regularity -- compute the Castelnuovo-Mumford regularity

Synopsis

Usage:

r = regularity C

r = regularity(C, Weights => w)
Inputs:
- C, a chain complex, an ideal, or a module,
Optional inputs:
- Weights => a list, default value null, a weight vector w, see betti(...,Weights=>...);
Outputs:
- r, an integer,

Description

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

Ways to use regularity :

regularity(BettiTally) -- see BettiTally -- the class of all Betti tallies
regularity(ChainComplex)
regularity(Ideal)
regularity(Module)

For the programmer

The object regularity is a method function with options.

regularity -- compute the Castelnuovo-Mumford regularity

Synopsis

Description

See also

Ways to use regularity :

For the programmer