Consider a differential graded algebra (A, d) sitting in homological degrees 0, 1, 2, … and with d : A_n —> A_{n – 1}. Then the cohomology H(A) is also a differential graded algebra (with zero differential of course).

We say that (A, d) is *strictly commutative* if xy = (-1)^e yx with e = deg(x)deg(y) and x^2 = 0 when x has odd degree. In this case H(A) is a strictly commutative differential graded algebra.

We say that (A, d) is a strictly commutative differential graded algebra *endowed with divided powers* if for every homogeneous element x of A in even degree d we have divided powers γ_n(x) of degree nd satisfying the usual rules for divided powers, and satisfying the compatibility

d(γ_n(x)) = d(x) γ_{n – 1}(x), for all n > 1

with the differential. Then H(A) is a strictly commutative differential graded algebra endowed with divided powers, right?

Wrong! Can you spot the mistake?

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What are “the usual rules of divided powers”? Does x have to be primitive?

See this paper Definition 2.1. No for your last question.