# cliffordModuleToCIResolution -- transforms a Clifford module to a resolution over a complete intersection ring

## Synopsis

• Usage:
F=cliffordModuleToCIResolution(M,S,CI)
• Inputs:
• M, an instance of the type CliffordModule, a Clifford module on a hyperelliptic curve of genus g
• S, a ring, a polynomial ring in x_0..y_(g-1),z_1,z_2,s,t
• CI, a ring, a complete intersection S/ideal(q1,q2) where qq=s*q1+t*q2
• Outputs:
• F, , a resolution which represents a module over CI

## Description

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