Given an inhomogeneous, singly-graded, or multi-graded ideal or module, this function computes a non-minimal free resolution. If the input is an ideal $I \subset S$, it computes a non-minimal resolution of $S^1/I$.
A key benefit of this function is that it allows a much faster method for computing the betti numbers of the minimal free resolution. If that is your only interest (i.e. you don't need the complex itself), instead use minimalBetti. However, minimalBetti currently only works for single gradings, not multi-gradings.
i1 : I = Grassmannian(1,6, CoefficientRing => ZZ/101);
ZZ
o1 : Ideal of ---[p ..p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p ]
101 0,1 0,2 1,2 0,3 1,3 2,3 0,4 1,4 2,4 3,4 0,5 1,5 2,5 3,5 4,5 0,6 1,6 2,6 3,6 4,6 5,6
|
i2 : S = ring I o2 = S o2 : PolynomialRing |
i3 : elapsedTime C = res(I, FastNonminimal => true)
-- 6.57719 seconds elapsed
1 35 241 841 1781 2464 2294 1432 576 135 14
o3 = S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- 0
0 1 2 3 4 5 6 7 8 9 10 11
o3 : ChainComplex
|
i4 : elapsedTime C1 = res ideal(I_*)
-- 2.40984 seconds elapsed
1 35 140 385 819 1080 819 385 140 35 1
o4 = S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- 0
0 1 2 3 4 5 6 7 8 9 10 11
o4 : ChainComplex
|
i5 : betti(C, Minimize => true) == betti C1 o5 = true |
For a non-minimal resolution, betti gives the actual Betti numbers, and using the betti(...,Minimize=>...) option gives the ranks in a minimal resolution (which is itself not computed).
i6 : betti C
0 1 2 3 4 5 6 7 8 9 10
o6 = total: 1 35 241 841 1781 2464 2294 1432 576 135 14
0: 1 . . . . . . . . . .
1: . 35 140 290 402 402 293 152 53 11 1
2: . . 101 514 1174 1577 1365 780 287 62 6
3: . . . 37 204 479 621 480 221 56 6
4: . . . . 1 6 15 20 15 6 1
o6 : BettiTally
|
i7 : betti(C, Minimize => true)
0 1 2 3 4 5 6 7 8 9 10
o7 = total: 1 35 140 385 819 1080 819 385 140 35 1
0: 1 . . . . . . . . . .
1: . 35 140 189 84 . . . . . .
2: . . . 196 735 1080 735 196 . . .
3: . . . . . . 84 189 140 35 .
4: . . . . . . . . . . 1
o7 : BettiTally
|
As mentioned above, if you are just interested in the minimal betti numbers of the ideal or module, then use minimalBetti, as it avoids construction of the non-minimal free resolution.
i8 : minimalBetti I
0 1 2 3 4 5 6 7 8 9 10
o8 = total: 1 35 140 385 819 1080 819 385 140 35 1
0: 1 . . . . . . . . . .
1: . 35 140 189 84 . . . . . .
2: . . . 196 735 1080 735 196 . . .
3: . . . . . . 84 189 140 35 .
4: . . . . . . . . . . 1
o8 : BettiTally
|
If the resolution is not large, this function can be slower than the usual function resolution. But for larger examples, if one is only interested in the betti numbers, this function can be hundreds or thousands of times faster.
If the input module is not graded, or is multi-graded, this function still works. However, minimalBetti does not work in these cases. In the inhomogeneous case, the returned free resolution is often highly non minimal. Of course, there is no notion of minimal resolution in this case, but one can use pruneComplex to clean up the returned complex.
i9 : R = ZZ/101[a..f] o9 = R o9 : PolynomialRing |
i10 : I = ideal"a3-ab-c2,abc-d2-1, b3-b2-b"
3 2 2 3 2
o10 = ideal (a - a*b - c , a*b*c - d - 1, b - b - b)
o10 : Ideal of R
|
i11 : C = res(I, FastNonminimal => true)
1 12 28 24 7
o11 = R <-- R <-- R <-- R <-- R <-- 0
0 1 2 3 4 5
o11 : ChainComplex
|
i12 : needsPackage "PruneComplex" o12 = PruneComplex o12 : Package |
i13 : pruneComplex C
1 3 3 1
o13 = R <-- R <-- R <-- R
0 1 2 3
o13 : ChainComplex
|
If one has a specific Groebner basis on which one wants to base the Schreyer resolution, use Strategy=>5. This will not check that the input forms a Groebner basis, but if it does not, then the function will either produce non-sensical answers, or fail.
i14 : R = ZZ/101[a..c,x_1..x_9, MonomialOrder=>{3,9}]
o14 = R
o14 : PolynomialRing
|
i15 : I = ideal(a^2 - b^2 - x_1 * a*c - x_2 * b*c - x_3 * c^2,
a*b - x_4 * a*c - x_5 * b*c - x_6 *c^2,
b^2 - x_7 * a*c - x_8 * b*c - x_9 *c^2
)
2 2 2 2
o15 = ideal (a - b - a*c*x - b*c*x - c x , a*b - a*c*x - b*c*x - c x ,
1 2 3 4 5 6
-----------------------------------------------------------------------
2 2
b - a*c*x - b*c*x - c x )
7 8 9
o15 : Ideal of R
|
i16 : C = res(I, FastNonminimal => true, Strategy=>5)
1 3 2
o16 = R <-- R <-- R <-- 0
0 1 2 3
o16 : ChainComplex
|
i17 : C.dd
1 3
o17 = 0 : R <------------------------------------------------------------------------- R : 1
| a2-b2-acx_1-bcx_2-c2x_3 ab-acx_4-bcx_5-c2x_6 b2-acx_7-bcx_8-c2x_9 |
3 2
1 : R <------------------------------------------ R : 2
{2} | -b+cx_4 cx_7 |
{2} | a-cx_1+cx_5-cx_7 -b-cx_4+cx_8 |
{2} | -b-cx_2+cx_4-cx_8 a-cx_5+cx_7 |
2
2 : R <----- 0 : 3
0
o17 : ChainComplexMap
|
Note that Strategy=>4 or Strategy=>5 implies FastNonminimal.
i18 : C1 = res(ideal I_*, Strategy=>5)
1 3 2
o18 = R <-- R <-- R <-- 0
0 1 2 3
o18 : ChainComplex
|
i19 : C1.dd
1 3
o19 = 0 : R <------------------------------------------------------------------------- R : 1
| a2-b2-acx_1-bcx_2-c2x_3 ab-acx_4-bcx_5-c2x_6 b2-acx_7-bcx_8-c2x_9 |
3 2
1 : R <------------------------------------------ R : 2
{2} | -b+cx_4 cx_7 |
{2} | a-cx_1+cx_5-cx_7 -b-cx_4+cx_8 |
{2} | -b-cx_2+cx_4-cx_8 a-cx_5+cx_7 |
2
2 : R <----- 0 : 3
0
o19 : ChainComplexMap
|
Released in M2 1.9, still experimental. Only works over finite prime fields. Uses quite a lot of memory. For inhomogeneous ideals or modules, the monomial order must be a degree order. For multi-graded ideals or modules, minimalBetti is not yet implemented.
The object FastNonminimal is a symbol.