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PencilsOfQuadrics :: cliffordModuleToMatrixFactorization

cliffordModuleToMatrixFactorization -- reads off a matrix factorization from a Clifford module

Synopsis

Description

Part of the series of explicit functors giving category equivalences:

cliffordModule

cliffordModuleToCIResolution

cliffordModuleToMatrixFactorization

ciModuleToMatrixFactorization

ciModuleToCliffordModule

A Clifford module M on the Clifford algebra C:=Cliff(qq) of a quadratic form qq has keys evenOperator and oddOperator, the list of the even operators uEv_i : M_0 \to M_1 and the odd operators uOdd_i : M_1 \to M_0, which form a representation of C.

From this representation we read off a matrix factorization (M1, M2) of qq.

i1 : kk=ZZ/101;
i2 : g=1;
i3 : rNP=randNicePencil(kk,g);
i4 : qq=rNP.quadraticForm;
i5 : S=rNP.qqRing;
i6 : cM=cliffordModule(rNP.matFact1,rNP.matFact2,rNP.baseRing)

o6 = CliffordModule{...6...}

o6 : CliffordModule
i7 : (M1,M2)=cliffordModuleToMatrixFactorization(cM,S);
i8 : r=rank source M1

o8 = 8
i9 : M1*M2 - qq*id_(S^r) == 0

o9 = true
i10 : M1 == rNP.matFact1

o10 = true
i11 : M2 == rNP.matFact2

o11 = true
i12 : cMu=cliffordModule(rNP.matFactu1,rNP.matFactu2,rNP.baseRing)

o12 = CliffordModule{...6...}

o12 : CliffordModule
i13 : (Mu1,Mu2)=cliffordModuleToMatrixFactorization(cMu,S);
i14 : ru=rank source Mu1

o14 = 4
i15 : Mu1*Mu2 - qq*id_(S^ru) == 0

o15 = true
i16 : Mu1 == rNP.matFactu1

o16 = true
i17 : Mu2 == rNP.matFactu2

o17 = true

See also

Ways to use cliffordModuleToMatrixFactorization :

For the programmer

The object cliffordModuleToMatrixFactorization is a method function.