Part of the series of explicit functors giving category equivalences:
cliffordModule
cliffordModuleToCIResolution
cliffordModuleToMatrixFactorization
ciModuleToMatrixFactorization
ciModuleToCliffordModule
From Clifford module M on the Clifford algebra C:=Cliff(qq) of a quadratic form qq=s*q1+t*q2, we may construct a module over CI=P/ideal(q1,q2) where P is a polynomial ring in x_0..y_{(g-1)},z_1,z_2. This function returns a part of its minimal free resolution over CI. This function uses cliffordModuleToMatrixFactorization.
i1 : kk=ZZ/101; |
i2 : g=1; |
i3 : rNP=randNicePencil(kk,g); |
i4 : qq=rNP.quadraticForm; |
i5 : S=rNP.qqRing; |
i6 : P=kk[drop(gens S,-2)] o6 = P o6 : PolynomialRing |
i7 : qs=sub(diff(matrix{{S_(2*g+2), S_(2*g+3)}}, qq), P)
o7 = | x_0y_0-z_1^2 -5x_0^2+12x_0z_1-18y_0z_1-30z_1^2-12x_0z_2+10y_0z_2+6z_1z_2-48z_2^2 |
1 2
o7 : Matrix P <--- P
|
i8 : CI=P/ideal qs o8 = CI o8 : QuotientRing |
i9 : cM=cliffordModule(rNP.matFact1,rNP.matFact2,rNP.baseRing)
o9 = CliffordModule{...6...}
o9 : CliffordModule
|
i10 : betti (F=cliffordModuleToCIResolution(cM,S,CI))
0 1 2 3 4 5
o10 = total: 20 16 12 8 5 5
0: 20 16 12 8 4 1
1: . . . . 1 4
o10 : BettiTally
|
i11 : cMu=cliffordModule(rNP.matFactu1,rNP.matFactu2,rNP.baseRing)
o11 = CliffordModule{...6...}
o11 : CliffordModule
|
i12 : f=cMu.hyperellipticBranchEquation
3 2 2 3 4
o12 = - 12s t - 50s t - 16s*t + 47t
o12 : kk[s, t]
|
i13 : L=randomLineBundle(0,f); |
i14 : betti (FL=cliffordModuleToCIResolution(tensorProduct(cM,L),S,CI))
0 1 2 3 4 5
o14 = total: 12 8 4 4 8 12
-1: 12 8 4 . . .
0: . . . 4 8 12
o14 : BettiTally
|
i15 : betti (FuL=cliffordModuleToCIResolution(tensorProduct(cMu,L),S,CI))
0 1 2 3 4 5
o15 = total: 7 5 3 2 3 5
-1: 7 5 3 1 . .
0: . . . 1 3 5
o15 : BettiTally
|
The object cliffordModuleToCIResolution is a method function.