# partialRegularities -- calculates Castelnuovo-Mumford regularity in each component of a multigrading

## Synopsis

• Usage:
partialRegularities M
• Inputs:
• M, , a multigraded module
• F, , the minimal free resolution of a module
• Outputs:
• a list, with i-th coordinate the regularity in the i-th component of the grading

## Description

This function applies the definition of Castelnuovo-Mumford regularity to the complex obtained by resolving the module $M$ and forgetting all but the i-th coordinate of the twists appearing. Alternately, the minimal resolution of $M$ can be given directly.

 i1 : S = ZZ/101[x_0,x_1,y_0,y_1,z_0,z_1,Degrees=>{{1,0,0},{1,0,0},{0,1,0},{0,1,0},{0,0,1},{0,0,1}}] o1 = S o1 : PolynomialRing i2 : I = ideal(x_0*x_1*y_0*z_0^2, x_1^2*y_0^2*y_1^2*z_0^2, x_0^3*y_0*z_1, x_0^2*x_1*y_1*z_0*z_1, x_0*x_1^2*y_1^2*z_0^3, x_1^3*y_0^2*y_1*z_1^2) 2 2 2 2 2 3 2 2 2 3 3 2 2 o2 = ideal (x x y z , x y y z , x y z , x x y z z , x x y z , x y y z ) 0 1 0 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 1 0 1 0 1 1 o2 : Ideal of S i3 : M = S^1/I o3 = cokernel | x_0x_1y_0z_0^2 x_1^2y_0^2y_1^2z_0^2 x_0^3y_0z_1 x_0^2x_1y_1z_0z_1 x_0x_1^2y_1^2z_0^3 x_1^3y_0^2y_1z_1^2 | 1 o3 : S-module, quotient of S i4 : netList supportOfTor M +---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+ o4 = |{0, 0, 0}| | | | | | | | | | +---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+ |{2, 1, 2}|{3, 1, 1}|{3, 1, 2}|{2, 4, 2}|{3, 2, 3}|{3, 3, 2}| | | | | +---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+ |{4, 2, 2}|{4, 1, 3}|{3, 2, 3}|{3, 4, 2}|{3, 3, 3}|{4, 2, 4}|{6, 3, 2}|{5, 3, 3}|{4, 3, 4}|{3, 4, 4}| +---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+ |{4, 2, 3}|{4, 3, 4}|{6, 3, 3}|{5, 3, 4}|{4, 4, 4}| | | | | | +---------+---------+---------+---------+---------+---------+---------+---------+---------+---------+ i5 : partialRegularities M o5 = {4, 3, 2} o5 : List

In the bigraded case this element will always be contained in the multigraded regularity if its total degree is at least $\operatorname{reg} M$.

## Caveat

Changing the grading of $M$ and applying the command regularity will not yield the correct result because betti tables in Macaulay2 do not (at least at the time of this writing) accommodate rings with generators of degree 0.