Given a free complex $C$ over a standard graded polynomial ring, the regularity $r$ of $C$ is the smallest integer such that each basis element of $C_i$ has degree at most $i + r$.
i1 : R = ZZ/101[a..d]; |
i2 : I = ideal(b^2-a*c, b*c-a*d, c^2-b*d)
2 2
o2 = ideal (b - a*c, b*c - a*d, c - b*d)
o2 : Ideal of R
|
i3 : C = freeResolution(R^1/I)
1 3 2
o3 = R <-- R <-- R
0 1 2
o3 : Complex
|
i4 : betti C
0 1 2
o4 = total: 1 3 2
0: 1 . .
1: . 3 2
o4 : BettiTally
|
i5 : regularity C o5 = 1 |
i6 : regularity I o6 = 2 |
i7 : regularity (R^1/I) o7 = 1 |
i8 : assert(regularity C === regularity(R^1/I)) |
The regularity is the label of the last row of the Betti table of $C$.
i9 : betti C
0 1 2
o9 = total: 1 3 2
0: 1 . .
1: . 3 2
o9 : BettiTally
|
i10 : betti(C[3])
-3 -2 -1
o10 = total: 1 3 2
3: 1 . .
4: . 3 2
o10 : BettiTally
|
i11 : regularity(C[3]) o11 = 4 |
Here is a slightly more complicated example.
i12 : J = ideal(a^3, b^3, c^3, d^3, (a+b+c+d)^3); o12 : Ideal of R |
i13 : FJ = freeResolution J
1 5 17 20 7
o13 = R <-- R <-- R <-- R <-- R
0 1 2 3 4
o13 : Complex
|
i14 : betti FJ
0 1 2 3 4
o14 = total: 1 5 17 20 7
0: 1 . . . .
1: . . . . .
2: . 5 . . .
3: . . . . .
4: . . 16 10 1
5: . . 1 10 6
o14 : BettiTally
|
i15 : regularity FJ o15 = 5 |
Although Castelnuovo-Mumford regularity is defined in more general settings (e.g. toric varieties with multi-degrees) this method does not currently handle these extensions. Similarly, Castelnuovo-Mumford regularity can be defined for non-free complexes, but this method doesn't handle that case either.