zerothHomology -- The zeroth homology of a DG algebra as a ring

Usage:

HA0 = zerothHomology A
Inputs:
- A, an instance of the type DGAlgebra,
Outputs:
- HA0, a ring, HH_0(A), presented as a quotient of A.ring

Description

Returns HH_0(A) as a ring, computed by taking A.ring modulo the image of the degree-one part of the differential. For a Koszul complex KR = koszulComplexDGA I over R, this recovers the quotient ring R/I.

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

o1 = Q

o1 : PolynomialRing

i2 : I = ideal(x^2, y^2, z^2, x*y*z)

             2   2   2
o2 = ideal (x , y , z , x*y*z)

o2 : Ideal of Q

i3 : KQ = koszulComplexDGA I

o3 = {Ring => Q                          }
      Underlying algebra => Q[T   ..T   ]
                               1,1   1,4
                        2   2   2
      Differential => {x , y , z , x*y*z}

o3 : DGAlgebra

i4 : H0 = zerothHomology KQ

o4 = H0

o4 : QuotientRing

i5 : describe H0

              Q
o5 = -------------------
       2   2   2
     (z , y , x , x*y*z)

When A is an acyclic closure, zerothHomology A returns the quotient ring on which the DG algebra resolves the residue field.

i6 : R = Q / I

o6 = R

o6 : QuotientRing

i7 : A = acyclicClosure(R, EndDegree => 2)

o7 = {Ring => R                                                                                          }
      Underlying algebra => R[T   ..T   , T   ..T   , T   ..T   ]
                               1,1   1,3   2,1   2,4   3,1   3,3
      Differential => {x, y, z, x*T   , y*T   , z*T   , y*z*T   , y*z*T   , -y*z*T   T   , -y*z*T   T   }
                                   1,1     1,2     1,3       1,1       2,1        1,1 1,3        1,1 1,2

o7 : DGAlgebra

i8 : zerothHomology A

      ZZ
o8 = ---
     101

o8 : QuotientRing

Ways to use zerothHomology:

zerothHomology(DGAlgebra)

For the programmer

The object zerothHomology is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:8875:0.

zerothHomology -- The zeroth homology of a DG algebra as a ring

Description

See also

Ways to use zerothHomology:

For the programmer