Suppose that f_1..f_c is a homogeneous regular sequence of forms of the same degree in a polynomial ring S and M is a high syzygy module over S/(f_1,..,f_c) = R(c), and mf = (d,h) is the output of matrixFactorization(M,ff). If the complexity of M is c', then M has a finite free resolution over R = S/(f_1,..,f_{(c-c')}) (and, more generally, has complexity c-d over S/(f_1,..,f_{(c-d)}) for d>=c').
The complex A is the minimal finite free resolution of M over A, constructed as an iterated Koszul extension, made from the maps in bMaps mf and psiMaps mf, as described in Eisenbud-Peeva.
i1 : setRandomSeed 0 o1 = 0 |
i2 : S = ZZ/101[a,b,c]; |
i3 : ff = matrix"a3,b3";
1 2
o3 : Matrix S <--- S
|
i4 : R = S/ideal ff; |
i5 : M = highSyzygy (R^1/ideal vars R); |
i6 : mf = matrixFactorization (ff, M)
o6 = {{4} | -c b 0 a2 0 0 0 0 0 |, {5} | 0 a2 0 -b 0 0 0 0 0 0
{4} | a 0 b 0 0 0 0 0 0 | {5} | 0 0 a2 -c b2 0 0 0 0 0
{4} | 0 a c 0 0 0 0 0 -b2 | {5} | 0 0 0 a 0 b2 0 0 0 0
{3} | 0 0 a2 0 0 b2 0 0 0 | {6} | a c -b 0 0 0 0 0 0 0
{4} | 0 0 0 0 b -a 0 0 0 | {5} | 0 0 0 0 0 0 0 a b2 0
{4} | 0 0 0 0 -c 0 a b2 0 | {5} | 0 0 0 0 0 -a2 0 b 0 0
{4} | 0 0 0 0 0 c -b 0 a2 | {5} | 0 0 0 0 0 0 -a2 c 0 0
{6} | 0 0 0 0 0 0 0 0 c b
{6} | 0 0 0 0 a c -b 0 0 0
------------------------------------------------------------------------
0 |, {3} | 0 0 0 1 0 0 0 |}
0 | {4} | 0 0 0 0 1 0 0 |
0 | {4} | 0 0 0 0 0 1 0 |
0 | {4} | 0 0 0 0 0 0 1 |
0 | {4} | 0 1 0 0 0 0 0 |
0 | {4} | 0 0 -1 0 0 0 0 |
-b2 | {4} | 1 0 0 0 0 0 0 |
a |
0 |
o6 : List
|
i7 : G = makeFiniteResolution(ff,mf)
7 12 5
o7 = S <-- S <-- S
0 1 2
o7 : ChainComplex
|
i8 : F = res pushForward(map(R,S),M)
7 12 5
o8 = S <-- S <-- S <-- 0
0 1 2 3
o8 : ChainComplex
|
i9 : G.dd_1
o9 = {4} | -c b 0 a2 0 0 0 0 0 0 0 0 |
{4} | a 0 b 0 0 0 0 0 0 0 0 0 |
{4} | 0 a c 0 0 0 0 0 -b2 0 0 0 |
{3} | 0 0 a2 0 0 b2 0 0 0 0 0 0 |
{4} | 0 0 0 0 b -a 0 0 0 -a3 0 0 |
{4} | 0 0 0 0 -c 0 a b2 0 0 -a3 0 |
{4} | 0 0 0 0 0 c -b 0 a2 0 0 -a3 |
7 12
o9 : Matrix S <--- S
|
i10 : F.dd_1
o10 = {3} | a2 b2 0 0 0 0 0 0 0 0 0 0 |
{4} | 0 -a 0 0 0 b 0 0 0 0 0 0 |
{4} | 0 0 a 0 0 -c 0 0 b2 0 0 0 |
{4} | 0 c -b 0 0 0 a2 0 0 0 0 0 |
{4} | b 0 0 a 0 0 0 0 0 b3 0 0 |
{4} | -c 0 0 0 a 0 b2 0 0 0 b3 0 |
{4} | 0 0 0 -c -b 0 0 a2 0 0 0 b3 |
7 12
o10 : Matrix S <--- S
|
i11 : G.dd_2
o11 = {5} | 0 -b3 0 0 0 |
{5} | 0 -b2c 0 0 -a2b2 |
{5} | 0 ab2 0 0 0 |
{6} | 0 0 0 0 b3 |
{5} | -a3 0 0 0 0 |
{5} | 0 -a3 0 0 0 |
{5} | 0 0 -a3 0 0 |
{6} | 0 0 0 -a3 0 |
{6} | 0 0 0 0 -a3 |
{7} | -b a 0 0 0 |
{7} | c 0 -a -b2 0 |
{7} | 0 -c b 0 -a2 |
12 5
o11 : Matrix S <--- S
|
i12 : F.dd_2
o12 = {5} | b3 0 0 0 0 |
{5} | -a2b 0 0 0 0 |
{5} | -a2c 0 0 -a2b2 0 |
{5} | 0 -b3 0 0 0 |
{5} | 0 0 -b3 0 0 |
{5} | -a3 0 0 0 0 |
{6} | 0 0 0 -b3 0 |
{6} | 0 0 0 0 -b3 |
{6} | 0 0 0 a3 0 |
{7} | -b a 0 0 0 |
{7} | c 0 a b2 0 |
{7} | 0 -c -b 0 a2 |
12 5
o12 : Matrix S <--- S
|
If the complexity of M is not maximal, then the finite resolution takes place over an intermediate complete intersection:
i13 : S = ZZ/101[a,b,c,d] o13 = S o13 : PolynomialRing |
i14 : ff1 = matrix"a3,b3,c3,d3"
o14 = | a3 b3 c3 d3 |
1 4
o14 : Matrix S <--- S
|
i15 : ff =ff1*random(source ff1, source ff1)
o15 = | 24a3-36b3-30c3-29d3 19a3+19b3-10c3-29d3 -8a3-22b3-29c3-24d3
-----------------------------------------------------------------------
-38a3-16b3+39c3+21d3 |
1 4
o15 : Matrix S <--- S
|
i16 : R = S/ideal ff o16 = R o16 : QuotientRing |
i17 : M = highSyzygy (R^1/ideal"a2b2")
o17 = cokernel {6} | b2 0 -a2 0 |
{7} | a b 0 0 |
{7} | 0 0 b a |
3
o17 : R-module, quotient of R
|
i18 : complexity M o18 = 2 |
i19 : mf = matrixFactorization (ff, M)
o19 = {{7} | -a -36b 0 a |, {8} | 35a2 48b 0 -33b 0 |, {6} | 0 36
{6} | b2 a2 0 0 | {8} | -35b2 -35a 0 0 0 | {7} | -36 0
{7} | 0 0 b a | {8} | 0 0 33b2 33a -33b2 | {7} | 1 0
{8} | 0 0 -43a2 -33b 0 |
-----------------------------------------------------------------------
0 |}
36 |
0 |
o19 : List
|
i20 : complexity mf o20 = 2 |
i21 : BRanks mf
o21 = {{2, 2}, {1, 2}}
o21 : List
|
i22 : G = makeFiniteResolution(ff,mf); |
i23 : codim ring G o23 = 2 |
i24 : R1 = ring G o24 = R1 o24 : QuotientRing |
i25 : F = res prune pushForward(map(R,R1),M)
3 5 2
o25 = R1 <-- R1 <-- R1 <-- 0
0 1 2 3
o25 : ChainComplex
|
i26 : betti F
0 1 2
o26 = total: 3 5 2
6: 1 . .
7: 2 4 .
8: . . .
9: . 1 2
o26 : BettiTally
|
i27 : betti G
0 1 2
o27 = total: 3 5 2
6: 1 . .
7: 2 4 .
8: . . .
9: . 1 2
o27 : BettiTally
|
The object makeFiniteResolution is a method function.