2 Flashcards
Define: variety, affine
which topologies used? Compare properties under the different topologies
pg 80-82
2.1.1 Y locally closed subset of variety => Closure in analytic = Closure in Zariski. implies: Zariski dense open subvarieties remain dense in analytic topology
2.1.4 Every variety has a smooth dense subvariety
2.1.5 Connected in Zariski <=> Connected in analytic
2.1.6 In the analytic topology, every variety is locally compact, locally contractible, and of finite c-soft dimension.
Define: Noetherian topological space, prove every variety is one
dcc
pg 81
Prove: If X is a variety and U1, U2 < X are two locally closed affine subvarieties, then U1 \cap U2 is an affine subvariety.
pg 81 Cartesian square
What is Nagata’s compactification theorem?
pg 82
Define: quasi-finite morphism, finite morphism
morphism finite iff proper and quasi-finite iff closed and quasi-finite (by 3rd characterization of proper on pg 21)
Noether Normalization Lemma?
pg 82
Prove: If f: X –> Y is a finite morphism, then o^f_*: Sh(X, k) –> Sh(Y, k) is an exact functor.
pg 82
Define/discuss: smooth of relative dimension m-n, smooth morphism, etale morphism
pg 83
Discuss generic smoothness of morphisms of varieties
If f: X –> Y and X smooth, then exists nonempty Zariski open subset U < Y s.t. f is smooth from f^-1(U) –> U
pg 83
What is the relationship between etale morphisms and covering maps?
pg 83 Lemma 2.1.14
Define: divisor, divisor with simple normal crossings
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumford). Both are derived from the notion of divisibility in the integers and algebraic number fields.
pg 84
Discuss resolution of singularities
pg 84-85
Define differentiable locally trivial fibration and discuss presence in algebraic geometry
pg 85
Define when map is transverse to divisor w/ snc Z. Transverse locally trivial fibration? Generalization of Ehressmann?
pg 85
Discuss the relationship between the fundamental group of a smooth, connected variety and a Zariski open subset of that variety. Proof?
pg 86
Prove:Prop 2.1.23
pg86-87
Show a smooth morphism of relative dim d is a topological submersion of relative dim 2d
pg 87
What is smooth base change? Proof?
pg 88
Define: Tate module, nth Tate twist
pg 88
Examples of Tate twists?
pg 89
f:X–>Y a smooth morphism of rel dim d. What can be said about the rel orientation sheaf? Proof
How does this help with figuring out f^! ?
pg 89-91
What is the dualizing complex for a smooth variety of dim n?
pg 91
How does smooth pullback interact with sheaf operations?
pg 91
Define: smooth pair, discuss restriction with supports functor for smooth pairs
pg 92
Define: stratification, strata, closure partial order
A stratification of X is a finite collection of disjoint smooth, connected, locally closed subvarieties s.t. their union is X and s.t. the closure of a strata is a union of strata.
pg 93-94
Define: filtration of X by smooth varieties
relation to stratification?
What varieties admit filtration?
refinement?
Finite collection of disjoint, smooth, connected, locally closed subvarieties whose union is X and s.t. the elements can be ordered s1, s2, … , sk in such a way that Xs1 U Xs2 U … U Xsi is always a closed subset of X. (affine paving if these are affine sets)
Any stratification is a filtration. Any filtration admits a refinement that is a stratification.
All varieties admit a filtration
pg 93
Define: weakly constructible and constructible (sheaf and derived)
Sheaf weakly constructible if on each stratum F is a local system (finite type = constructible)
An object in derived bounded category is (weakly ) constructible if each cohomology shead has the same property
94
What is modified dimension of support over a field?
What can be said about it w.r.t. s.e.s of sheaves and distinguished triangles in derived cat?
Over a field - no need to modify. just dimension of support
pg 94
Prove: f* preserves constructibility
pg 96
Prove: if h:Y –> X is inclusion of locally closed sub, h_! preserves constructibility
pg 96
Prove: If U open in X and F constructible on U and X - U, then F constructible on U
pg 95
How do 6 functors interact with constructibility?
All preserve constructibility
pg 96
Define: good stratification
Importance?
If pushforward of the inclusion j of a strata Xs into X always sends a local system of finite type on Xs to a constructible sheaf on X
Can ignore here - in algebraic setting always good
Allows easier proof that 4 functors associated to inclusion of locally closed subvariety all preserve constructibility
pg 97-98
Define: normal crossings stratification
is this a good stratification?
Given a divisor with simple normal crossings, we get a stratification.
pg 99, 100-101
Define: normal crossings coordinate chart
Z divisor with simple normal crossings, Z1 … Zk components. J < {1,…k} say J = {i1, …, ij}. For any point of X_J, there is an analytic open set V < X_…
pg 99
Discuss base change and the affine line
Define Z finite/surjective/etale over X
The main point: consider the cartesian square on 103 - find some conditions under which base change maps are isomorphisms.
2.5.3 If Z closed and finite over Y and U = Y \Z. Then if F|U is a local system, base change is an iso.
Discuss Artin’s vanishing theorem
proof?
For affine varieties, H^k(X, k) = 0 for all k > n (you would expect it to be 2n)
Thm. X affine variety, F constructible sheaf on X. Then H^k(X, F) is a f.g. k-module and it vanishes unless 0 < k < dimm supp F. In particular, = unless k < dim X.
Discuss the cohomology of a constructible sheaf on X
Thm 2.7.4, 2.7.5
pg 112 -113
Discuss Verdier duality with respect to constructible sheaves
- Dualizing complex is constructible, so Verdier duality functor restricts to a functor between (derived) constructible sheaves
- Evaluation map F –> D(D(F)) is an isomorphism - i.e. natural isomorphism D o D = id
- Relationship between stalk of dual complex and original stalk
- Fact that DD = id allows us to prove f^! preserves constructibility
Discuss how external tensor product and extension of scalars interact with 6 sheaf functors
Commute with all of them and Verdier duality
What is Noetherian induction? Why does it work? Examples?
Borcherds & Pramod
