Algebraic space

Abstract: ing, now let π : Y → W be a proper morphism of schemes and let αi : Mi → OY , i = 1, 2, be two fine saturated log structures on Y . Consider the functor (2.1) LMorY/W (M1, M2) : (Sch/W ) −→ (Sets) that on objects is defined by V %−→ { φ : (M1)V → (M2)V morphism of log structures } . Then the statement that W ×M×M(X) M (X) is an algebraic space10 essentially is a special case of the following proposition. Proposition 2.9. LMorY/W (M1, M2) is represented by an algebraic space locally of finite type over W . The proof is provided, after some preparations, at the end of this subsection. This then finishes the proof of Proposition 2.3 in the case S = Spec k with the trivial log structure. In the general case we have in addition two morphisms of log structures ψi : πMS → Mi, π : C → W , the projection, and we need to restrict to those φ : (M1)V → (M2)V compatible with ψi. But by Proposition 2.9 composition with ψ1 defines a morphism of algebraic spaces LMorY/W (M1, M2) −→ LMorY/W (πMS , M2). Now W ×M×M(X) M (X) arises as the fibre product with the morphism W −→ LMorY/W (πMS , M2) defined by ψ2, and is hence represented by an algebraic space. This finishes the proof of Proposition 2.3 also in the general case. Remark 2.10. One problem in showing representability of LMorY/W (M1, M2) is that it is non-separated, essentially because the induced map M1 → M2 cannot be determined by its restriction to an open dense subset. As an example (cf. [Ol2], Remark 3.12) consider the log structure M on A1 = Spec k[x] with chart N −→ k[x], (a, b) %−→ x. The map N2 → N2, (a, b) %→ (b, a) induces a non-trivial automorphism of M that restricts to the identity on A1 \ {0}. To find an etale cover of the algebraic space representing LMorY/W (M1, M2) in Proposition 2.9 we thus first restrict the map φ. To this end let w → W be a geometric point and let φw : (M1)w → (M2)w be a choice of φ over one geometric fibre Yw of Y → W . Now since Mi are fine sheaves the choice at a geometric point x → Y determines φ at any generization y of x. Moreover, if y specializes to some other point z such that the generization map Mi,z → Mi,y is an isomorphism, then φ is also determined at z. Iterating the generization-specialization process we are lead to the following definition. 10As for algebraic stacks we have to drop the condition of quasi-separatedness from the definition of algebraic spaces ([Kt], Ch.II, Definition 1.1). Licensed to Brown Univ. Prepared on Fri Oct 4 09:27:34 EDT 2013 for download from IP 128.148.231.12. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 474 MARK GROSS AND BERND SIEBERT Definition 2.11. Let M be a fine sheaf on a scheme Y and let A ⊂ |Y | be a set of geometric points. We say x ∈ |Y | has property (Agen) with respect to M if there exists a sequence y1, z1, . . . , yr, zr ∈ |Y | for some r with the following properties: y1 ∈ A, zr = x, yi ∈ cl(zi), i = 1, . . . , r, yi ∈ cl(zi−1) and Myi → Mzi−1 is an isomorphism, i = 2, . . . , r. Thus the giving of φ on a closed subset A ⊂ |Y | then determines φ also on the subset UA := { x ∈ |Y | ∣x fulfills (Agen) } of |Y |. Note that by definition UA is closed under generization. Since M is a fine sheaf it is also immediate that UA is a constructible subset of |Y |, and hence UA ⊂ |Y | is open. Since the statement of Proposition 2.9 is local in W and by properness of Y → W we may assume any point of |Y | fulfills (Agen) for A = Yw with respect both to M1 and to M2, that is, UYw = Y . Then for any V → W there is at most one φ : (M1)V → (M2)V compatible with φw under sequences of generization maps. Let us call such φ (or a lift φ to a morphism of log structures) compatible with φw, and similarly for any A ⊂ |Y |. Note that φw may not extend to Y , but it may do so after certain base changes. We first treat the representability problem locally on Y , that is, for Y = W . Lemma 2.12. Let Y = W and suppose that there exists a closed subset A ⊂ Y such that any x ∈ |Y | fulfills (Agen) with respect to both Mi (Definition 2.11). Let φA : (M1)A −→ (M2)A be a homomorphism of sheaves of monoids. Then the functor LMor φA Y : (Y ′ f −→Y ) %−→ { φ : (Y ′, fM2)→(Y ′, fM1) ∣φ is compatible with φA } is represented by a scheme LMorA Y of finite type and affine over Y . Proof. It is sufficient to prove the statement on an etale open cover of Y , since we can then use descent for affine morphisms ([SGA1], VIII, Theorem 2.1) to obtain a scheme over Y . Thus we can assume that we in fact have charts ψi : Pi → Γ(Y, Mi) for the two log structures. We can also assume that φA is induced by a homomorphism of monoids φA : P1 → P2. Let p1, . . . , pn ∈ P1 be a generating set for P1 as a monoid. Consider the sheaf of finitely generated OY -algebras FY := OY [P gp 1 ]/〈α1(ψ1(pi)) − zα2(ψ2(φA(pi))) | 1 ≤ i ≤ n〉. Then the desired scheme is LMorA Y := SpecFY . To see that this is the correct scheme, suppose f : Y ′ → Y is given. We wish to show that giving a commutative diagram of schemes