof?

A

1.7.4
pg 59

55
Q

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

A

pg 60

56
Q

Discuss pullbacks along deck transformations w.r.t Monodromy

pushforwards?

A

pg 61 1.7.18
pg 62

57
Q

What are the internal monodromy actions?

A

pg 62 - 63

58
Q

Define derived categories of local systems

A

pg 64

59
Q

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

A

p65

60
Q

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

Proof? Implications

A

pg 65-66

61
Q

Prove Loc(X, k) closed under extensions

A

pg 66

62
Q

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

Proof strategy?

A

The are equivalent

pg 67-68

Summer School (4 decks)

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