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By Tammo tom Dieck

This e-book is a jewel– it explains very important, valuable and deep issues in Algebraic Topology that you simply won`t locate in different places, rigorously and in detail."""" Prof. Günter M. Ziegler, TU Berlin

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The needed ingredient to do this is an R-algebra homomorphism Φ0 : RG → RG ⊗R RG such that the following diagram commutes: RG o ∼ = RU ⊗R RG ⊗1 RG Φ0 G RG ⊗R RG 1⊗ RG o 9 ∼ = RG ⊗R R. Define Φ0 to be the diagonal map: Φ0 : RG → RG ⊗R RG ∼ = R(G × G) a x → a x ⊗ x∼ = x∈G ax (x, x). x∈G x x∈G x 41 Then it is clear that Φ0 is an R-algebra homomorphism which makes the diagram commute. Remarks: 1. Observe that F ⊗R F is again a DG-algebra over R. 2. We have initially defined our augmentation : RG → R.

1 Applications of Tate’s Theorem The General Setup Let P be a (polynomial) ring and let J = (g1 , . . , gm ) ⊆ I = (f1 , . . , fn ) ⊆ P be ideals of P generated by Koszul regular sequences. Let n gj = aji fi i=1 A = (aji ), aji ∈ P σ = (σ1 , . . , σm ) τ = (τ1 , . . , τn ). 1 provides a projective resolution of PI over PJ :   • ∂ ∂  Γ•P (σ) ⊗ P + aji τi (τ ), ∂ = fi J J ∂τi ∂σj P i,j i J where |R| = 0 |τi | = 1, exterior variables |σj | = 2, divided power variables. 1. Let K be a field.

Gm be Koszul regular sequences such that the ideal J = (g1 , . . , gm ) generated by the gj is contained in the ideal I = (f1 , . . , fn ) generated by the fi . Write gj = ni=1 aji fi , 1 ≤ j ≤ m, with aji ∈ R. Let R = RJ and I = JI , and let aji and f i denote the J-residues of aji and fi . 6) R τ1 , . . , τn ; σ1 , . . , σm with exterior variables τi of degree 1 and divided power variables σj of degree 2, and with algebra differential d defined through dτi = f i n dσj = aji τi i=1 is acyclic, and therefore yields a free resolution of the R-module R .

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