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