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ToricHigherDirectImages for JSAG #4000

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92 changes: 52 additions & 40 deletions M2/Macaulay2/packages/ToricHigherDirectImages.m2
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Original file line number Diff line number Diff line change
Expand Up @@ -34,6 +34,7 @@ newPackage(
export {
-- types
-- methods
"frobeniusPushforward",
"frobeniusDirectImage",
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A cleaner way to declare a synonym would be to write

"frobeniusDirectImage" => "frobeniusPushforward"

(on line 38) and then get rid of anythin that mentions frobeniusDirectImage in the rest of code and documentation.

"nefContraction",
"nefRayContractions",
Expand All @@ -55,11 +56,12 @@ importFrom(Core, {"sortBy", "raw", "rawHilbertBasis"})
-- M - a module over a multigraded polynomial ring.
-- OUTPUT: the module (F_p)_* M, i.e. the pushforward of M under the p-th toric
-- Frobenius map.
frobeniusDirectImage = method();
frobeniusDirectImage (ZZ, Module) := Module => (p, M) -> (
frobeniusPushforward = method();
frobeniusDirectImage = frobeniusPushforward;
frobeniusPushforward (ZZ, Module) := Module => (p, M) -> (
if M == 0 then return M;
-- If it isn't free
if not isFreeModule M then coker frobeniusDirectImage_p presentation M else (
if not isFreeModule M then coker frobeniusPushforward_p presentation M else (
S := ring M;
X := variety S;
A := matrix rays X;
Expand All @@ -76,7 +78,7 @@ frobeniusDirectImage (ZZ, Module) := Module => (p, M) -> (
);
dSum := directSum apply(B, b -> S^b);
-- We should remember our choice of Cartier divisors.
dSum.cache.frobeniusDirectImage = D;
dSum.cache.frobeniusPushforward = D;
dSum
)
)
Expand All @@ -86,22 +88,22 @@ frobeniusDirectImage (ZZ, Module) := Module => (p, M) -> (
-- INPUT: p - an integer
-- f - a map between multigraded modules f : M -> N.
-- OUTPUT: the map between modules (F_p)_* f : (F_p)_* M -> (F_p)_* N.
frobeniusDirectImage (ZZ, Matrix) := (p, f) -> (
frobeniusPushforward (ZZ, Matrix) := (p, f) -> (
S := ring f;
-- easy way to access the variety which these modules are over.
X := variety S;
-- these index the characters splitting the source and target.
pts := toList (toList (dim X:{0})..toList (dim X:{p-1}));
-- our map will be a block matrix
blocksize := #pts;
src := frobeniusDirectImage_p source f;
tgt := frobeniusDirectImage_p target f;
src := frobeniusPushforward_p source f;
tgt := frobeniusPushforward_p target f;
A := matrix rays X;
-- this is the main reason to keep track of the divisors.
-- we need them to make sure that the map is sending terms to the correct place.
-- e.g. the trivial summand should map to the trivial summand.
Dsrc := src.cache.frobeniusDirectImage;
Dtgt := tgt.cache.frobeniusDirectImage;
Dsrc := src.cache.frobeniusPushforward;
Dtgt := tgt.cache.frobeniusPushforward;
-- to make the direct image of the map
matrix table(numrows f, numcols f, (r,c) -> (
-- we want to convert the entries of the input map
Expand Down Expand Up @@ -141,10 +143,10 @@ frobeniusDirectImage (ZZ, Matrix) := (p, f) -> (
-- OUTPUT: the pushforward of the complex C, by applying the functor to each differential.
-- NOTE: since the Frobenius is a finite map, the pushforward is an exact functor, i.e.
-- there are no higher direct images.
frobeniusDirectImage (ZZ, Complex) := Complex => (p, K) -> (
frobeniusPushforward (ZZ, Complex) := Complex => (p, K) -> (
complex (
if length K == 0 then map(frobeniusDirectImage_p K_0, 0, 0)
else apply(length K, i -> frobeniusDirectImage(p, K.dd_(i+1)))
if length K == 0 then map(frobeniusPushforward_p K_0, 0, 0)
else apply(length K, i -> frobeniusPushforward(p, K.dd_(i+1)))
)
)

Expand Down Expand Up @@ -559,13 +561,13 @@ doc ///
Text
2) When $\varphi \colon Y \rightarrow Y$ is a toric Frobenius map, i.e.
induced from the natural morphism on the dense torus given by raising all
of the coordinates to the same power, then the method \texttt{frobeniusDirectImage}
of the coordinates to the same power, then the method \texttt{frobeniusPushforward}
allows one to compute pushforwards of any coherent sheaf. For instance,
the pushforward of the trivial bundle on $\mathbb{P}^2$ by the second Frobenius map
splits into 4 line bundles.
Example
S = ring Y;
frobeniusDirectImage(2,S^1)
frobeniusPushforward(2,S^1)
Text
@SUBSECTION "References"@
Text
Expand All @@ -585,13 +587,17 @@ doc ///

doc ///
Key
frobeniusPushforward
frobeniusDirectImage
(frobeniusPushforward, ZZ, Module)
(frobeniusDirectImage, ZZ, Module)
Headline
compute the pushforward of a module under the $p$th toric Frobenius map
Usage
frobeniusDirectImage(p, M)
frobeniusDirectImage_p M
frobeniusPushforward(p, M)
frobeniusPushforward_p M
frobeniusDirectImage(p,M)
frobeniusDirectImage_p M
Inputs
p : ZZ
M : Module
Expand All @@ -611,43 +617,46 @@ doc ///
Example
X = hirzebruchSurface 1;
R = ring X;
frobeniusDirectImage(4, R^1)
frobeniusPushforward(4, R^1)
Text
We can pushforward many kinds of modules. Here is the pushforward of an ideal.
Example
I = module ideal {R_0, R_2};
prune frobeniusDirectImage(2, I)
prune frobeniusPushforward(2, I)
Text
Here is the pushforward of a free module.
Example
F = R^{{1,0},{0,1}};
frobeniusDirectImage(2, F)
frobeniusPushforward(2, F)
Text
Here is the pushforward of a torsion module.
Example
M = R^1/(module I)
prune frobeniusDirectImage(2, M)
prune frobeniusPushforward(2, M)
Text
As mentioned, $p$ is not related to the characteristic of the field, and the
outputs will be the same modules over a different coefficient ring.
Example
Y = hirzebruchSurface(1, CoefficientRing => ZZ/2);
S = ring Y;
I' = module ideal {S_0, S_2};
prune frobeniusDirectImage(2, I')
prune frobeniusPushforward(2, I')
SeeAlso
(frobeniusDirectImage, ZZ, Matrix)
(frobeniusDirectImage, ZZ, Complex)
(frobeniusPushforward, ZZ, Matrix)
(frobeniusPushforward, ZZ, Complex)
///

doc ///
Key
(frobeniusPushforward, ZZ, Matrix)
(frobeniusDirectImage, ZZ, Matrix)
Headline
compute the pushforward of map of modules under the $p$th toric Frobenius map
Usage
frobeniusDirectImage(p, M)
frobeniusDirectImage_p M
frobeniusPushforward(p, M)
frobeniusPushforward_p M
frobeniusDirectImage(p, M)
frobeniusDirectImage_p M
Inputs
p : ZZ
f : Matrix
Expand All @@ -670,20 +679,23 @@ doc ///
X = hirzebruchSurface 1;
R = ring X;
M = koszul_1 vars R
frobeniusDirectImage_2 M
frobeniusPushforward_2 M
SeeAlso
(frobeniusDirectImage, ZZ, Module)
(frobeniusDirectImage, ZZ, Complex)
(frobeniusPushforward, ZZ, Module)
(frobeniusPushforward, ZZ, Complex)
///

doc ///
Key
(frobeniusPushforward, ZZ, Complex)
(frobeniusDirectImage, ZZ, Complex)
Headline
compute the pushforward of a complex of modules under the $p$th toric Frobenius map
Usage
frobeniusDirectImage(p, C)
frobeniusDirectImage_p C
frobeniusPushforward(p, C)
frobeniusPushforward_p C
frobeniusDirectImage(p, C)
frobeniusDirectImage_p C
Inputs
p : ZZ
C : Complex
Expand All @@ -706,10 +718,10 @@ doc ///
X = hirzebruchSurface 1;
R = ring X;
C = complex koszul vars R
frobeniusDirectImage_2 C
frobeniusPushforward_2 C
SeeAlso
(frobeniusDirectImage, ZZ, Module)
(frobeniusDirectImage, ZZ, Matrix)
(frobeniusPushforward, ZZ, Module)
(frobeniusPushforward, ZZ, Matrix)
///

doc ///
Expand Down Expand Up @@ -931,9 +943,9 @@ doc ///
RL = prune phi_*^1 L
annihilator RL
SeeAlso
(frobeniusDirectImage, ZZ, Module)
(frobeniusDirectImage, ZZ, Matrix)
(frobeniusDirectImage, ZZ, Complex)
(frobeniusPushforward, ZZ, Module)
(frobeniusPushforward, ZZ, Matrix)
(frobeniusPushforward, ZZ, Complex)
///

------------------------------------------------------------------------------
Expand All @@ -943,7 +955,7 @@ doc ///
TEST ///
X = toricProjectiveSpace 2;
S = ring X;
FS = frobeniusDirectImage_2 S^1;
FS = frobeniusPushforward_2 S^1;
assert(degrees FS == {{0}, {1}, {1}, {1}})
///

Expand All @@ -952,14 +964,14 @@ X = smoothFanoToricVariety(3,7);
d = dim X;
p = 4;
S = ring X;
FS = frobeniusDirectImage_p S^1
FS = frobeniusPushforward_p S^1
assert(rank FS == p^d)
///

TEST ///
X = hirzebruchSurface 3;
S = ring X;
M = frobeniusDirectImage_2 koszul_1 vars S;
M = frobeniusPushforward_2 koszul_1 vars S;
M' = map(S^{{0, 0}, {1, -1}, {-1, 0}, {1, -1}},
S^{{-1, 0}, {0, -1}, {-1, 0}, {1, -1}, {1, -1}, {2, -1}, {1, -1}, {3, -1},
{-1, 0}, {0, -1}, {-1, 0}, {1, -1}, {0, -1}, {1, -1}, {-1, -1}, {1, -1}},
Expand All @@ -976,7 +988,7 @@ d = dim X;
p = 2;
S = ring X;
C = complex koszul vars S;
FC = frobeniusDirectImage_p C;
FC = frobeniusPushforward_p C;
assert(isWellDefined FC)
assert(length FC == length C)
assert(all(length C, i -> rank FC_i == (rank C_i) * p^d))
Expand Down