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N ) Indeed, since Spec Z = Spec Z ∪ Spec AN and Z = F±1 [+[2] | x + 0 = x = 0 + x, x + (−x) = 0], all we have to check is that AN is a finitely presented F±1 -algebra. We show this by presenting an explicit presentation [p] of AN over F±1 . Namely, AN is generated over F±1 by (sp )p|N , where p runs through all prime divisors of N, and sn denotes the averaging operation sn := (1/n){1} + · · ·+ (1/n){n}, subject to following idempotency, symmetry and cancellation relations: sn ({1}, {1}, . . , {1}) = {1} sn ({1}, {2}, .

This correspondence between chain complexes and simplicial objects is called Dold–Kan correspondence. 6. ) Since the simplicial objects play a fundamental role in the homotopic algebra parts of our work, we would like to fix some notations, mostly consistent with those of [GZ]. We denote by [n] the standard finite ordered set {0, 1, . . , n}, endowed with its natural linear order. Notice that [n] is an n + 1-element ordered set, while n = {1, 2, . . , n} is an n-element unordered set. One shouldn’t confuse these two notations.

Indeed, if Λ = T | E is any presentation of Λ, then Λ S can be constructed as S, T | E . Notice that this Λ S is something like a “very non-commutative” polynomial algebra over S: not only the indeterminates from S are not required to commute between themselves, they are even not required to commute with operations from Λ! e. e. central homomorphisms Λ → Σ. The graded underlying set functor Σ → Σ still admits a left adjoint, which will be denoted by S → Λ{S}. If Λ = T | E , then Λ{S} = T, S | E, [S, T ] , where [S, T ] denotes the set of all commutativity relations [s, t], s ∈ S, t ∈ T .

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