# regularityMA -- Compute regularity from decomposition

## Synopsis

• Usage:
regularityMA R
regularityMA B
regularityMA M
• Inputs:
• Optional inputs:
• CoefficientField => ..., default value ZZ/101, Option to set the coefficient field.
• Decomposition => ..., default value new HashTable from {}, Optional argument of regularityMA to specify the decomposition.
• Outputs:
• an integer, list of the regularity and list of the summands achieving the maximum

## Description

Compute the regularity of K[B] from the decomposition of the homogeneous monomial algebra K[B].

We assume that B=<b_{1},...,b_{r}> is homogeneous and minimally generated by b_{1},...,b_{r}, that is, there is a group homomorphism \phi : G(B) \to \mathbb{Z} such that \phi(b_{i}) = 1 for all i.

In the case of a monomial curve an ad hoc formula for the regularity of the components is used (if R or B is given).

Specifying R:

 i1 : a=5 o1 = 5 i2 : B={{a, 0}, {0, a}, {1, a-1}, {a-1, 1}} o2 = {{5, 0}, {0, 5}, {1, 4}, {4, 1}} o2 : List i3 : R=QQ[x_0..x_3,Degrees=>B] o3 = R o3 : PolynomialRing i4 : regularityMA R 2 2 o4 = {3, {{ideal (x , x ), | 3 |}, {ideal (x , x ), | 2 |}}} 1 0 | 2 | 1 0 | 3 | o4 : List

Specifying a monomial algebra:

 i5 : a=5 o5 = 5 i6 : B={{a, 0}, {0, a}, {1, a-1}, {a-1, 1}} o6 = {{5, 0}, {0, 5}, {1, 4}, {4, 1}} o6 : List i7 : M=monomialAlgebra B ZZ o7 = ---[x ..x ] 101 0 3 o7 : MonomialAlgebra generated by {{5, 0}, {0, 5}, {1, 4}, {4, 1}} i8 : regularityMA M 2 2 o8 = {3, {{ideal (x , x ), | 3 |}, {ideal (x , x ), | 2 |}}} 1 0 | 2 | 1 0 | 3 | o8 : List

Specifying the decomposition dc:

 i9 : a=5 o9 = 5 i10 : B={{a, 0}, {0, a}, {1, a-1}, {a-1, 1}} o10 = {{5, 0}, {0, 5}, {1, 4}, {4, 1}} o10 : List i11 : R=QQ[x_0..x_3,Degrees=>B] o11 = R o11 : PolynomialRing i12 : dc=decomposeMonomialAlgebra R o12 = HashTable{| -1 | => {ideal 1, | 4 |} } | 1 | | 1 | 2 | -2 | => {ideal (x , x ), | 3 |} | 2 | 1 0 | 2 | 0 => {ideal 1, 0} | 1 | => {ideal 1, | 1 |} | -1 | | 4 | 2 | 2 | => {ideal (x , x ), | 2 |} | -2 | 1 0 | 3 | o12 : HashTable i13 : regularityMA(B,Decomposition=>dc) 2 2 o13 = {3, {{ideal (x , x ), | 3 |}, {ideal (x , x ), | 2 |}}} 1 0 | 2 | 1 0 | 3 | o13 : List

Specifying B:

 i14 : a=5 o14 = 5 i15 : B={{a, 0}, {0, a}, {1, a-1}, {a-1, 1}} o15 = {{5, 0}, {0, 5}, {1, 4}, {4, 1}} o15 : List i16 : regularityMA B 2 2 o16 = {3, {{ideal (x , x ), | 3 |}, {ideal (x , x ), | 2 |}}} 1 0 | 2 | 1 0 | 3 | o16 : List

Compare to

 i17 : I=ker map(QQ[s,t],QQ[x_0..x_3],matrix {{s^a,t^a,s*t^(a-1),s^(a-1)*t}}) 4 3 3 2 2 2 2 3 3 4 o17 = ideal (x x - x x , x - x x , x x - x x , x x - x x , x x - x ) 0 1 2 3 2 1 3 0 2 1 3 0 2 1 3 0 2 3 o17 : Ideal of QQ[x ..x ] 0 3 i18 : -1+regularity I o18 = 3

## Ways to use regularityMA :

• "regularityMA(List)"
• "regularityMA(MonomialAlgebra)"
• "regularityMA(PolynomialRing)"

## For the programmer

The object regularityMA is .