Macaulay2 » Documentation
Packages » PencilsOfQuadrics :: cliffordModuleToMatrixFactorization
next | previous | forward | backward | up | index | toc

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

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 : setRandomSeed 0
 -- setting random seed to 0

o2 = 0
i3 : g=1;
i4 : rNP=randNicePencil(kk,g);
i5 : qq=rNP.quadraticForm;
i6 : S=rNP.qqRing;
i7 : cM=cliffordModule(rNP.matFact1,rNP.matFact2,rNP.baseRing)

o7 = CliffordModule{...6...}

o7 : CliffordModule
i8 : (M1,M2)=cliffordModuleToMatrixFactorization(cM,S);
i9 : r=rank source M1

o9 = 8
i10 : M1*M2 - qq*id_(S^r) == 0

o10 = true
i11 : M1 == rNP.matFact1

o11 = true
i12 : M2 == rNP.matFact2

o12 = true
i13 : cMu=cliffordModule(rNP.matFactu1,rNP.matFactu2,rNP.baseRing)

o13 = CliffordModule{...6...}

o13 : CliffordModule
i14 : (Mu1,Mu2)=cliffordModuleToMatrixFactorization(cMu,S);
i15 : ru=rank source Mu1

o15 = 4
i16 : Mu1*Mu2 - qq*id_(S^ru) == 0

o16 = true
i17 : Mu1 == rNP.matFactu1

o17 = true
i18 : Mu2 == rNP.matFactu2

o18 = true

See also

Ways to use cliffordModuleToMatrixFactorization:

  • cliffordModuleToMatrixFactorization(CliffordModule,Ring)

For the programmer

The object cliffordModuleToMatrixFactorization is a method function.


The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/PencilsOfQuadrics.m2:3260:0.