## <#GAPDoc Label="HorrocksMumford">
## <Subsection Label="HorrocksMumford">
## <Heading>Horrocks Mumford bundle</Heading>
## This example computes the global sections module of the Horrocks-Mumford bundle.
## <Example><![CDATA[
## gap> LoadPackage( "GradedRingForHomalg" );;
## gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x0..x4";;
## gap> S := GradedRing( R );;
## gap> A := KoszulDualRing( S, "e0..e4" );;
## gap> LoadPackage( "GradedModules" );;
## gap> mat := HomalgMatrix( "[ \
## > e1*e4, e2*e0, e3*e1, e4*e2, e0*e3, \
## > e2*e3, e3*e4, e4*e0, e0*e1, e1*e2 \
## > ]",
## > 2, 5, A );
## <A 2 x 5 matrix over a graded ring>
## gap> phi := GradedMap( mat, "free", "free", "left", A );;
## gap> IsMorphism( phi );
## true
## gap> M := GuessModuleOfGlobalSectionsFromATateMap( 2, phi );
## #I GuessModuleOfGlobalSectionsFromATateMap uses a heuristic for efficiency;
## please check the correctness of the following result
##
## <A graded left module presented by yet unknown relations for 19 generators>
## gap> IsPure( M );
## true
## gap> Rank( M );
## 2
## gap> Display( BettiTable( Resolution( M ) ) );
## total: 19 35 20 2
## --------------------
## 3: 4 . . .
## 4: 15 35 20 .
## 5: . . . 2
## --------------------
## degree: 0 1 2 3
## gap> Display( BettiTable( TateResolution( M, -5, 5 ) ) );
## total: 100 37 14 10 5 2 5 10 14 37 100 ? ? ? ?
## ----------|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
## 4: 100 35 4 . . . . . . . . 0 0 0 0
## 3: * . 2 10 10 5 . . . . . . 0 0 0
## 2: * * . . . . . 2 . . . . . 0 0
## 1: * * * . . . . . . 5 10 10 2 . 0
## 0: * * * * . . . . . . . . 4 35 100
## ----------|---|---|---|---|---|---|---|---|---|---|---|---|---|---S
## twist: -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5
## -------------------------------------------------------------------
## Euler: 100 35 2 -10 -10 -5 0 2 0 -5 -10 -10 2 35 100
## gap> M;
## <A graded reflexive non-projective rank 2 left module presented by 99 \
## relations for 19 generators>
## gap> P := ElementOfGrothendieckGroup( M );
## ( 2*O_{P^4} - 1*O_{P^3} - 4*O_{P^2} - 2*O_{P^1} ) -> P^4
## gap> P!.DisplayTwistedCoefficients := true;
## true
## gap> P;
## ( 2*O(-3) - 10*O(-2) + 15*O(-1) - 5*O(0) ) -> P^4
## gap> chi := HilbertPolynomial( M );
## 1/12*t^4+2/3*t^3-1/12*t^2-17/3*t-5
## gap> c := ChernPolynomial( M );
## ( 2 | 1-h+4*h^2 ) -> P^4
## gap> ChernPolynomial( M * S^3 );
## ( 2 | 1+5*h+10*h^2 ) -> P^4
## gap> ch := ChernCharacter( M );
## [ 2-u-7*u^2/2!+11*u^3/3!+17*u^4/4! ] -> P^4
## gap> HilbertPolynomial( ch );
## 1/12*t^4+2/3*t^3-1/12*t^2-17/3*t-5
## gap> List( [ -8 .. 7 ], i -> Value( chi, i ) );
## [ 35, 2, -10, -10, -5, 0, 2, 0, -5, -10, -10, 2, 35, 100, 210, 380 ]
## gap> HF := HilbertFunction( M );
## function( t ) ... end
## gap> List( [ 0 .. 7 ], HF );
## [ 0, 0, 0, 4, 35, 100, 210, 380 ]
## gap> IndexOfRegularity( M );
## 4
## gap> DataOfHilbertFunction( M );
## [ [ [ 4 ], [ 3 ] ], 1/12*t^4+2/3*t^3-1/12*t^2-17/3*t-5 ]
## ]]></Example>
## </Subsection>
## <#/GAPDoc>
LoadPackage( "GradedRingForHomalg", false );;
R := HomalgFieldOfRationalsInDefaultCAS( ) * "x0..x4";;
S := GradedRing( R );;
A := KoszulDualRing( S, "e0..e4" );;
LoadPackage( "GradedModules", false );;
## [EFS, Example 7.3]:
## A famous Beilinson monad was discovered by Horrocks and Mumford [HM]:
mat := HomalgMatrix( "[ \
e1*e4, e2*e0, e3*e1, e4*e2, e0*e3, \
e2*e3, e3*e4, e4*e0, e0*e1, e1*e2 \
]",
2, 5, A );
phi := GradedMap( mat, "free", "free", "left", A );;
IsMorphism( phi );
M := GuessModuleOfGlobalSectionsFromATateMap( 2, phi );
IsPure( M );
Rank( M );
Display( BettiTable( Resolution( M ) ) );
CT := BettiTable( TateResolution( M, -4, 6 ) );
Display( CT );
chi := HilbertPolynomial( M );
c := ChernPolynomial( M );
ch := ChernCharacter( M );
Assert( 0, HilbertPolynomial( ch ) = chi );
Assert( 0,
List( [ -8 .. 7 ], i -> Value( chi, i ) ) =
[ 35, 2, -10, -10, -5, 0, 2, 0, -5, -10, -10, 2, 35, 100, 210, 380 ] );
Assert( 0, DataOfHilbertFunction( M )[ 1 ] = [ [ 4 ], [ 3 ] ] );
HF := HilbertFunction( M );
Assert( 0,
List( [ 0 .. 7 ], HF ) =
[ 0, 0, 0, 4, 35, 100, 210, 380 ] );
