# supportOfTor -- computes multidegrees in the support of Tor_i(M,k), where k is the residue field

## Synopsis

• Usage:
supportOfTor M
• Inputs:
• M, , a multigraded module
• F, , the minimal resolution of a module
• Outputs:
• L, a list, where L_i contains multidegrees for $\operatorname{Tor}_i(M,k)$

## Description

This function computes a minimal free resolution of the (pruned) module $M$, reduces it by the maximal ideal, and returns a list of the unique degrees that occur at each step.

 i1 : S = multigradedPolynomialRing {1,2} o1 = S o1 : PolynomialRing i2 : B = irrelevantIdeal S o2 = ideal (x x , x x , x x , x x , x x , x x ) 0,1 1,2 0,0 1,2 0,1 1,1 0,0 1,1 0,1 1,0 0,0 1,0 o2 : Ideal of S i3 : M = S^1/B o3 = cokernel | x_(0,1)x_(1,2) x_(0,0)x_(1,2) x_(0,1)x_(1,1) x_(0,0)x_(1,1) x_(0,1)x_(1,0) x_(0,0)x_(1,0) | 1 o3 : S-module, quotient of S i4 : F = res prune M 1 6 9 5 1 o4 = S <-- S <-- S <-- S <-- S <-- 0 0 1 2 3 4 5 o4 : ChainComplex i5 : multigraded betti F 0 1 2 3 4 o5 = 0: 1 . . . . 2: . 6ab . . . 3: . . 3a2b+6ab2 . . 4: . . . 3a2b2+2ab3 . 5: . . . . a2b3 o5 : MultigradedBettiTally i6 : supportOfTor M o6 = {{{0, 0}}, {{1, 1}}, {{2, 1}, {1, 2}}, {{2, 2}, {1, 3}}, {{2, 3}}} o6 : List i7 : netList supportOfTor M +------+------+ o7 = |{0, 0}| | +------+------+ |{1, 1}| | +------+------+ |{2, 1}|{1, 2}| +------+------+ |{2, 2}|{1, 3}| +------+------+ |{2, 3}| | +------+------+

Alternately, the minimal free resolution can be given directly.

 i8 : netList supportOfTor F +------+------+ o8 = |{0, 0}| | +------+------+ |{1, 1}| | +------+------+ |{2, 1}|{1, 2}| +------+------+ |{2, 2}|{1, 3}| +------+------+ |{2, 3}| | +------+------+

## Ways to use supportOfTor :

• "supportOfTor(ChainComplex)"
• "supportOfTor(Module)"

## For the programmer

The object supportOfTor is .