# Die Ausdehnungslehre by Grassmann H. By Grassmann H.

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Then A≥1 ⊕ A0 is an object in K if and only if A≥1 is an A0 -module and Sq j , P j , β on A≥1 are A0 -linear and the multiplication in A≥1 is A0 -bilinear. 2. Power algebras 9 Proof. Given an object A in K we see that x ∈ A0 satisﬁes Sq j x = 0, P j x = 0, βx = 0 for j > 0 since A is an unstable A-module. Moreover (K1) implies that Sq j , P j , β for j > 0 are A0 -linear. Finally (K2) implies for | x |= 0 that x = xp . 13). In the next section we use power algebras to describe a category isomorphic to K0 .

9). Hence also A = Eβ (V ) ⊗ H is an object in Kp . We now deﬁne for x ∈ A1 the power operation γx : Aq → Apq by the formula γx (y) = ϑq + ϑq (−1)j ω(q−2j)(p−1) (x) · P j (y) j (2) (−1)j ω(q−2j)(p−1)−1 (x) · βP j (y). 3)(4). Now one can check that A = Eβ (V ) ⊗ H is a well-deﬁned Bockstein power algebra natural for maps under H in β-Alg0 so that H is a well-deﬁned unitary extended Bockstein power algebra. 4) below. 8). Let p = 2. 11)) x1−j · Sq j (y) γx (y) = j = = Sq 1 (y) + x · Sq 0 (y) y 2 + x · y.

For the σn -space Z n and for σ ∈ σn we have the map σ : Z n → Z n which carries x to σ · x. 3) σ∗ = sign(σ) : Hn (Z n ) → Hn (Z n ). These properties of Z n are crucial for the deﬁnition of the power maps in Part II. 4 Proposition. 3) exist. Proof. We shall need the following categories and functors; compare the Appendix of this section and Goerss-Jardine [GJ]. Let Set and Mod be the category of sets and R-modules respectively and let ∆Set and ∆ Mod be the corresponding categories of simplicial objects in Set and Mod respectively.