Get Algebraic Topology, Aarhus 1978: Proceedings of a Symposium PDF

By J. L. Dupont, I. H. Madsen

ISBN-10: 354009721X

ISBN-13: 9783540097211

College of Aarhus, 50. Anniversary, eleven September 1978

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Additional resources for Algebraic Topology, Aarhus 1978: Proceedings of a Symposium held at Aarhus, Denmark, August 7-12, 1978

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The collection of all Wnm becomes a direct system via the homomorphisms v and i:  Wnm  i  Wnm+1  G _ _ v v  m  Wn+1  i G  m+1 Wn+1 Let σ : W (k) −→ W (k) denote the ring endomorphism induced by F . ) Definition. Let E be the ring of “noncommutative polynomials” over W (k) in two variables F and V , subject to the following relations: • F · ξ = σ(ξ) · F ∀ξ ∈ W (k) • V · σ(ξ) = ξ · V ∀ξ ∈ W (k) • FV = V F = p Note that E is a free left, or right, module over W (k) with basis {. . , V 2 , V, 1, F, F 2, .

So the ring homomorphism u must be given by 1/p2 1/p u(x) = i(x0 ) + p · i(x1 ) + p2 · i(x2 ) + .... In particular u is unique, but we must verify that this formula does define a ring homomorphism. For this, let m be the maximal ideal of R, which contains p, and calculate: n 1/p u(x) ≡ i(x0 ) + p · i(x1 ) + . . + pn · i(x1/p ) n −n n −n = i(x0p )p + p · i(xp1 )p −n n−1 = Φn i(x0p ), . . , i(xnp ) . −n mod mn+1 , −n + . . + pn · i(xpn ) It is enough to show that this defines a ring homomorphism W (k) → R/mn+1 for any n, because R is complete noetherian and hence R = lim R/mn+1 .

Definition. The Artin-Hasse exponential is the morphism E given by WZ(p) −→ ΛZ(p) , x → E(x, t) := n n≥0 F (xn · tp ). 4. There exists unique polynomials sn ∈ Z[x0 , . . , xn , y0 , . . , yn ] such that E(x, t) · E(y, t) = E(s(x, y), t) with s = (s0 , s1 , . ). Moreover, the morphism s : WZ × WZ → WZ defines the structure of a commutative group scheme over Z. Proof. 3. For the “moreover” part we must produce the unit section and the inversion morphism of WZ . The former is defined as 0 = (0, 0, .

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Algebraic Topology, Aarhus 1978: Proceedings of a Symposium held at Aarhus, Denmark, August 7-12, 1978 by J. L. Dupont, I. H. Madsen

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