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 a module M, the regularity is the regularity of a free minimal resolution of M.
i1 : R=ZZ/32003[a..d];
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i2 : I=ideal(a^20,b^20,a*c^19-b*d^19);
o2 : Ideal of R
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i3 : regularity I
o3 = 399
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The regularity is the label of the last row in the betti diagram of a chain complex.
i4 : J=ideal(a^3,a^2*b,a*b^6,a^2*c);
o4 : Ideal of R
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i5 : C=resolution J
1 4 4 1
o5 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o5 : ChainComplex
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i6 : betti C
0 1 2 3
o6 = total: 1 4 4 1
0: 1 . . .
1: . . . .
2: . 3 3 1
3: . . . .
4: . . . .
5: . . . .
6: . 1 1 .
o6 : BettiTally
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i7 : regularity C
o7 = 6
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