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TateResolution -- TateResolution of a module over an exterior algebra

Synopsis

• Usage:
F = TateResolution(M,lower,upper)
• Inputs:
• M, ,
• lower, an integer,
• upper, an integer, lower and upper bounds for the resolution
• Outputs:
• F, ,

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 .