For a free chain complex C, the regularity r is the smallest number so that each basis element of C_i has degree at most i+r. For an ideal I, regularity is one plus the regularity of the minimal free resolution of the quotient of the ambient ring by I. For a module M, regularity is the regularity of a minimal free resolution of M.
i1 : R = ZZ/32003[a..d]; |
i2 : I = ideal(a^20, b^20, a*c^19-b*d^19); o2 : Ideal of R |
i3 : C = resolution I
1 3 23 40 19
o3 = R <-- R <-- R <-- R <-- R <-- 0
0 1 2 3 4 5
o3 : ChainComplex
|
i4 : regularity C o4 = 398 |
i5 : regularity comodule I o5 = 398 |
i6 : regularity I o6 = 399 |
i7 : regularity module I o7 = 399 |
The regularity is the label of the last row in the Betti diagram of a chain complex. However, this depends on the total degree weights in the Betti tally, which are computed based on the heft vector of the underlying ring. To adjust this vector, a vector w whose length is the same as the degree length of the ring can be provided using the option Weights. The dot products of w with the multidegrees in the tally will be used in the resulting computation.
i8 : C = resolution ideal(a^3, a^2*b, a*b^6, a^2*c); |
i9 : betti C
0 1 2 3
o9 = total: 1 4 4 1
0: 1 . . .
1: . . . .
2: . 3 3 1
3: . . . .
4: . . . .
5: . . . .
6: . 1 1 .
o9 : BettiTally
|
i10 : regularity C o10 = 6 |
i11 : betti(C, Weights => {2})
0 1 2 3
o11 = total: 1 4 4 1
0: 1 . . .
1: . . . .
2: . . . .
3: . . . .
4: . . . .
5: . 3 . .
6: . . 3 .
7: . . . 1
8: . . . .
9: . . . .
10: . . . .
11: . . . .
12: . . . .
13: . 1 . .
14: . . 1 .
o11 : BettiTally
|
i12 : regularity(C, Weights => {2})
o12 = 14
|
The object regularity is a method function with options.