3 Flashcards
Define the perverse t-structure and its heart
pg 142
How does perverse t-structure interact with extensions? Proof?
Stable under extensions
pg 143
Describe the perverse t-structure for local systems in terms of the natural t-structure
what is heart?
pf?
Heart is shifted local systems
pg 143
Describe how perverse t-structure interacts with functors coming from open and closed embeddings.
3.1.4 - 3.1.6 pg 144
Discuss how perverse sheaves on a closed subvariety of X sit inside the perverse sheaves on X
3.1.10
i_* induces an equiv of categories with cat of perverse sheaves supporteed on Z <– a Serre subcategory
pg 146
How does Verdier duality functor interact with perverse t-structure?
If k a field, t-exact
pg 147
How does RHom interact with perverse t-structure?
pg148
How do derived tensor and hom interact with perverse t-structure on local systems?
right/left exact - exact if L locally free
pg 148
Discuss how extension of scalars interacts with perverse t-structure
right exact, exact if k’ flat as a k module
pg 149
How does external tensor product interact with perverse t-structure?
right exact, exact if k is a field
pg 149
Define: intermediate extension functor, intersection cohomology complex, intersection cohomology
intermediate extension - pg 151
intersection cohohomology complex - specific type of intermediate extension - pg 154-155
Discuss properties of the intermediate extension functor, alternate characterizations
Lemma 3.3.3 Fully faithful - unique perverse sheaf on X s.t.
-Supported on Y closure
-Restriction to Y is isomorphic to F
-It has no nonzero subobjects or quotient objects supported on Y closure \ Y
Lemma 3.3.4 pullback conditions
Lemma 3.3.5 takes injectives maps to injective maps, surjective maps to surjective maps
Not exact, image not a Serre subcategory
Discuss maximal subobjects and quotients of a perverse sheaf supported on a closed subvariety. Related s.e.s?
Lemma 3.3.7, Lemma 3.3.8
Discuss alternative descriptions of IC complexes
Lemma 3.3.11 - Support is closure of strata
- Restriction to strata is shifted local system
Lemma 3.3.12
pg 155
How does Verdier duality functor interact with IC complex?
Lemma 3.3.13
pg155
How does external tensor product interact with IC complex?
pg 156
Prove: If X is a smooth, connected variety of dim n, then Loc(X, k)[n] is a Serre subcat of Perv(X,k)
pg 157
Prove: Every perverse sheaf admits a finite filtration whose subquotients are IC complexes
Notice here we don’t say simple IC complexes - see Thm 3.4.5
pg 158
Discuss steps in showing Perv(X,k) is Noetherian. What if k is a field?
If field, IC simple objects, artinian
pg 158-160
Discuss characterization of perverse t-structure using affine open sets
Thm 3.5.3 F in <=0 same as derived global sections on any affine open in <=0 for natural t-structure - i.e. cohomology in degree > 0 vanishes
Thm 3.5.7 F in >= 0 same as derived global sections with compact support on any affine open in >= 0 for natural t-structure - i.e. cohomology with compact support vanishes in degrees < 0
Discuss perverse t-exactness of functors for affine morphisms
f_* right t-exact, f_! left exact
In particular, if we are looking at a locally closed embedding that is also an affine morphism, then h_! and h_* are t-exact
pg 167 - 168
For a smooth morphism f define the functors f upper and lower dagger. Discuss perverse counterparts. Prove exactness results and adjuntion. Discuss faithfulness
upper is t-exact
lower is left exact
168-169
If f is a smooth surjective morphism, upper dagger is faithful - fully faithful if f has connected fibers
pg 172
Discuss smooth descent
Given f:X –> Y and a perverse sheaf on X, how can we tell if it comes from a perverse sheaf on Y? If it is, how can we recover the original sheaf on Y?
Descent data: thm 3.7.4 for smooth surjective morphisms, upper dagger is an equivalence of categories - Perv(Y,k) and Desc(f, k)
pg 173 - 180
Define: semismall, small morphisms and their stratified counterparts
Discuss key theorems related to these morphisms
Thm. 3.8.4 If f:X –> Y is a proper, semismall morphism and X is smooth, connected, then the pushforward of any shifted local system of finite type is in Perv(Y,k)
Thm. 3.8.9 For a proper stratified semismall morphism, pushforwad is t-exact for perverse t-structure
pg 181-184
What is the Decomposition theorem?
If f is a proper morphism of varieties, then pushforward perserves semisimple complexes
pg 185
What is Relative hard Lefschetz theorem? Relationship with decomposition theorem?
pg 185 - 190
