By Michel Raynaud

ISBN-10: 3540052836

ISBN-13: 9783540052838

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Yn−1) for certain elements of A with xi θ yi , i = 1, . . , n−1, then t(y, x1 , . . , xn−1 ) [θ, θ] t(y, y1, . . , yn−1 ) for any y ∈ A with x θ y. Applying this to the sixth variable of the above equation, we get mi+1 (mi (y, y, y, x), y, x, mi(y, y, y, x)) [θ, θ] mi+1 (mi (y, y, y, y), y, x, mi(x, x, x, x)) = mi+1 (y, y, x, x). Thus qi+1 (x, y, y) [θ, θ] mi+1 (y, y, x, x). Now suppose i is even and that (1) holds for i. Then qi+1 (x, y, y) [θ, θ] mi+1 (mi (y, y, x, x), x, y, mi(y, y, x, x)) as before.

This exercise will show that this correspondence is not as nice as one might hope. Namely if G is a group and H is a normal subgroup of G then [H, H] is the derived subgroup of H. In particular it depends only on H and not on G. However for loops this is not the case; [H, H] is not determined from just H. One needs the whole congruence associated with H to determine [H, H]. To see this let Z4 = {0, 1, 2, 3} denote the group of integers under addition modulo 4. Let G = Z4 ×Z4 and define a binary operation on G by a, b · c, d = a + c, b + d unless b = d = 1 EXERCISES 45 in which case the operation is defined by the following table: · 01 11 21 31 01 12 02 22 32 11 02 22 32 12 21 22 32 12 02 31 32 12 02 22 Show that G = G, · is a loop with identity element 0, 0 and that the second projection is a homomorphism.

7. The starting assumptions are the same as in the previous exercise. Now suppose that α and β are congruences of A and that either α is Abelian, or that p satisfies the other Mal’cev equation p(x, y, y) ≈ x. Prove that [α, β] = 0A if and only if p : D → A is a homomorphism, where D is the subalgebra of A3 consisting of the elements x, y, z with x α y β z. 7. 8. Let A and B be algebras in a modular variety and suppose that the congruences of A permute, and the congruences of B permute. Prove that the congruences of A × B permute.

### Anneaux locaux henseliens by Michel Raynaud

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