TateResolution -- TateResolution of a module over an exterior algebra

Synopsis

Usage:

F = TateResolution(M,lower,upper)
Inputs:
- M, a module,
- lower, an integer,
- upper, an integer, lower and upper bounds for the resolution
Outputs:
- F, a chain complex,

Description

Forms an interval, lower..upper, of a doubly infinite free resolution of a a Cohen-Macaulay module over a Gorenstein ring, such as any module over an exterior algebra (actually, any module over any ring.)

i1 : E = ZZ/101[a,b,c, SkewCommutative=>true]

o1 = E

o1 : PolynomialRing, 3 skew commutative variable(s)

i2 : M = coker map(E^2, E^{-1}, matrix"ab;bc")

o2 = cokernel | ab |
              | bc |

                            2
o2 : E-module, quotient of E

i3 : presentation M

o3 = | ab |
     | bc |

             2      1
o3 : Matrix E  <-- E

i4 : TateResolution(M,-2,7)

      9      5      2      1      2      4      7      11      16      22
o4 = E  <-- E  <-- E  <-- E  <-- E  <-- E  <-- E  <-- E   <-- E   <-- E   <-- 0
                                                                               
     -2     -1     0      1      2      3      4      5       6       7       8

o4 : ChainComplex

Caveat

In a previous version of this script, this command returned a betti table; now use "betti TateResolution" instead.

Ways to use TateResolution :

TateResolution(Module)
TateResolution(Module,ZZ)
TateResolution(Module,ZZ,ZZ)

For the programmer

The object TateResolution is a method function.