# ciModuleToCliffordModule -- transforms a module over a complete intersection of 2 quadrics into a Clifford Module.

## Synopsis

• Usage:
C = ciModuleToCliffordModule M
• Inputs:
• M, , module over a complete intersection of 2 quadrics
• Outputs:

## Description

Part of the series of explicit functors giving category equivalences:

cliffordModule

cliffordModuleToCIResolution

cliffordModuleToMatrixFactorization

ciModuleToMatrixFactorization

ciModuleToCliffordModule

This function uses ciModuleToMatrixFactorization, and then calls cliffordModule

 i1 : n = 4 o1 = 4 i2 : c = 2 o2 = 2 i3 : kk = ZZ/101 o3 = kk o3 : QuotientRing i4 : U = kk[x_0..x_(n-1)] o4 = U o4 : PolynomialRing i5 : qq = matrix{{x_0^2+x_1^2,x_0*x_1}} o5 = | x_0^2+x_1^2 x_0x_1 | 1 2 o5 : Matrix U <--- U i6 : qq = random(U^1, U^{2:-2}) o6 = | 24x_0^2-36x_0x_1+19x_1^2-30x_0x_2+19x_1x_2-29x_2^2-29x_0x_3-10x_1x_3- ------------------------------------------------------------------------ 8x_2x_3-22x_3^2 -29x_0^2-24x_0x_1+39x_1^2-38x_0x_2+21x_1x_2+19x_2^2-16x_ ------------------------------------------------------------------------ 0x_3+34x_1x_3-47x_2x_3-39x_3^2 | 1 2 o6 : Matrix U <--- U i7 : Ubar = U/ideal qq o7 = Ubar o7 : QuotientRing i8 : M = coker vars Ubar o8 = cokernel | x_0 x_1 x_2 x_3 | 1 o8 : Ubar-module, quotient of Ubar i9 : M = coker random(Ubar^2, Ubar^{-1,-2,-2}) o9 = cokernel | -18x_0-43x_1-28x_2+38x_3 -16x_1^2+16x_0x_2+39x_1x_2-38x_2^2-48x_0x_3+48x_1x_3+46x_2x_3+22x_3^2 7x_1^2+22x_0x_2+43x_1x_2+33x_2^2-47x_0x_3+36x_1x_3-28x_2x_3-47x_3^2 | | -13x_0-15x_1-47x_2+2x_3 15x_1^2+45x_0x_2-17x_1x_2+40x_2^2+47x_0x_3+35x_1x_3+x_2x_3-23x_3^2 -23x_1^2-34x_0x_2-11x_1x_2+11x_2^2+19x_0x_3+11x_1x_3-3x_2x_3-7x_3^2 | 2 o9 : Ubar-module, quotient of Ubar i10 : C = ciModuleToCliffordModule M o10 = CliffordModule{...6...} o10 : CliffordModule i11 : keys C o11 = {hyperellipticBranchEquation, oddCenter, evenOperators, symmetricM, ----------------------------------------------------------------------- evenCenter, oddOperators} o11 : List i12 : C.evenOperators o12 = {{-1} | 0 0 0 -48t -38t2 -30t2 24st-23t2 -22t2 -3t2 {0} | 0 0 -6 15 -29s+49t 3t 46s-40t -46t -40t {0} | 0 0 41 21 -47s-4t -20s+38t 47s-16t 13s+28t 18s-44t {0} | 0 0 -29 4 7t -11s-36t -47s+35t -46s-27t 6s+39t {0} | 0 0 7 -34 30s+41t 28s-23t 44s-21t -11s-47t -47s+42t {0} | 0 0 12 20 24s+50t 28s+30t -10s+24t 28s-24t -37s+40t {0} | 0 0 0 -44 -40s+16t -49s-41t 32s+25t -43s-3t -7s+45t {0} | 0 0 0 0 0 0 0 0 0 {1} | 0 0 0 0 0 0 0 0 0 {1} | 0 0 0 0 0 0 0 0 0 {2} | 0 0 0 0 0 0 0 0 0 {2} | 0 0 0 0 0 0 0 0 0 ----------------------------------------------------------------------- -31t2 27st2+40t3 0 |, {-1} | 0 0 43t 42t 39st-6t2 -11t -29st-40t2 0 | {0} | 0 0 -43 -44 36s-9t -6s+12t 29s2+22st+12t2 0 | {0} | 0 0 -21 37 5s-4t -44s+26t -30s2+44st+13t2 0 | {0} | 0 0 33 -46 28s-32t 24s-45t -27s2-44st-41t2 0 | {0} | 0 0 4 -5 46s+31t 26s+18t 35st-49t2 0 | {0} | 0 0 -16 28 40s-t 28s+9t 49st-26t2 0 | {0} | 0 0 0 47 -44s 0 0 -39s2t-48st2 | {0} | 0 0 0 0 0 0 0 28st+15t2 | {1} | 0 0 0 0 0 0 0 40st-33t2 | {1} | 0 0 0 0 0 0 0 13s-10t | {2} | 0 0 0 0 0 0 0 -31s+37t | {2} | 0 0 0 0 0 ----------------------------------------------------------------------- 22st+27t2 -9st+13t2 45st+24t2 -22st+29t2 43st-44t2 -7st2+3t3 -32s-41t -7s+26t -23s+14t 28s-38t -11s+9t -28st+16t2 19s-34t 41s-34t -4s+10t -50s-37t 26s-5t -27s2-50st+25t2 s-32t -16s+27t 37s+24t -7s-23t 50s-16t -40s2+16t2 -50s-47t -46s-15t 25s-43t 34s+25t 15s-34t -s2-4st+10t2 -s+50t 27s 26s -23s+27t 3s 39st -23s-35t 35s 13s -45s+14t -45s 43st 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ----------------------------------------------------------------------- 0 |, {-1} | 0 24s+14t 0 0 0 0 0 0 | {0} | 24 -41 0 0 0 0 0 0 | {0} | -36 10 0 0 0 0 0 0 | {0} | 19 -23 0 0 0 0 0 0 | {0} | 0 -36 0 0 0 0 0 0 | {0} | -30 6 0 0 0 0 0 0 | {0} | 0 32 0 0 0 0 0 9s2t-10st2+2t3 | {0} | 0 0 35 37 49s+46t -9s-4t -3s-43t 3st-38t2 | {1} | 0 0 0 0 16 22 27 -47st+2t2 | {1} | 0 0 0 0 32 -47 -21 30s+40t | {2} | 0 0 0 0 0 0 0 14s+24t | {2} | 0 0 0 0 0 0 0 ----------------------------------------------------------------------- 0 0 0 0 0 |, {-1} | 0 -29s+25t 0 0 0 0 0 0 0 | {0} | -29 -22 0 0 0 0 0 0 0 | {0} | -24 38 0 0 0 0 0 0 0 | {0} | 39 -5 0 0 0 0 0 0 0 | {0} | 0 49 0 0 0 0 0 0 0 | {0} | -38 47 0 0 0 0 0 0 0 | {0} | 0 -24 0 0 -48s-35t -11s-39t 49s-20t 32st-14t2 0 | {0} | 0 0 -1 -28 3 -28 30 10t 0 | {1} | 0 0 0 0 2 -26 -9 30t 0 | {1} | 0 0 0 0 0 0 0 -20 0 | {2} | 0 0 0 0 0 0 0 27 0 | {2} | 0 0 0 0 ----------------------------------------------------------------------- 0 0 0 0 0 0 0 0 |} 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 | -2s+31t 31s+47t -8s+15t 28s+43t 8s-47t -49s-3t 37st-35t2 0 | -29 -38 17 -39 30 -8 37t 0 | -31 -50 -29 -34 21 -48 -42t 0 | 0 0 0 0 0 0 -48 0 | 0 0 0 0 0 0 -20 0 | o12 : List i13 : C.symmetricM o13 = | 0 0 -15s-41t -19s-40t | | 0 0 36s-5t -8s+17t | | -15s-41t 36s-5t 0 0 | | -19s-40t -8s+17t 0 0 | 4 4 o13 : Matrix (kk[s, t]) <--- (kk[s, t]) i14 : C.evenCenter o14 = {1} | 4s2-4st-12t2 0 0 0 {1} | 0 4s2-4st-12t2 0 0 {1} | 0 0 -39s2+20st+3t2 -35s2-8st+8t2 {1} | 0 0 33s2-st-29t2 -25s2-31st-22t2 {2} | 0 0 10s-17t 48s-28t {2} | 0 0 32s+22t -40s-36t {2} | 0 0 -35s-40t -s+43t {2} | 0 0 2s+48t 29s+22t {2} | 0 0 19s+22t 16s-4t {2} | 0 0 -37s+11t -17s+27t {3} | 0 0 0 0 {0} | 0 0 0 0 ----------------------------------------------------------------------- 0 0 0 0 0 0 -25s3-42s2t-14st2-27t3 43s3+23s2t+32st2+35t3 -48s3+16s2t+3st2-12t3 -27s3+18s2t-16st2+2t3 44s3+20s2t+8st2-36t3 29s3+23s2t+5st2-30t3 -11s2+8st-6t2 -15s2-5st-7t2 9s2+41st+44t2 11s2+46st+39t2 17s2+11st+18t2 -5s2+st-18t2 20s2+16st+14t2 -33s2-11st+24t2 38s2-st-19t2 6s2+39st-33t2 -24s2-8st-32t2 -45s2-40st+22t2 46s2+23st+10t2 -33s2-11st+9t2 -29s2-19st-29t2 20s2+38st+45t2 4s2+35st+23t2 -11s2-15st-15t2 0 0 0 0 0 0 ----------------------------------------------------------------------- 0 0 0 0 0 0 49s3+42s2t-19st2+8t3 -12s3+29s2t+33st2+6t3 28s3+8s2t+10st2-23t3 50s3-47s2t-33st2+43t3 -11s3+13s2t+36st2+29t3 -8s3-12s2t-17st2+46t3 32s2+11st-30t2 -40s2+15st-43t2 38s2-30st+28t2 4s2+36st-47t2 27s2-47st+48t2 19s2-17st-42t2 -5s2-25st-15t2 16s2-27st 42s2-39st-21t2 6s2-16st+35t2 -42s2-12st-8t2 -49s2+49st-29t2 -25s2+45st+43t2 -44s2-14st+17t2 24s2-38st-48t2 -23s2-23st+21t2 -46s2-49st-27t2 22s2+7st-49t2 0 0 0 0 0 0 ----------------------------------------------------------------------- 0 0 | 0 0 | -43s4-5s3t+49s2t2+21st3+15t4 0 | 15s4-40s3t-35s2t2-33st3-11t4 0 | 16s3-9s2t+26st2-48t3 0 | -32s3+18s2t-20st2+20t3 0 | 30s3+2s2t+38st2+11t3 0 | -33s3+29s2t+26st2+3t3 0 | 12s3-20s2t+11st2+39t3 0 | 23s3+49s2t+9st2+17t3 0 | 4s2-4st-12t2 0 | 0 4s2-4st-12t2 | 12 12 o14 : Matrix (kk[s, t]) <--- (kk[s, t])