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ISBN-10: 2729827889

ISBN-13: 9782729827885

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Extra info for 30 ouvrages de mathématiques qui ont changé le monde

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R) (2) Now we use the following fact: As M is a complete intersection R-module (it is the quotient of R by a regular sequence), it has Poincaré duality in the sense that mult there is a map M → Q such that Γ : M ⊗ M −→ M → Q is a perfect pairing. Take elements ν, μj ∈ M, j = 1, . . , n, such that Γ ν, xj = 0, j = 1, . . , n, Γ μj , xk = δj k , j , k = 1, . . , n, Γ (ν, 1) = 1, Γ μj , 1 = 0, Γ (ν, Q) = Γ μj , Q = 0, for any quadratic Q ∈ R. Since the elements ν, μj are in the kernel of (2) and they are linearly independent, we get dim Tor rS (M, Q0 ) ≥ n + 1.

So, r = 3 and n = 3. It remains to deal with the case n = 3, r = 3, and do is an isomorphism of the odd degree elements onto Λ2 V even . Let x1 , x2 , x3 be the even-degree generators, of degrees d1 ≤ d2 ≤ d3 , respectively. The degrees of x12 , x1 x2 , x22 , x1 x3 , x2 x3 , x32 are the six numbers 2d1 ≤ d1 + d2 ≤ 2d2 , d1 + d3 ≤ d2 + d3 ≤ 2d3 . We have two cases: • Case 2d2 ≤ d1 +d3 . We can arrange the odd generators y1 , . . , y6 with increasing degree and so that do y1 = x12 , do y2 = x1 x2 , do y3 = x22 , do y4 = x1 x3 , do y5 = x2 x3 , do y6 = x32 .

Continuing the computation, the even-degree closed elements in Λ(x1 , x2 , x3 , y1 , y2 , y3 ) are either polynomials on the xi ’s or a multiple of the element x22 y1 y2 − x1 x2 y1 y3 + x12 y2 y3 = d(y1 y2 y3 ), which is exact. Therefore, we can again manage to arrange that dy4 = x1 x3 . • Case 2d2 > d1 + d3 . Then, we have that do y3 = x1 x3 and do y4 = x22 . As before, we can arrange dy3 = x1 x3 . Now the even-degree closed elements in Λ(x1 , x2 , x3 , y1 , y2 , y3 ) are polynomials on the xi ’s or a multiple of x3 y1 y2 − x2 y1 y3 + x1 y2 y3 .