Complexe cotangent et deformations I by L. Illusie

By L. Illusie

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M }. For i = 1, . . , m, write Φi = (Φ1i (X1 ), . . , Φs−1 (X1 )). Let ρ > 0 be small enough such that for i 1 ≤ i ≤ m, 1 ≤ j ≤ s − 1, each Φji (X1 ) converges in 0 < |X1 | < ρ. We define j s Vρ∗ (R) := ∪m i=1 {x ∈ C | 0 < |x1 | < ρ, xj+1 = Φi (x1 ), j = 1, . . , s − 1}. Theorem 1. We have Vρ∗ (R) = Vρ (R). Proof. We prove this by induction on s. For i = 1, . . , s − 1, recall that hi is d1 the initial of ri . If s = 2, we have r1 (X1 , X2 ) = h1 (X1 ) i=1 (X2 − Φ1i (X1 )). So ∗ Vρ (R) = Vρ (R) clearly holds.

The first four equations of the cyclic 5-roots system C5 (x) = 0, define solution curves: ⎧ x0 + x1 + x2 + x3 + x4 = 0 ⎪ ⎪ ⎨ x0 x1 + x0 x4 + x1 x2 + x2 x3 + x3 x4 = 0 (14) f (x) = x0 x1 x2 + x0 x1 x4 + x0 x3 x4 + x1 x2 x3 + x2 x3 x4 = 0 ⎪ ⎪ ⎩ x0 x1 x2 x3 + x0 x1 x2 x4 + x0 x1 x3 x4 + x0 x2 x3 x4 + x1 x2 x3 x4 = 0. where v = (1, 1, 1, 1, 1). As the first four equations of C5 are homogeneous, the first four equations of C5 coincide with the first four equations of inv (C5 )(x) = 0. Because these four equations are homogeneous, we have lines of solutions.

Let ςi,j be i j j ςi,j the ramification index of Φi and (T , Xj+1 = ϕi (T )), where ϕji ∈ C T , be the corresponding Puiseux parametrization of Φji . Let ςi be the least common multiple of {ςi,1 , . . , ςi,s−1 }. Let gij = ϕji (T = T ςi /ςi,j ). We call the set GR := {(X1 = T ςi , X2 = gi1 (T ), . . , Xs = gis−1 (T )), i = 1, . . , M } a system of Puiseux parametrizations of R. Theorem 3. We have lim0 (W (R)) = GR (T = 0). Proof. It follows directly from Theorem 2 and Definition 2. Remark 1. The limit points of W (R) at X1 = α = 0 can be reduced to the computation of lim0 (W (R)) by a coordinate transformation X1 = X1 + α.

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