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koszulComplexDGA(Ideal) -- The Koszul complex on the generators of an ideal, as a DG algebra

Description

Given generators f_1, ..., f_r of I, the output is a DG algebra over R whose underlying graded algebra is R[T_(1,1), ..., T_(1,r)] on r exterior generators in hom-degree 1, with differential d(T_(1,i)) = f_i extended by the Leibniz rule. The variable naming convention and Variable option mirror koszulComplexDGA(Ring).

i1 : R = ZZ/101[a, b, c]

o1 = R

o1 : PolynomialRing
 
i2 : I = ideal(a^3, b^3, c^3, a^2 * b^2 * c^2)

             3   3   3   2 2 2
o2 = ideal (a , b , c , a b c )

o2 : Ideal of R
 
i3 : A = koszulComplexDGA I

o3 = {Ring => R                           }
      Underlying algebra => R[T   ..T   ]
                               1,1   1,4
                        3   3   3   2 2 2
      Differential => {a , b , c , a b c }

o3 : DGAlgebra
 
i4 : gens A.natural

o4 = {T   , T   , T   , T   }
       1,1   1,2   1,3   1,4

o4 : List
 
i5 : flatten entries matrix A.diff

       3   3   3   2 2 2
o5 = {a , b , c , a b c , a, b, c}

o5 : List

The resulting complex coincides with koszul applied to gens I, up to monomial order:

i6 : complexA = toComplex A

      1      4      6      4      1
o6 = R  <-- R  <-- R  <-- R  <-- R
                                  
     0      1      2      3      4

o6 : Complex
 
i7 : ranks = apply(5, i -> numgens prune HH_i complexA)

o7 = {1, 3, 0, 0, 0}

o7 : List
 
i8 : ranks == apply(5, i -> numgens prune HH_i koszul gens I)

o8 = true

In particular, I contains the redundant generator a^2 b^2 c^2 (it is in (a^3, b^3, c^3)), so the Koszul complex is not exact at H_1; nonzero higher Koszul homology encodes the relations and syzygies between the chosen generators.

See also

Ways to use this method:

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/DGAlgebras/doc.m2:978:0.