- Page 2 and 3: The dissertation of Alice Medvedev
- Page 4 and 5: Abstract Minimal sets in ACFA by Al
- Page 6 and 7: To friends and colleagues who have
- Page 8 and 9: 3.3.1 Localizations and quasi-coher
- Page 10 and 11: Chapter 1 Introduction 1.1 Backgrou
- Page 12 and 13: ise to a difference-Zariski topolog
- Page 14 and 15: ACFA satisfies a strong form of the
- Page 16 and 17: 1.1.5 Non-orthogonality Another mod
- Page 18 and 19: • (field-like) p is non-orthogona
- Page 20 and 21: Theorem 1. Sb is never mixed. Furth
- Page 22 and 23: sets of a special form. Nevertheles
- Page 24 and 25: Definition 12. A curve is a project
- Page 26 and 27: Several times we encounter a defina
- Page 28 and 29: • (A) There is a non-trivial morp
- Page 30 and 31: such that θ = ψ ◦ φ. We show t
- Page 32 and 33: In this diagram, the horizontal arr
- Page 34 and 35: for some difference polynomials pi,
- Page 36 and 37: • Each V ′ i is not Zariski-den
- Page 38 and 39: Proof. It is obvious that the dimen
- Page 40 and 41: that people in A1 still have friend
- Page 42 and 43: in ACFA: [CH99] gives the following
- Page 44 and 45: 2.4 Stronger conclusions under extr
- Page 46 and 47: We pause for a chapter to construct
- Page 48 and 49: properties: • If f : X → Y is a
- Page 50 and 51: 3.1.2 Commutative diagram for jets
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3.1.4 Three rings To finish functor

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So w := a ⊗ 1 + j (bj ⊗ 1)r j

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localizations. The purpose of this

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Theorem 6. For a scheme X associate

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Then the fiber of Spec(Sym(J )) ove

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so q(x) ∈ ker(e) = N. Hence, I(B)

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of jet spaces factors as J m Spec(S

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• Let f, g : X → Y be the morp

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Proof. By “R/I”, we mean the im

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4.1 Part 1, in which the group corr

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Starting diagram (black) Unnecessar

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The new group G1 and some group mor

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G Γ Gσ . . . . . . . . . . . . .

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This construction can be repeated a

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Lemma 4.4. Unless ψ1 is an isomorp

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4.2 Part 2, in which the intermedia

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4.2.2 Elliptic curves If G is an el

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So now our diagram is φ P1 −−

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only such morphisms are x ↦→ a

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and this identification is uniform

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• Conjugating µ m ab by (+a)m e

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So we need to show that {φ(b) | π

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size nm of its fibers is genericall

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1. ζ(g) = ζ(h), which implies 2.

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Theorem 10. The minimal set defined

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4.4.3 Ga The additive group is rele

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Bibliography [AM69] M. F. Atiyah an