acyclicClosure -- Tate's construction of a semifree acyclic DG algebra extension

Tate's construction. Starting from A, acyclicClosure iterates killCycles from StartDegree up through EndDegree. At each step n, a minimal set of cycle representatives of H_n(A) is computed and a new DG algebra generator is adjoined for each, placed in hom-degree n + 1 and whose differential is the chosen cycle. Adjoining follows the graded-commutative convention: generators of odd hom-degree are exterior (skew-commutative, square-zero) while generators of even hom-degree are polynomial, so the result is a free DG algebra in the sense of Tate/Gulliksen. If A is the Koszul complex on a regular sequence, no homology exists to kill and acyclicClosure returns A unchanged; otherwise new generators appear, recording the deviations of A.

Variable-naming convention. Both the existing generators of A and the newly adjoined generators use a doubly-indexed naming scheme base_(i, j), where i is the homological degree (the first entry of the multi-degree) and j is a 1-indexed counter among generators at that hom-degree. Counters continue past those already present in A. For example, if A already has three generators T_(1,1), T_(1,2), T_(1,3) in hom-degree 1 and two new hom-degree-2 cycles need killing, the output has new generators T_(2,1) and T_(2,2). The Variable option only affects the base symbol for new generators; existing names are preserved.

Over a complete intersection R = k[a, b, c]/(a^3, b^3, c^3), the Koszul complex has three nonzero hom-degree-2 cycles (one per relation) which must be killed, introducing three new polynomial generators T_(2,1), T_(2,2), T_(2,3):

Here the new generators satisfy d(T_(2,i)) = a_i^2 T_(1,i), the cycle representatives of H_2 for each element of the regular sequence. The result is quasi-isomorphic to R/(a,b,c) in hom-degrees <= 2.

For larger EndDegree, the construction keeps going; over a c.i. no further generators appear because the Koszul-plus-polynomial DG algebra is already a resolution of the residue field (this is Tate's theorem):

Over a non-c.i., higher deviations are nontrivial and acyclicClosure adjoins further generators at each stage. Tallying by hom-degree gives the deviation sequence of the ring (excluding hom-degree 0, where the ring itself sits):

The hom-degree-1 count of 2 comes from the two Koszul exterior generators (one per variable of S); hom-degree 2 has three polynomial generators killing the three relations; and hom-degrees 3 and beyond have nontrivial deviations, signaling that S is not a complete intersection. Over a complete intersection, all hom-degrees >= 3 would be empty.

To choose a different base symbol for the newly adjoined generators, use the Variable option. Existing generators of A keep their names:

For a DG algebra A whose degree-zero part already has inhomogeneous structure (e.g. one built via freeDGAlgebra with a hand-written differential that is not square-zero), the package assumes d^2 = 0 without checking; the resulting acyclicClosure is only meaningful when this holds.