Let $k$ be a field, $S$ a $\ZZ^r$-graded polynomial ring over $k$, and $M$ a finitely generated, $\ZZ^r$-graded $S$-module. Write $M_{\geq d}$ for the truncation $\oplus_{d'\geq d} M_d'$ of $M$ at $d$ (where $d'\geq d$ if $d'_i\geq d_i$ for all $i$). The main purpose of this package is to find the degrees $d\in\ZZ^r$ so that $M_{\geq d}$ has a linear resolution, i.e. satisfies the function isLinearComplex. No sufficient finite search space is known, so the result may not be complete.
i1 : S = ZZ/101[x_1..x_4,Degrees=>{{1,0},{1,0},{0,1},{0,1}}]
o1 = S
o1 : PolynomialRing
|
i2 : I = ideal(x_1^3*x_3, x_2*x_3*x_4, x_3^4*x_4, x_4*x_2^2, x_1^3*x_2^3, x_3^3)
3 4 2 3 3 3
o2 = ideal (x x , x x x , x x , x x , x x , x )
1 3 2 3 4 3 4 2 4 1 2 3
o2 : Ideal of S
|
i3 : M = S^1/I
o3 = cokernel | x_1^3x_3 x_2x_3x_4 x_3^4x_4 x_2^2x_4 x_1^3x_2^3 x_3^3 |
1
o3 : S-module, quotient of S
|
i4 : regularity M o4 = 5 |
i5 : r = coarseMultigradedRegularity M
o5 = {6, 4}
o5 : List
|
i6 : L = linearTruncations({{0,0}, r}, M)
o6 = {{2, 3}, {3, 2}}
o6 : List
|
i7 : apply(L, i -> isLinearComplex res truncate(i,M))
o7 = {true, true}
o7 : List
|
If $M_{\geq d}$ has a linear truncation then $M_{\geq d'}$ has a linear truncation for all $d'\geq d$, so the function linearTruncations gives the minimal such multidegrees in a given range, using the function findRegion. The functions linearTruncationsBound and regularityBound estimate the linear truncation region and the multigraded regularity region of $M$, respectively, without calculating cohomology or truncations.
If the ring $S$ is standard $\ZZ$-graded then $M_{\geq d}$ has a linear resolution if and only if $d\geq\operatorname{reg} M$, where $\operatorname{reg} M$ is the Castelnuovo-Mumford regularity of $M$.
This documentation describes version 1.0 of LinearTruncations.
The source code from which this documentation is derived is in the file LinearTruncations.m2.
The object LinearTruncations is a package.