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).<\/p>\n
We say that (A, d) is strictly commutative<\/em> 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.<\/p>\n We say that (A, d) is a strictly commutative differential graded algebra endowed with divided powers<\/em> 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<\/p>\n d(γ_n(x)) = d(x) γ_{n – 1}(x), for all n > 1<\/p>\n with the differential. Then H(A) is a strictly commutative differential graded algebra endowed with divided powers, right?<\/p>\n Wrong! Can you spot the mistake?<\/p>\n","protected":false},"excerpt":{"rendered":" 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 … Continue reading