1 Flashcards
Define: presheaf, sheaf, stalk, germ, support, morphisms
pg14-15
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
Discuss sheafification
left adjoint to forgetful functor
pg 16
What is the support of an object in derived cat of sheaves?
closure of support of all cohomologies -pg17
Does Sh(X) have enough injectives? Projectives? Proof?
yes–use that mod has enough inj
no pg 17
Discuss hypercohomology, relation to singular cohomology?
From global sections functor, in nice situations is equal to singular pg18-19
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.
Discuss 4 sheaf functors coming from a continuous map
pullback
push-forward
proper push-forward
proper pullback <— tricky one
Discuss exactness of f, f_, f_!
Derived functors?
f* exact, other two left exact
pg 21
DIscuss examples of pushforwards, pullbacks in case of one point space
pg 22
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
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
Define: flabby and c-soft sheaves,
Purpose?
restriction to open set is surjective
restriction to compact set is surjective
pg 23
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
Induced map in hypercohomology?
pg 24
Discuss/prove different commutative diagrams coming from cartesian squares
When are the different morphisms isomorphisms? state without proof
pg 25-26
1.9
Prove proper base change theorems
1.2.13 and 1.2.15
State and prove open base change
pg 28
Define: locally close subset
examples?
pg 29
Prove properties of the extension by zero functor
pg 29-30
What is the right adjoint of extension by zero functor? Properties?
restriction with supports in Y
pg 30
Prove: If j is an open embedding, j^! F = j* F (Lemma 1.3.3)
pg 30
h^! is left exact (Lemma 1.3.4)
pg 30
prove h_! h^! adjunction
pg 30
Induced maps in hypercohomology with compact support?
pg 31
Prove isomorphims among different pushforward/pullback functors hh_ 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
