Safe Haskell	None
Language	Haskell2010

Math.RootLoci.Geometry.Cohomology

Contents

The non-equivariant case
The equivariant case
Conversion between different bases

Description

Bases in the cohomology of the spaces appearing in the computations.

We have three different spaces:

Q^n = P^1 x P^1 x ... x P^1 (n times; m = length lambda)
Q^m = P^1 x P^1 x ... x P^1 x P^1 (m times, m = sum lambda >= n)
```
P^m = P(Sym^m C^2)
```

Furthermore, we have GL2 acting naturally on these spaces.

Synopsis

newtype U = U Int
newtype H = H Int
newtype G = G Int
newtype US = US [U]
newtype HS = HS [H]
data Omega ab = Omega ![Int] !ab
data Eta ab = Eta ![Int] !ab
data Gam ab = Gam !Int !ab
class Functor f => Equivariant f where
- injectMonom :: x -> f x
- projectMonom :: f x -> x
injectZMod :: (Equivariant f, ChernBase base, Ord (f base)) => ZMod base -> ZMod (f base)
forgetGamma :: Ord base => ZMod (Gam base) -> ZMod base
forgetEquiv :: ChernBase base => ZMod (Gam base) -> ZMod G
convertOmega :: (Ord ab, Ord cd) => (ZMod ab -> ZMod cd) -> ZMod (Omega ab) -> ZMod (Omega cd)
convertEta :: (Ord ab, Ord cd) => (ZMod ab -> ZMod cd) -> ZMod (Eta ab) -> ZMod (Eta cd)
convertGam :: (Ord ab, Ord cd) => (ZMod ab -> ZMod cd) -> ZMod (Gam ab) -> ZMod (Gam cd)
convertEach :: forall f x y ab cd. (Functor f, Ord ab, Ord cd, Ord (f ab), Ord (f cd), Ord x) => (forall y. f y -> x) -> (forall y. f y -> y) -> (forall y. x -> y -> f y) -> (ZMod ab -> ZMod cd) -> ZMod (f ab) -> ZMod (f cd)
unsafeEtaToOmega :: (Eq coeff, Num coeff, Ord ab) => FreeMod coeff (Eta ab) -> FreeMod coeff (Omega ab)
unsafeOmegaToEta :: (Eq coeff, Num coeff, Ord ab) => FreeMod coeff (Omega ab) -> FreeMod coeff (Eta ab)

The non-equivariant case

newtype U Source #

a (ring) generator of H^*(Q^n) (note that u_i^2 = 0)

Constructors

U Int

Instances

Instances details

Eq U Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods (==) :: U -> U -> Bool # (/=) :: U -> U -> Bool #
Ord U Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods compare :: U -> U -> Ordering # (<) :: U -> U -> Bool # (<=) :: U -> U -> Bool # (>) :: U -> U -> Bool # (>=) :: U -> U -> Bool # max :: U -> U -> U # min :: U -> U -> U #
Show U Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods showsPrec :: Int -> U -> ShowS # show :: U -> String # showList :: [U] -> ShowS #
Pretty U Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods pretty :: U -> String # prettyInParens :: U -> String #
Graded U Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods grade :: U -> Int Source #

newtype H Source #

(a ring) generator of H^*(Q^m) (note that h_i^2 = 0)

Constructors

H Int

Instances

Instances details

Eq H Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods (==) :: H -> H -> Bool # (/=) :: H -> H -> Bool #
Ord H Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods compare :: H -> H -> Ordering # (<) :: H -> H -> Bool # (<=) :: H -> H -> Bool # (>) :: H -> H -> Bool # (>=) :: H -> H -> Bool # max :: H -> H -> H # min :: H -> H -> H #
Show H Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods showsPrec :: Int -> H -> ShowS # show :: H -> String # showList :: [H] -> ShowS #
Pretty H Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods pretty :: H -> String # prettyInParens :: H -> String #
Graded H Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods grade :: H -> Int Source #

newtype G Source #

the generator of H^*(P^n) (with g^(n+1) = 0)

Constructors

G Int

Instances

Instances details

Eq G Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods (==) :: G -> G -> Bool # (/=) :: G -> G -> Bool #
Ord G Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods compare :: G -> G -> Ordering # (<) :: G -> G -> Bool # (<=) :: G -> G -> Bool # (>) :: G -> G -> Bool # (>=) :: G -> G -> Bool # max :: G -> G -> G # min :: G -> G -> G #
Show G Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods showsPrec :: Int -> G -> ShowS # show :: G -> String # showList :: [G] -> ShowS #
Semigroup G Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods (<>) :: G -> G -> G # sconcat :: NonEmpty G -> G # stimes :: Integral b => b -> G -> G #
Monoid G Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods mempty :: G # mappend :: G -> G -> G # mconcat :: [G] -> G #
Pretty G Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods pretty :: G -> String # prettyInParens :: G -> String #
Graded G Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods grade :: G -> Int Source #

newtype US Source #

A monomial in u_i (encoded as a subset of [1..n], as u_i^2=0)

Constructors

US [U]

Instances

Instances details

Eq US Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods (==) :: US -> US -> Bool # (/=) :: US -> US -> Bool #
Ord US Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods compare :: US -> US -> Ordering # (<) :: US -> US -> Bool # (<=) :: US -> US -> Bool # (>) :: US -> US -> Bool # (>=) :: US -> US -> Bool # max :: US -> US -> US # min :: US -> US -> US #
Show US Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods showsPrec :: Int -> US -> ShowS # show :: US -> String # showList :: [US] -> ShowS #
Semigroup US Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods (<>) :: US -> US -> US # sconcat :: NonEmpty US -> US # stimes :: Integral b => b -> US -> US #
Monoid US Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods mempty :: US # mappend :: US -> US -> US # mconcat :: [US] -> US #
Pretty US Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods pretty :: US -> String # prettyInParens :: US -> String #
Graded US Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods grade :: US -> Int Source #

newtype HS Source #

A monomial in h_i (encoded as a subset of [1..m], as h_i^2=0)

Constructors

HS [H]

Instances

Instances details

Eq HS Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods (==) :: HS -> HS -> Bool # (/=) :: HS -> HS -> Bool #
Ord HS Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods compare :: HS -> HS -> Ordering # (<) :: HS -> HS -> Bool # (<=) :: HS -> HS -> Bool # (>) :: HS -> HS -> Bool # (>=) :: HS -> HS -> Bool # max :: HS -> HS -> HS # min :: HS -> HS -> HS #
Show HS Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods showsPrec :: Int -> HS -> ShowS # show :: HS -> String # showList :: [HS] -> ShowS #
Semigroup HS Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods (<>) :: HS -> HS -> HS # sconcat :: NonEmpty HS -> HS # stimes :: Integral b => b -> HS -> HS #
Monoid HS Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods mempty :: HS # mappend :: HS -> HS -> HS # mconcat :: [HS] -> HS #
Pretty HS Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods pretty :: HS -> String # prettyInParens :: HS -> String #
Graded HS Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods grade :: HS -> Int Source #

The equivariant case

data Omega ab Source #

A monomial generator of Z[alpha,beta;u1,u2,...,u_nd]/(...), the cohomology ring of Q^n.

The encoding is that the list is the list of indices of u which appear.

Constructors

Omega ![Int] !ab

Instances

Instances details

Functor Omega Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods fmap :: (a -> b) -> Omega a -> Omega b # (<$) :: a -> Omega b -> Omega a #
Equivariant Omega Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods injectMonom :: x -> Omega x Source # projectMonom :: Omega x -> x Source #
Eq ab => Eq (Omega ab) Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods (==) :: Omega ab -> Omega ab -> Bool # (/=) :: Omega ab -> Omega ab -> Bool #
Ord ab => Ord (Omega ab) Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods compare :: Omega ab -> Omega ab -> Ordering # (<) :: Omega ab -> Omega ab -> Bool # (<=) :: Omega ab -> Omega ab -> Bool # (>) :: Omega ab -> Omega ab -> Bool # (>=) :: Omega ab -> Omega ab -> Bool # max :: Omega ab -> Omega ab -> Omega ab # min :: Omega ab -> Omega ab -> Omega ab #
Show ab => Show (Omega ab) Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods showsPrec :: Int -> Omega ab -> ShowS # show :: Omega ab -> String # showList :: [Omega ab] -> ShowS #
Semigroup ab => Semigroup (Omega ab) Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods (<>) :: Omega ab -> Omega ab -> Omega ab # sconcat :: NonEmpty (Omega ab) -> Omega ab # stimes :: Integral b => b -> Omega ab -> Omega ab #
Monoid ab => Monoid (Omega ab) Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods mempty :: Omega ab # mappend :: Omega ab -> Omega ab -> Omega ab # mconcat :: [Omega ab] -> Omega ab #
(Pretty ab, Monoid ab, Eq ab) => Pretty (Omega ab) Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods pretty :: Omega ab -> String # prettyInParens :: Omega ab -> String #
Graded ab => Graded (Omega ab) Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods grade :: Omega ab -> Int Source #

data Eta ab Source #

A monomial generator of Z[alpha,beta;eta1,eta2...eta_m]/(...), he cohomology ring of Q^m.

The encoding is that the list is the list of indices of eta which appear.

Constructors

Eta ![Int] !ab

Instances

Instances details

Functor Eta Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods fmap :: (a -> b) -> Eta a -> Eta b # (<$) :: a -> Eta b -> Eta a #
Equivariant Eta Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods injectMonom :: x -> Eta x Source # projectMonom :: Eta x -> x Source #
Eq ab => Eq (Eta ab) Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods (==) :: Eta ab -> Eta ab -> Bool # (/=) :: Eta ab -> Eta ab -> Bool #
Ord ab => Ord (Eta ab) Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods compare :: Eta ab -> Eta ab -> Ordering # (<) :: Eta ab -> Eta ab -> Bool # (<=) :: Eta ab -> Eta ab -> Bool # (>) :: Eta ab -> Eta ab -> Bool # (>=) :: Eta ab -> Eta ab -> Bool # max :: Eta ab -> Eta ab -> Eta ab # min :: Eta ab -> Eta ab -> Eta ab #
Show ab => Show (Eta ab) Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods showsPrec :: Int -> Eta ab -> ShowS # show :: Eta ab -> String # showList :: [Eta ab] -> ShowS #
Semigroup ab => Semigroup (Eta ab) Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods (<>) :: Eta ab -> Eta ab -> Eta ab # sconcat :: NonEmpty (Eta ab) -> Eta ab # stimes :: Integral b => b -> Eta ab -> Eta ab #
Monoid ab => Monoid (Eta ab) Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods mempty :: Eta ab # mappend :: Eta ab -> Eta ab -> Eta ab # mconcat :: [Eta ab] -> Eta ab #
(Pretty ab, Monoid ab, Eq ab) => Pretty (Eta ab) Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods pretty :: Eta ab -> String # prettyInParens :: Eta ab -> String #
Graded ab => Graded (Eta ab) Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods grade :: Eta ab -> Int Source #

data Gam ab Source #

A monomial generator of Z[alpha,beta;gamma]/(...), the cohomology ring of P^m.

Constructors

Gam !Int !ab

Instances

Instances details

Functor Gam Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods fmap :: (a -> b) -> Gam a -> Gam b # (<$) :: a -> Gam b -> Gam a #
Equivariant Gam Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods injectMonom :: x -> Gam x Source # projectMonom :: Gam x -> x Source #
Eq ab => Eq (Gam ab) Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods (==) :: Gam ab -> Gam ab -> Bool # (/=) :: Gam ab -> Gam ab -> Bool #
Ord ab => Ord (Gam ab) Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods compare :: Gam ab -> Gam ab -> Ordering # (<) :: Gam ab -> Gam ab -> Bool # (<=) :: Gam ab -> Gam ab -> Bool # (>) :: Gam ab -> Gam ab -> Bool # (>=) :: Gam ab -> Gam ab -> Bool # max :: Gam ab -> Gam ab -> Gam ab # min :: Gam ab -> Gam ab -> Gam ab #
Show ab => Show (Gam ab) Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods showsPrec :: Int -> Gam ab -> ShowS # show :: Gam ab -> String # showList :: [Gam ab] -> ShowS #
Semigroup ab => Semigroup (Gam ab) Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods (<>) :: Gam ab -> Gam ab -> Gam ab # sconcat :: NonEmpty (Gam ab) -> Gam ab # stimes :: Integral b => b -> Gam ab -> Gam ab #
Monoid ab => Monoid (Gam ab) Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods mempty :: Gam ab # mappend :: Gam ab -> Gam ab -> Gam ab # mconcat :: [Gam ab] -> Gam ab #
(Pretty ab, Monoid ab, Eq ab) => Pretty (Gam ab) Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods pretty :: Gam ab -> String # prettyInParens :: Gam ab -> String #
Graded ab => Graded (Gam ab) Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods grade :: Gam ab -> Int Source #

class Functor f => Equivariant f where Source #

Class of monomial bases which form modules over the H^*(BGL2)

Methods

injectMonom :: x -> f x Source #

projectMonom :: f x -> x Source #

Instances

Instances details

Equivariant Gam Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods injectMonom :: x -> Gam x Source # projectMonom :: Gam x -> x Source #
Equivariant Eta Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods injectMonom :: x -> Eta x Source # projectMonom :: Eta x -> x Source #
Equivariant Omega Source #
Instance details Defined in Math.RootLoci.Geometry.Cohomology Methods injectMonom :: x -> Omega x Source # projectMonom :: Omega x -> x Source #

injectZMod :: (Equivariant f, ChernBase base, Ord (f base)) => ZMod base -> ZMod (f base) Source #

forgetGamma :: Ord base => ZMod (Gam base) -> ZMod base Source #

forgetEquiv :: ChernBase base => ZMod (Gam base) -> ZMod G Source #

Conversion between different bases

convertOmega :: (Ord ab, Ord cd) => (ZMod ab -> ZMod cd) -> ZMod (Omega ab) -> ZMod (Omega cd) Source #

convertEta :: (Ord ab, Ord cd) => (ZMod ab -> ZMod cd) -> ZMod (Eta ab) -> ZMod (Eta cd) Source #

convertGam :: (Ord ab, Ord cd) => (ZMod ab -> ZMod cd) -> ZMod (Gam ab) -> ZMod (Gam cd) Source #

convertEach :: forall f x y ab cd. (Functor f, Ord ab, Ord cd, Ord (f ab), Ord (f cd), Ord x) => (forall y. f y -> x) -> (forall y. f y -> y) -> (forall y. x -> y -> f y) -> (ZMod ab -> ZMod cd) -> ZMod (f ab) -> ZMod (f cd) Source #

A generic function which can convert the GL2 representations

unsafeEtaToOmega :: (Eq coeff, Num coeff, Ord ab) => FreeMod coeff (Eta ab) -> FreeMod coeff (Omega ab) Source #

This is a hack to reuse the same pushforward code

unsafeOmegaToEta :: (Eq coeff, Num coeff, Ord ab) => FreeMod coeff (Omega ab) -> FreeMod coeff (Eta ab) Source #