A minicourse on Formal Geometry Doan Trung Cuong 30.10.2008 Abstract We introduce some very first notions in Formal Geometry. Two theorems which will be presented are the comparison theorem and the existence theorem. The talk relies mainly on the lectures of L. Illusie [3]. 1 Formal schemes 1.1 Formal spectrum 1.1.1 Adic ring Let A be a Noetherian topological ring, i. e., there is a linear topology on A whose an n open neighbourhood base of 0 ∈ A is provided by {I }n where I ⊂ A is an ideal. A is called an adic ring if it is separated and complete. In this case I is called an ideal of definition of A (see 1.2.5). 1.1.2 Formal spectrum A n+1 Fix an adic ring A. Denote Xn = /I . These affine Noetherian schemes have the A ∞ same underlying space as |X0| = | /I|. Moreover, over this space {On}n=0 is an inverse system of sheaves with the natural homomorphisms. Taking the limit one obtains a ringed space (X, O ) where X = |X | and O = lim O . This ringed space is called X 0 X ←−n Xn the formal spectrum of A and denoted by Spf A. 1.1.3 Remark. (i) X consists of all open primes of A. (ii) By the definition we have O (U) = lim Γ(U, O ) for any open subset U ⊆ X. X ←−n Xn In particular, OX(X) = A. (iii) For each f ∈ A, let Df = {open p ∈ Spec A : f 6∈ p}. Then O (D ) = lim A /In+1A = A X f ←− f f cf n the completion of Af with respect to the topology defined by IAf . (iv) Let x ∈ X be a point defined by an open ideal p ∈ Spec A. We have OX,x = lim A which is a local ring with the maximal ideal pO (for this one need the ←−f(x)6=0 cf X,x Noetherianness assumption). In general, OX,x is not a complete ring. 1 1.2 Formal schemes 1.2.1 Affine formal schemes A ringed space (X, OX) which is isomorphic to the formal spectrum Spf A for some adic ring A is called an affine formal spectrum. 1.2.2 Locally Noetherian formal schemes These are ringed spaces (X, OX) which is locally an affine formal scheme, i. e., for any point x ∈ X, there is an open neighbourhood U 3 x such that (U, OX |U ) is an affine formal scheme. 1.2.3 Morphisms A morphism between two locally Noetherian formal schemes f : X −→ Y is a morphism between ringed spaces which is local and for any affine open subset U = Spf A ⊆ Y, −1 f : Γ(U, OY) = A −→ Γ(f (U), OX) is a continous homomorphism between two topological rings. In particular, we also have Mor(Spf A, Spf B) = Homcont(B, A). 1.2.4 Coherent sheaves Over a ringed space one has already the notion of coherent sheaves. On a formal scheme (X, OX) this notion is more intuitive. An OX-module E is a coherent sheaf if and only if for any affine open subset U = Spf A ⊆ X, E | ' lim M/I^n+1M for a U ←−n finitely generated A-module M where I is an ideal of definition of A. In particular, OX is a coherent sheaf on (X, OX). 1.2.5 Ideals of definition An ideal of definition of a formal scheme (X, OX) is a coherent ideal I ⊂ OX such that for any afine open subset U = Spf A ⊆ X, I | ' lim I/I]n for an ideal of definition I of U ←−n A. It is easy to see that I is an ideal of definition if and only if (X, OX/I) is a scheme. In fact, ideals of definition always exist and there is a biggest one, i. e., an coherent ideal J such that (X, OX/J) is a reduced scheme. Then a coherent ideal I ⊂ OX is an ideal of definition iff J ⊇ I ⊇ Jn for an n > 0. 1.2.6 Thickening Formal schemes are easily described as colimits of chains of nilpotent thickenings. Fix n+1 an ideal of definition I of a formal scheme (X, OX) and denote Xn = (X, OX/I ). One gets a direct system of schemes X• = (X0 → X1 → ...) where each |Xn| = X and O = lim O (see also 1.1.2). X ←−n Xn Let un : Xn → X be the natural morphism of ringed spaces. Any OX-module ∗ ∞ F defines an inverse system of sheaves F• = (Fn = unF )n=0 where Fn is an OXn - module. Siminarly, a morphim of formal schemes f : X −→ Y defines a sequence of 2 morphism f• : X• −→ Y• consists of fn : Xn −→ Yn such that the following diagram is commutative un Xn −−−→ X f f ny y Yn −−−→ Y. vn ∗ This is because if J is an ideal of definition of fY then f (J)OX ⊆ I for an ideal of definition I of X. 1.2.7 Adic morphism ∗ If f (J)OX is an ideal of definition of X for an ideal of definition J of Y then f is called an adic morphism and X is called a Y-adic formal scheme. It is easy to see that f is adic iff the above diagram is cartesian. 1.2.8 Example. Let A be an adic ring with an ideal of definition. Denote A[T ] = A[T1,...,Tr]. (i) The ring of formal power series A[[T ]] is the completion of A[T ] with respect to the topology defined by the ideal I ⊕TA[T ] ⊂ A[T ]. Then the map Spf A[[T ]] −→ Spf A is not adic. The formal scheme Spf A[[T ]] is called the formal r-disc over Spf A and is r denoted by DSpf A. (ii) The ring of restricted formal power series A{T } is the completion of A[T ] with respect to the topology defined by the ideal IA[T ] ⊂ A[T ] of polynomials with coefficents in I. The map Spf A{T } −→ Spf A is adic and Spf A{T } is called the affine formal r r-space over Spf A. It is denoted by ASpf A. It is worth noting that the underlying space r r of ASpf A is the same as of the affine space Spec A/I[T ] = ASpec A/I . 1.3 Completion of a scheme along a closed subscheme Formal schemes come naturally from the usual theory of schemes. Let X be a locally Noetherian scheme and X0 ⊆ X be a closed subscheme of X which is defined by an n+1 ideal I ⊆ OX. Again one denotes Xn = Spec OX/I and obtains a chain of thinkening ˆ X• = (X0 → X1 → ...). Taking the colimit of X• we get a formal scheme X where 0 |Xˆ| = |X | and O ˆ = lim O . One calls Xˆ the completion of X along the closed X ←−n Xn subscheme X0. 1.3.1 Remark. (i) A formal scheme is called algebraizable if it is the completion of some locally Noetherian scheme. (ii) If X = Spec A and X0 = Spec A/I then Xˆ = Spf Aˆ where Aˆ is the completion of A with respect to the topology defined by I. (iii) There are commutative diagrams in X /X n? @ ?? ÑÑ ?? ÑÑ un ?? ÑÑ i ? ÑÑ Xˆ where in is the canonical closed embedding and i is flat. An OX-module F defines an ∗ ∞ inverse system of sheaves {Fn = inF }n=0 where Fn is an OXn -module. By taking the 3 ∗ limit one gets an O ˆ -module Fˆ = lim F . If F is coherent then in fact, Fˆ ' i F X ←−n n which is called the completion of F . Similarly, any morphism f : X −→ Y such that X0 ⊆ f −1(Y 0) (hence, there is a 0 0 restriction f |X0 : X −→ Y ) also defines an inverse system of morphisms of scheme f• : X• −→ Y• such that the diagram fn Xn −−−→ Yn x x fm Xm −−−→ Ym ˆ ˆ ˆ n 6 m, is commutative. Hence f defines a morphism f : X −→ Y by taking the limit. Note that fˆ is adic iff f −1(Y 0) = X0. 2 The comparison theorem 2.1 Base change maps 2.1.1 General case Let’s consider a commutative diagram of ringed spaces X0 −−−→h X 0 f f y y Y 0 −−−→ Y g ∗ q and let F be an OX-module. Denote a = f ◦ h. There are maps γ : g R f∗F −→ q 0 ∗ R f∗h F which are called base change maps. γ is defined via the adjunction formula q q 0 ∗ by γ1 : R f∗F −→ g∗R f∗h F which is the composition of two other maps q q ∗ α : R f∗F −→ R a∗h F, q ∗ q 0 ∗ β : R a∗h F −→ g∗R f∗h F. ∗ ∗ To construct α, choose an h∗-acyclic resolution C(h F ) of h F , via the adjunction map ∗ ∗ + F → h∗h F → h∗C(h F ) one gets a map in D (X) ∗ F −→ Rh∗h F. q q ∗ Applying Rf∗ to this map one gets α : R f∗F −→ R a∗h F . For an open V ⊆ Y , let V 0 = g−1(V ) and U 0 = a−1(V ). The second map β is the associated map to the restriction q 0 ∗ 0 0 q 0 ∗ H (U , h F ) −→ H (V ,R f∗(h F ). 4 2.1.2 Completion morphism Let f : X −→ Y be a morphism of Noetherian schemes, Y 0 ⊂ Y a closed subscheme and X0 = f −1(Y −1).