# tateResolution -- finite piece of the Tate resolution

## Synopsis

• Usage:
tateResolution(m,E,l,h)
• Inputs:
• m, , a presentation matrix for a module
• E, , exterior algebra
• l, an integer, lower cohomological degree
• h, an integer, upper bound on the cohomological degree
• Outputs:
• , a finite piece of the Tate resolution

## Description

This function takes as input a presentation matrix m of a finitely generated graded S-module M an exterior algebra E and two integers l and h. If r is the regularity of M, then this function computes the piece of the Tate resolution from cohomological degree l to cohomological degree max(r+2,h). For instance, for the homogeneous coordinate ring of a point in the projective plane:
 i1 : S = ZZ/32003[x_0..x_2];  i2 : E = ZZ/32003[e_0..e_2, SkewCommutative=>true]; i3 : m = matrix{{x_0,x_1}}; 1 2 o3 : Matrix S <--- S i4 : regularity coker m o4 = 0 i5 : T = tateResolution(m,E,-2,4) 1 1 1 1 1 1 1 o5 = E <-- E <-- E <-- E <-- E <-- E <-- E 0 1 2 3 4 5 6 o5 : ChainComplex i6 : betti T 0 1 2 3 4 5 6 o6 = total: 1 1 1 1 1 1 1 -4: 1 1 1 1 1 1 1 o6 : BettiTally i7 : T.dd_1 o7 = {-4} | e_2 | 1 1 o7 : Matrix E <--- E