regularityBound -- bounds the multigraded regularity of a module

Synopsis

Usage:

regularityBound M
Inputs:
- M, a module, 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.

Ways to use regularityBound :

regularityBound(Module)

For the programmer

The object regularityBound is a method function.