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By Uwe Jannsen

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Let A be a local ring with maximal ideal m and factor field k = A/m. 1 Let x be the closed point of X = Spec(A). A is called henselian, if the following equivalent conditions hold. , ⟨g0 , h0 ⟩ = k[X]), then there are monic polynomials g, h ∈ A[X] with f = g · h, g = g0 and h = h0 . Here let f = f mod m in k[X]; similarly for g and h). , ⟨g, h⟩ = A[X]). (a′ ) If f ∈ A[X] and f = g0 · h0 where g0 is monic and g0 and h0 are coprime, then there exist g, h ∈ A[X] with monic g, f = g · h, g = g0 and h = h0 .

Hence we obtain a uniquely determined morphism K → A with the desired property. (b) ⇒ (a): Consider a diagram (1) A❅ ❅❅ ❅❅f ❅❅ ❅1 B . cC ⑦⑦ ⑦ ⑦ ⑦⑦ g ⑦⑦ In the associated diagram (2) UAP ♥♥♥ y PPPPP PPfP pA ♥♥ P ♥ ♥ 1 2 f p PPPP ♥ A P9G q ♥♥♥ GA×B UG C PPP ♥ ♥ gpB ♥♥ PPP ♥ ♥ 4 PPP 3 ♥ pB ♥♥♥g qB PPPP PP9  ♥♥♥♥♥ qA ♥♥♥♥ K B let K be the difference kernel of f pA and gpB , and let qA = pA q and qB = qB q. Then all triangles 1 to 4 are commutative. We claim that K forms a fiber product of diagram (1) via qA and qB .

F p E n ⊇ F p+1 E n ⊇ . . and isomorphisms ∼ p,q E∞ → F p E p+q /F p+1 E p+q for all p, q ∈ Z. , { 0 for p >> 0, p n F E = E n for p << 0. Some spectral sequences begin with E2p,q ; then E1p,q does not exist, and all Erp,q are subquotients of E2p,q . We now explain how to operate with spectral sequences. 1) The layers: For each r one considers the Er -layer of all terms Erp,q and their differentials q E1 -layer: d1 • • • • • • • • p q • E2 -layer: • • • • • d2 • • • p q • dr Er -layer: r • dr r−1 p One has dr dr = 0 and Er+1 = ker dr /im dr .

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Étale cohomology [Lecture notes] by Uwe Jannsen

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