# regularityBound -- bounds the multigraded regularity of a module

## Synopsis

• Usage:
regularityBound M
• Inputs:
• M, , a multigraded module with no $H^0_B$ or $H^1_B$
• Outputs:
• a list, containing multidegrees in the regularity of M

## Description

Using only the multigraded betti numbers of a $\ZZ^r$-graded module $M$, this function identifies a subset of the multigraded regularity of a module $M$ over the coordinate ring $S$ of a product of projective spaces, in the sense of Maclagan and Smith. It assumes that the local cohomology groups $H^0_B(M)$ and $H^1_B(M)$ vanish, where $B$ is the irrelevant ideal of $S$.

 i1 : (S,E) = productOfProjectiveSpaces {1,2} o1 = (S, E) o1 : Sequence i2 : I = ideal(x_(0,0)*x_(1,0),x_(1,1)^3) 3 o2 = ideal (x x , x ) 0,0 1,0 1,1 o2 : Ideal of S i3 : M = S^1/I o3 = cokernel | x_(0,0)x_(1,0) x_(1,1)^3 | 1 o3 : S-module, quotient of S i4 : regularityBound M o4 = {{0, 2}} o4 : List i5 : needsPackage "VirtualResolutions" o5 = VirtualResolutions o5 : Package i6 : multigradedRegularity(S,M) o6 = {{0, 2}} o6 : List

The output is often but not always {partialRegularities M}.

## Caveat

In general regularityBound will not give the minimal elements of $\operatorname{reg} M$ but will be faster than computing cohomology.

## See also

• partialRegularities -- calculates Castelnuovo-Mumford regularity in each component of a multigrading
• isQuasiLinear -- checks whether degrees in the resolution of a truncation are at most those of the irrelevant ideal
• multigradedRegularity -- computes the minimal elements of the multigraded regularity of a module over a multigraded ring

## Ways to use regularityBound :

• "regularityBound(Module)"

## For the programmer

The object regularityBound is .