next | previous | forward | backward | up | index | toc

# stronglyLinearStrand -- computes the strongly linear strand of the minimal free resolution of a finitely generated graded module over a multigraded polynomial ring, provided the module is generated in a single degree.

## Synopsis

• Usage:
stronglyLinearStrand M
• Inputs:
• M, , a finitely generated graded module over a multigraded polynomial ring that is generated in a single degree
• Outputs:
• , the strongly linear strand of the minimal free resolution of M

## Description

The strongly linear strand of the minimal free resolution of a multigraded module $M$ is defined in the paper "Linear strands of multigraded free resolutions" by Brown-Erman. It is, roughly speaking,the largest subcomplex of the minimal free resolution of $M$ that is linear, in the sense that its differentials are matrices of linear forms. The method we use for computing the strongly linear strand uses BGG, mirroring Corollary 7.11 in Eisenbud's textbook "The geometry of syzygies".

 i1 : S = ZZ/101[x_0, x_1, x_2, Degrees => {1,1,2}] o1 = S o1 : PolynomialRing i2 : M = coker matrix{{x_0, x_2 - x_1^2}} o2 = cokernel | x_0 -x_1^2+x_2 | 1 o2 : S-module, quotient of S i3 : L = stronglyLinearStrand(M) ZZ 1 ZZ 1 o3 = (---[x ..x ]) <-- (---[x ..x ]) 101 0 2 101 0 2 0 1 o3 : Complex i4 : L == koszulComplex {x_0} o4 = true