getDegNModule -- Extract the degree-n strand of a graded algebra as a module over its degree-zero piece

Usage:

MN = getDegNModule(N, R, A)
Inputs:
- N, an integer, the homological degree to extract
- R, a ring, the degree-zero piece of A, i.e.\ A_0
- A, a ring, a graded R-algebra with A_0 = R
Outputs:
- MN, a module, the degree-N component of A, presented as an R-module

Description

Given a graded algebra A over R with A_0 = R, the strand A_N is an R-module. This function returns that strand with a minimal presentation by monomials of internal degree N modulo the relations of A. It is most often applied to a homology algebra HA = HH A together with its degree-zero subring HA_0 = zerothHomology A.

i1 : Q = ZZ/101[x,y,z]

o1 = Q

o1 : PolynomialRing

i2 : I = ideal(x^3, y^3, z^3)

             3   3   3
o2 = ideal (x , y , z )

o2 : Ideal of Q

i3 : R = Q / I

o3 = R

o3 : QuotientRing

i4 : KR = koszulComplexDGA R

o4 = {Ring => R                          }
      Underlying algebra => R[T   ..T   ]
                               1,1   1,3
      Differential => {x, y, z}

o4 : DGAlgebra

i5 : HA = HH KR;
Finding easy relations           :  -- used 0.0202323s (cpu); 0.0169848s (thread); 0s (gc)

i6 : H0 = zerothHomology KR

o6 = H0

o6 : QuotientRing

i7 : M1 = getDegNModule(1, H0, HA)

       3
o7 = H0

o7 : H0-module, free

i8 : M2 = getDegNModule(2, H0, HA)

       3
o8 = H0

o8 : H0-module, free

The ranks of these strands are the Koszul Betti numbers of R:

i9 : apply(0..3, n -> numgens getDegNModule(n, H0, HA))

o9 = (1, 3, 3, 1)

o9 : Sequence

Ways to use getDegNModule:

getDegNModule(ZZ,Ring,Ring)

For the programmer

The object getDegNModule is a method function.

The source of this document is in DGAlgebras/doc.m2:8919:0.

getDegNModule -- Extract the degree-n strand of a graded algebra as a module over its degree-zero piece

Description

See also

Ways to use getDegNModule:

For the programmer