By James R. Milgram

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**Example text**

So we have to prove exactness at the middle term. For this, we use the fact that MTM(Gm) is a neutral Tannakian Q-linear category and all its non-zero objects have a The Motivic Vanishing Cycles and the Conservation Conjecture 47 strictly positive dimension (given by the trace of the identity). So to prove the exactness at the middle term we only need to show that dim(Log n+m+1 ) = dim(Log n ) + dim(Log m ). But this is true because dim(Log l ) = l + 1, which is an easy consequence of dim(K) = 2.

F I = f ! I. 2) (iii) Using adjunction, we get from the above pairing the desired natural / Ds Ψf (A) . 27. When A is constructible in SH(Xη ) , the morphism δf : Ψf Dη (A) / Ds Ψf (A) is an isomorphism. 11. First note that when A is constructible, Dη Dη (A) = A (by [3], chapter II). Thus we only / Ds Ψf Dη is need to prove that the natural transformation δf : Ψf an isomorphism when evaluated on constructible objects. 2). Now we have two specialization systems: Ψ and Ds ΨDη and a morphism δ? between them.

11 to a well chosen morphism between two specialization systems. 9): (a) (i) Ψ(a) , given by the formula: Ψf (A) = Ψf (A) ⊗ fs∗ Ψid F , (b) (ii) Ψ(b) , given by the formula: Ψf (A) = Ψf (A ⊗ fη∗ F ). One sees immediately that the composition in the statement of the theorem / Ψ(b) . Note also defines a morphism of specialization systems: Ψ(a) (a) (b) that Ψf and Ψf both commute with infinite sums. 11, we only need to consider the two special cases: k 42 J. Ayoub • f = en and A = I, • f = en and A = I.