Characteristic cycles and the conductor of direct image
aa r X i v : . [ m a t h . AG ] A p r Characteristic cycles and the conductor of direct image
Takeshi SaitoSeptember 17, 2018
Abstract
We prove the functoriality for proper push-forward of the characteristic cycles ofconstructible complexes by morphisms of smooth projective schemes over a perfectfield, under the assumption that the direct image of the singular support has thedimension at most that of the target of the morphism. The functoriality is deducedfrom a conductor formula which is a special case for morphisms to curves. Theconductor formula in the constant coefficient case gives the geometric case of aformula conjectured by Bloch.
Let k be a perfect field and Λ be a finite field of characteristic invertible in k . For aconstructible complex F of Λ-modules on a smooth scheme X over k , the characteristiccycle CC F is defined in [16, Definition 5.10] as a cycle supported on the singular support SS F defined by Beilinson in [2] as a closed conical subset of the cotangent bundle T ∗ X .We study the functoriality of characteristic cycles for proper push-forward.Let f : X → Y be a morphism of smooth projective schemes over k . Then, we provein Theorem 2.2.5 the equality(2 . CCRf ∗ F = f ! CC F conjectured in [17, Conjecture 1] under the assumption dim f ◦ SS F ≦ dim Y for the directimage f ◦ SS F ⊂ T ∗ Y . The precise definitions will be given in Subsection 2.1. We canslightly weaken the assumption, as is seen in Theorem 2.2.5. The formula (2.11) is analgebraic analogue of [12, Proposition 9.4.2] where functorial properties of characteristiccycles are studied in a transcendental context. In the case where Y = Spec k , the equality(2.11) is the index formula(2 . χ ( X ¯ k , F ) = ( CC F , T ∗ X X ) T ∗ X computing the Euler-Poincar´e characteristic as an intersection number proved in [16, The-orem 7.13].We deduce the functoriality (2.11) from the index formula (2.12) in Subsection 2.2 asfollows. By taking a projective embedding of Y and a good pencil, we reduce it to the casewhere Y is a projective smooth curve. By the index formula (2.12) applied to a generalfiber, the equality (2.11) is equivalent to a conductor formula(2 . − a y Rf ∗ F = ( CC F , df ) T ∗ X,X y proved in Theorem 2.2.3, where the left hand side denotes the Artin conductor at a closedpoint y ∈ Y of the direct image. In the case where F is the constant sheaf Λ, the right1and side equals the localized self-intersection product defined in [4] and the formula (2.17)specializes to the geometric case, Corollary 2.2.4, of the conductor formula conjectured in[4] by Bloch.Further the index formula implies that we have an equality (2.18) for the sums over y ∈ Y of the both sides in (2.17). To deduce (2.17) from (2.18) for the sums, it suffices toshow the existence of a covering of Y ´etale at a fixed point y killing the contributions ofthe other points.For the vanishing of the left hand side, the local acyclicity of f : X → Y relative to F is a sufficient condition. The SS F -transversality of f : X → Y defined in Definition1.3.3 and studied in Subsection 1.3 after some preliminaries in Subsection 1.2 is a strongercondition and is a sufficient condition for the vanishing of the right hand side. Thus,the proof of (2.17) is reduced to showing variants of the stable reduction theorem on theexistence of ramified covering of Y such that the base change of f : X → Y is locallyacyclic relatively to a modification F ′ of the pull-back and is SS F ′ -transversal.We show that f : X → Y is locally acyclic relatively to a modification of a perversesheaf F if the inertia action on the nearby cycles complex R Ψ F is trivial in Proposition1.1.2.2. This is rather a direct consequence of the relation of the direct image by the openimmersion of the generic fiber with the nearby cycles complex. As we work with torsioncoefficients, the condition is satisfied over a ramified covering of Y .Further, we show that the local acyclicity of f : X → Y relatively to F implies theexistence of a ramified covering such that the base change of f : X → Y is SS F ′ -transversalfor the pull-back F ′ of F in Corollary 1.5.4 of Theorem 1.5.2. Theorem 1.5.2 is deducedfrom a weaker version Proposition 1.4.4 which is proved by using the alteration [5, Theorem8.2]. In Proposition 1.4.4, the ramified covering may inseparable, while it is generically´etale in Theorem 1.5.2. Theorem 1.5.2 is proved by an argument similar to that in theproof of [6, Proposition 3.2] by using a consequence of the stable reduction theorem [18,Theorem 1.5].We also prove an index formula Proposition 2.3.3 for vanishing cycles complex.The author thanks A. Beilinson for the remark that Theorem 2.2.3 implies the geomet-ric case of the conductor formula conjectured by Bloch in [4] and for showing the proof ofLemma 2.2.6 in the characteristic zero case. The author thanks H. Haoyu for discussionon the subject of Subsection 2.3 and thanks H. Kato for pointing out an error in the proofof Proposition 2.3.3 in an earlier version.The research was supported by JSPS Grants-in-Aid for Scientific Research (A) 26247002. Contents C -transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 SS F -transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Alteration and transversality . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5 Potential transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 We fix some conventions on perverse sheaves. Let X be a noetherian scheme and let Λ bea finite field of characteristic ℓ invertible on X . We say that a complex F of Λ-modules onthe ´etale site of X is constructible if the cohomology sheaf H q F is constructible for everyinteger q and H q F = 0 except for finitely many q . Let D bc ( X, Λ) denote the category ofconstructible complexes of Λ-modules.First we recall the case where X is a scheme of finite type over a field k . Let Λ bea finite field of characteristic ℓ invertible in k . Then, the t -structure ( p D ≦ , p D ≧ ) on D bc ( X, Λ) relative to the middle perversity is defined in [3, 2.2.10] and the perverse sheaveson X form an abelian subcategory Perv( X, Λ) = p D ≦ ∩ p D ≧ . Next, we consider the case where X is a scheme of finite type over the spectrum S of a discrete valuation ring as in [11, 4.6]. Let s and η denote the closed point and thegeneric point of S respectively and let i : X s → X and j : X η → X be the closed immersionand the open immersion of the fibers. Let Λ be a finite field of characteristic ℓ invertibleon S . Then, we consider the t -structure on D bc ( X, Λ) obtained by gluing ([3, 1.4.10]) the t -structure ( p D ≦ , p D ≧ ) on D bc ( X s , Λ) and the t -structure ( p D ≦ − , p D ≧ − ) on D bc ( X η , Λ).In particular, a constructible complex
F ∈ D bc ( X, Λ) is contained in p D ≦ if and only if wehave i ∗ F ∈ p D ≦ and j ∗ F ∈ p D ≦ − .Note that if the t -structure on D bc ( X η , Λ) where X η is regarded as a scheme over η is ( p D ≦ , p D ≧ ), then that on D bc ( X η , Λ) where X η is regarded as a scheme over S is( p D ≦ − , p D ≧ − ). To distinguish them, we call the former the t -structure on X η over η andthe latter the t -structure on X η over S . We use the same terminology for perverse sheaveson X η .The functors j ! , Rj ∗ : D bc ( X η , Λ) → D bc ( X, Λ) are t -exact with respect to the t -structureon X η over S . This follows from [1, Th´eor`eme 3.1] by the argument in [11, 4.6 (a)]. Let F ∈
Perv( X η , Λ) be a perverse sheaf on X η over S . Then the intermediate extension j ! ∗ F ∈
Perv( X, Λ) is defined as the image j ! ∗ F = Im( j ! F → Rj ∗ F ) . We have p H q i ∗ Rj ∗ F = 0 for q = 0 , − j ! ∗ F → Rj ∗ F induces anisomorphism(1.1) i ∗ j ! ∗ F → p H − i ∗ Rj ∗ F . This is deduced similarly as [3, (4.1.12.1)] from a consequence [3, (4.1.11.1)] of the t -exactness of the functors j ! and Rj ∗ .Assume that S is strictly local. Let ¯ η be a geometric point above η and let ¯ j : X ¯ η → X η denote the canonical morphisms. Then, the nearby cycles functor R Ψ = i ∗ R ( j ¯ j ) ∗ ¯ j ∗ : D bc ( X η , Λ) → D bc ( X s , Λ)is t -exact with respect to the t -structure on X η over η [11, Corollaire 4.5].3 emma 1.1.1. Let S = Spec O K be the spectrum of a strictly local discrete valuationring and let s and η denote the closed and the generic point of S respectively. Let X be ascheme of finite type over S , and let i : X s → X and j : X η → X denote the immersions.Let F be a perverse sheaf of Λ -modules on X η over η . Then the morphism i ∗ Rj ∗ F → R Ψ F induces an isomorphism (1.2) p H i ∗ Rj ∗ F → ( R Ψ F ) I to the inertia fixed part as a perverse sheaf on X s .Proof. Let P ⊂ I denote the wild inertia subgroup. Then, since the functor taking the P -invariant parts is an exact functor, we have an isomorphism i ∗ Rj ∗ F → R Γ( I/P, ( R Ψ F ) P ).Since the profinite group I/P is cyclic, the assertion follows.We study the local acyclicity of a morphism to the spectrum of a discrete valuationring with respect to a perverse sheaf.
Proposition 1.1.2.
Let S = Spec O K be the spectrum of a discrete valuation ring andlet s and η denote the closed and the generic point of S respectively. Let X be a schemeof finite type over S , and let i : X s → X and j : X η → X denote the immersions. Let G be a perverse sheaf of Λ -modules on X . If X → S is locally acyclic relativelyto G , then G has no non-zero subquotient supported on the closed fiber and is isomorphicto j ! ∗ j ∗ G . For a perverse sheaf F of Λ -modules on X η over S , the following conditions areequivalent: (1) The morphism X → S is locally acyclic relatively to j ! ∗ F . (2) Let ¯ s be a geometric point above the closed point s ∈ S and let ¯ i : X ¯ s → X denotethe canonical morphism. Then, the canonical morphism (1.3) ¯ i ∗ j ! ∗ F → R Ψ F is an isomorphism. (3) The inertia group I of K acts trivially on the nearby cycles complex R Ψ F . (4) The formation of j ! ∗ F commutes with the pull-back by faithfully flat morphisms S ′ → S of the spectra of discrete valuation rings.Proof.
1. The local acyclicity is equivalent to the vanishing R Φ G = 0. Since the shiftedvanishing cycles functor R Φ[ −
1] : D bc ( X, Λ) → D bc ( X s , Λ) is t -exact [11, Corollaire 4.6], itis reduced to the case where G is a simple perverse sheaf. If G is supported on the closedfiber, we have R Φ G [ −
1] = G and the assertion follows.2. (1) ⇔ (2): The condition (2) is equivalent to that for every geometric point x of X s ,the canonical morphism j ! ∗ F x → R Γ( X ( x ) × S ( s ) ¯ η, F ) is an isomorphism.(2) ⇔ (3): By (1.1) and (1.2), the morphism (1.3) induces an isomorphism ¯ i ∗ j ! ∗ F → ( R Ψ F ) I .(2) ⇒ (4): Since the formation of nearby cycles complex R Ψ F commutes with basechange [6, Proposition 3.7], the isomorphism (1.3) implies the condition (4).(4) ⇒ (2): There exists a finite extension K ′ of K such that the inertia action I ′ ⊂ I on R Ψ F is trivial, since Λ is a finite field. Let j ′ : X K ′ → X S ′ denote the base changeof the open immersion j by S ′ = Spec O K ′ → S , let i ′ : X ¯ s → X S ′ denote the canonicalmorphism and let F ′ denote the pull-back of F on X K ′ . We factorize the morphism (1.3)4s the composition of ¯ i ∗ j ! ∗ F → ¯ i ′∗ j ′ ! ∗ F ′ → R Ψ F . By (3) ⇒ (2) already proven, the secondarrow is an isomorphism. The condition (4) implies that the first arrow is an isomorphism.Hence the composition (1.3) is an isomorphism.Finally, we consider the case where X is a scheme of finite type over a regular noetherianconnected scheme S of dimension 1. Let Λ be a finite field of characteristic ℓ invertibleon S . Then the t -structure ( p D ≦ , p D ≧ ) on D bc ( X, Λ) is defined as the intersection of theinverse images of the t -structures ( p D ≦ , p D ≧ ) on D bc ( X × S S s , Λ) for the base changesby the localizations S s → S at closed points s ∈ S . If Y = S is a smooth curve over afield k and if f : X → Y is a morphism of schemes of finite type over k , the t -structure on D bc ( X, Λ) defined above is the same as that defined by considering X as a scheme of finitetype over k . Corollary 1.1.3.
Let S be a regular noetherian scheme of dimension . Let X be a schemeof finite type over S and F be a perverse sheaf of Λ -modules on X . Let V ⊂ S be a denseopen subscheme such that the base change X V → V is universally locally acyclic relativelyto the restriction F V of F .Then, there exists a finite faithfully flat and generically ´etale morphism S ′ → S ofregular schemes such that the base change X ′ → S ′ is locally acyclic relatively to j ′ ! ∗ F V ′ where F V ′ denotes the pull-back of F on V ′ = V × S S ′ and j ′ : X ′ V ′ → X ′ denote the basechange.Proof. By Proposition 1.1.2.2 (1) ⇒ (4) and weak approximation, it suffices to considerlocally on a neighborhood of each point of the complement S V . Since the coefficientfield Λ is finite, the assertion follows from Proposition 1.1.2.2 (3) ⇒ (1). C -transversality We introduce some terminology on proper intersection.
Lemma 1.2.1.
Let f : C → X and h : W → X be morphisms of schemes of finite typeover a field k . Assume that C is irreducible of dimension n and that h is locally ofcomplete intersection of relative virtual dimension d . Then every irreducible component of h ∗ C = C × X W is of dimension ≧ n + d .Proof. Since the assertion is local on W , we may decompose h = gi as the composition ofa smooth morphism g with a regular immersion of codimension c . Since the assertion isclear for g , we may assume that h = i is a regular immersion. Then, it follows from [8,Proposition (5.1.7)]. Definition 1.2.2.
Let f : C → X and h : W → X be morphisms of schemes of finite typeover a field k . Assume that every irreducible component of C is of dimension n and that h is locally of complete intersection of relative virtual dimension d . We say that h : W → X meets f : C → X properly if h ∗ C = C × X W is of dimension n + d . By Lemma 1.2.1, the condition that h ∗ C = C × X W is of dimension n + d is equivalentto the condition that every irreducible component of h ∗ C = C × X W is of dimension n + d . Lemma 1.2.3.
Let f : C → X be a morphism of schemes of finite type over a field k . Assume that X is equidimensional of dimension m and that C is equidimensional ofdimension n ≧ m . We consider the following conditions: Every morphism h : W → X locally of complete intersection meets C properly. (2) For every closed point x of X , the fiber C × X x is of the dimension n − m . We have (2) ⇒ (1) . Assume that the condition (2) is satisfied and let h : W → X bea morphism locally of complete intersection of relative virtual dimension d of schemes offinite type over k . Then C × X W is equidimensional of dimension n + d and the morphism C × X W → W satisfies the condition (2) and hence (1) . If X is regular, we have (1) ⇒ (2) . Assume that X = P is a projective space and let c be an integer. Then, the linearsubspaces V ⊂ P of codimension c such that the immersion V → P meets C properly forma dense open subset of the Grassmannian variety G .Proof.
1. Assume that the condition (2) is satisfied and let h : W → X be a morphismlocally of complete intersection of relative dimension d . Then, we have dim C × X W ≦ dim W + n − m = n + d . Hence, C × X W is equidimensional of dimension n + d by Lemma1.2.1. The rest is clear.2. If X is regular and x is a closed point, the closed immersion i : x → X is a regularimmersion of codimension m and hence the condition (1) implies that dim C × X x = n − m .3. Let V ⊂ P × G be the universal family of linear subspaces of codimension c andwe consider the cartesian diagram C V / / (cid:3) (cid:15) (cid:15) } } ⑤⑤⑤⑤⑤⑤⑤⑤ C (cid:15) (cid:15) G V o o / / P . Then, since the projection V → P is smooth of relative dimension dim G − c , we havedim C V = dim G + n − c . Hence the open subset of G consisting of V such that dim C × P V ≦ n − c is dense.Recall that a closed subset C of a vector bundle E on a scheme X is said to be conical if it is stable under the action of the multiplicative group. For a closed conical subset C ⊂ E , the intersection B = C ∩ X with the 0-section identified with a closed subset of X is called the base of C . We say that a morphism f : X → Y of noetherian schemes isfinite (resp. proper) on a closed subset Z ⊂ X if its restriction Z → Y is finite (resp.proper) with respect to a closed subscheme structure of Z ⊂ X . Definition 1.2.4.
Let f : X → Y be a morphism of smooth schemes over a field k andlet C ⊂ T ∗ X be a closed conical subset.
1. ([2, 1.2])
We say that f : X → Y is C - transversal if the inverse image of C by thecanonical morphism X × Y T ∗ Y → T ∗ X is a subset of the -section X × Y T ∗ Y Y ⊂ X × Y T ∗ Y . Assume that every irreducible component of X is of dimension n and that everyirreducible component of C is of dimension n . Assume that every irreducible component of Y is of dimension m ≦ n . We say that f : X → Y is properly C -transversal if f : X → Y is C -transversal and if for every closed point y of Y , the fiber C × Y y is of dimension n − m . Definition 1.2.5.
Let h : W → X be a morphism of smooth schemes over a field k andlet C ⊂ T ∗ X be a closed conical subset. Let K ⊂ W × X T ∗ X be the inverse image of the -section T ∗ W W ⊂ T ∗ W by the canonical morphism W × X T ∗ X → T ∗ W .
6. ([2, 1.2])
We say that h : W → X is C - transversal if the intersection ( W × X C ) ∩ K ⊂ W × X T ∗ X is a subset of the -section W × X T ∗ X X .If h : W → X is C -transversal, a closed conical subset h ◦ C ⊂ T ∗ W is defined to be theimage of h ∗ C = W × X C by W × Y T ∗ X → T ∗ W .
2. ([16, Definition 7.1])
Assume that every irreducible component of X is of dimension n and that every irreducible component of C is of dimension n . Assume that every irreduciblecomponent of W is of dimension m . We say that h : W → X is properly C -transversal if h : W → X is C -transversal and if h : W → X meets C → X properly. If h : W → X is C -transversal, the morphism W × X T ∗ X → T ∗ W is finite on h ∗ C = W × X C and hence h ◦ C ⊂ T ∗ W is a closed subset by [2, Lemma 1.2 (ii)]. For a morphism r : X → Y of smooth schemes proper on the base B = C ∩ T ∗ X X ⊂ X of a closed conicalsubset C ⊂ T ∗ X , the closed conical subset r ◦ C ⊂ T ∗ Y is defined to be the image bythe projection X × Y T ∗ Y → T ∗ Y of the inverse image of C by the canonical morphism X × Y T ∗ Y → T ∗ X . Lemma 1.2.6.
Let f : X → Y be a smooth morphism of smooth schemes over a field k and let C ⊂ T ∗ X be a closed conical subset. Let X h ←−−− W f y (cid:3) y g Y i ←−−− Z be a cartesian diagram of smooth schemes over k . Assume that f : X → Y is C -transversal (resp. properly C -transversal). Then, h : W → X is C -transversal (resp. properly C -transversal) and g : W → Z is h ◦ C -transversal (resp. properly h ◦ C -transversal). Assume that f : X → Y is proper on the base of C . Then, i : Z → Y is f ◦ C -transversal if and only if h : W → X is C -transversal. If these equivalent conditions aresatisfied, we have i ◦ f ◦ C = g ◦ h ◦ C .Proof.
1. The assertion for the transversality is proved in [16, Lemma 3.9.2]. The propertransversality of h : W → X follows from the transversality and Lemma 1.2.3 applied to C → Y and Z → Y . The proper h ◦ C -transversality of g : W → Z follows from that h ∗ C → h ◦ C is finite.2. We consider the commutative diagram T ∗ X ←−−− W × X T ∗ X dh −−−→ T ∗ W x (cid:3) x x X × Y T ∗ Y ←−−− W × Y T ∗ Y g ∗ ( di ) −−−→ W × Z T ∗ Z y (cid:3) y g ∗ (cid:3) y T ∗ Y ←−−− Z × Y T ∗ Y di −−−→ T ∗ Z with cartesian squares indicated by (cid:3) . The upper vertical arrows are injections. Since dh induces an isomorphism W × X T ∗ X/Y → T ∗ W/Z for the relative cotangent bundles and f : X → Y is smooth, the upper right square is also cartesian.7et K and K ′ be the inverse image of the 0-sections by dh : W × X T ∗ X → T ∗ W and di : Z × Y T ∗ Y → T ∗ Z respectively. Since the upper right square is cartesian, K isidentified with the inverse image of the 0-section by g ∗ ( di ) : W × Y T ∗ Y → W × Z T ∗ Z which equals g − ∗ ( K ′ ) ⊂ W × Y T ∗ Y .Since the lower left square is cartesian, the pull-back Z × Y f ◦ C is the image g ∗ ( C ′ ) of C ′ = ( W × X C ) ∩ ( W × Y T ∗ Y ). Hence the condition that ( Z × Y f ◦ C ) ∩ K ′ = g ∗ ( C ′ ) ∩ K ′ = g ∗ ( C ′ ∩ g − ∗ ( K ′ )) is a subset of the 0-section is equivalent to the condition that( W × X C ) ∩ K = C ′ ∩ g − ∗ ( K ′ ) is a subset of the 0-section.If these conditions are satisfied, the equality i ◦ f ◦ C = g ◦ h ◦ C follows from the cartesiandiagram. Lemma 1.2.7.
Let P be a projective space of dimension n and let C ⊂ T ∗ P be a closedconical subset. Let P ∨ be the dual projective space, let Q ⊂ P × P ∨ be the universal family ofhyperplanes and let (1.4) P p ←−−− Q p ∨ −−−→ P ∨ be the projections. Let C ∨ = p ∨◦ p ◦ C be the Legendre transform. Let V ⊂ P be a linearsubspace and let V ∨ ⊂ P ∨ be the dual subspace. Then the immersion V → P is C -transversal if and only if V ∨ → P ∨ is C ∨ -transversal. Assume that every irreducible component of C is of dimension n = dim P andlet ≦ c ≦ n be an integer. Then, the linear subspaces V ⊂ P of codimension c suchthat the immersion V → P is properly C -transversal form a dense open subset of theGrassmannian variety G .Proof.
1. The C -transversality of V → P means P ( T ∗ V P ) ∩ P ( C ) = ∅ and similarly for the C ∨ -transversality of V ∨ → P ∨ . Then, the assertion follows from P ( T ∗ V P ) = P ( T ∗ V ∨ P ∨ )and P ( C ) = P ( C ∨ ) under the identification P ( T ∗ P ) = Q = P ( T ∗ P ∨ ).2. Since the condition is an open condition on V , it suffices to show the existence.By induction on c , it is reduced to the case c = 1. By 1, the hyperplanes H such thatthe immersion H → P is C -transversal is parametrized by the complement of the image p ∨ ( P ( C )) $ P ∨ . Hence, the assertion follows from this and Lemma 1.2.3.3. SS F -transversality For the definitions and basic properties of the singular support of a constructible complexon a smooth scheme over a field, we refer to [2] and [16]. Let k be a field and let Λ be afinite field of characteristic ℓ invertible in k . Let X be a smooth scheme over k such thatevery irreducible component is of dimension n and let F be a constructible complex on X . The singular support SS F is defined in [2] as a closed conical subset of the cotangentbundle T ∗ X . By [2, Theorem 1.3 (ii)], every irreducible component C a of the singularsupport SS F = C = [ a C a is of dimension n = dim X .Further if k is perfect, the characteristic cycle CC F = X a m a C a
8s defined as a linear combination with Z -coefficients in [16, Definition 5.10]. It is charac-terized by the Milnor formula(1.5) − dim tot φ u ( F , f ) = ( CC F , df ) T ∗ U,u for morphisms f : U → Y to smooth curves Y defined on an ´etale neighborhood U of anisolated characteristic point u . For more detail on the notation, we refer to [16, Section5.2]. Lemma 1.3.1.
Let h : W → X be a morphism of smooth schemes over a field k . Let F be a constructible complex of Λ -modules on X and let C denote the singular support SS F .If h : W → X is properly C -transversal, we have SSh ∗ F = h ◦ SS F . Proof.
By [2, Theorem 1.4 (iii)], we may assume that k is perfect. Suppose dim W =dim X + d . If F is a perverse sheaf on X , then h ∗ F [ d ] is a perverse sheaf on W by theassumption that h : W → X is C -transversal and by [16, Lemma 8.6.5]. Hence by [2,Theorem 1.4 (ii)], we may assume that F is a perverse sheaf. By [16, Proposition 5.14.2],we have CC F ≧ CC F equals the singular support SS F . Also wehave ( − d CCh ∗ F ≧ CCh ∗ F equals the singular support SSh ∗ F .By the assumption that h : W → X is properly C -transversal and by [16, Theorem7.6], we have CCh ∗ F = h ! CC F . Hence by the positivity [7, Proposition 7.1 (a)], thesingular support SSh ∗ F equals the support h ◦ SS F of h ! CC F . Lemma 1.3.2.
Let k be a field and f : X → Y be a morphism of schemes of finite typeover k . Assume that Y is smooth over k . Let F be a constructible complex of Λ -moduleson X . Let X i −−−→ P i ′ y y g P ′ g ′ −−−→ Y be a commutative diagram of schemes over k such that i and i ′ are closed immersions andthe schemes P and P ′ are smooth over k . Let C = SSi ∗ F ⊂ X × P T ∗ P ⊂ T ∗ P and C ′ = SSi ′∗ F ⊂ X × P ′ T ∗ P ′ ⊂ T ∗ P ′ denote the singular supports of the direct images.Then, P → Y is C -transversal (resp. properly C -transversal) if and only if P ′ → Y is C ′ -transversal (resp. properly C ′ -transversal).Proof. By factorizing P → Y as the composition of the graph P → P × Y and theprojection P × Y , we may assume that P → Y is smooth. Similarly, we may assume that P ′ → Y is smooth. By considering the morphism ( i, i ′ ) : X → P × Y P ′ , we may assumethat there exists a smooth morphism P ′ → P compatible with the immersions of X andthe morphisms to Y . Since the assertion is ´etale local on P , we may assume that thereexists a section s : P → P ′ compatible with the immersions of X and the morphisms to Y . Then, we have C ′ = s ◦ C and the assertion follows from [16, Lemma 3.8].Lemma 1.3.2 allows us to make the following definition.9 efinition 1.3.3. Let k be a field and f : X → Y be a morphism of schemes of finite typeover k . Assume that Y is smooth over k . Let F be a constructible complex of Λ -moduleson X .We say that f : X → Y is SS F -transversal (resp. properly SS F -transversal) if locallyon X there exist a closed immersion i : X → P to a smooth scheme P over k and amorphism g : P → Y over k such that f = g ◦ i and that g : P → Y is C -transversal (resp.properly C -transversal) for C = SSi ∗ F . In Definition 1.3.3, we obtain an equivalent condition by requiring that g is smooth .Let f : X → Y be a morphism of schemes of finite type over a field k such that Y issmooth over k and let F be a constructible complex of Λ-modules on X . For an opensubset U ⊂ X , we say f : X → Y is SS F -transversal (resp. properly SS F -transversal) on U if the restriction U → Y of f is SS F U -transversal (resp. properly SS F U -transversal) forthe restriction F U of F on U . Similarly, for an open subset V ⊂ Y , we say f : X → Y is SS F -transversal (resp. properly SS F -transversal) on V if the base change X × Y V → V of f is SS F X × Y V -transversal (resp. properly SS F X × Y V -transversal) for the restriction F X × Y V of F on X × Y V . Lemma 1.3.4.
Let f : X → Y be a morphism of schemes of finite type over a field k .Assume that Y is smooth over k . Let F be a constructible complex of Λ -modules on X . Assume that f : X → Y is smooth and that F is locally constant. Then, f : X → Y is properly SS F -transversal. Assume that f : X → Y is SS F -transversal. Or more weakly, suppose that thereexists a quasi-finite faithfully flat morphism Y ′ → Y of smooth schemes over k such that thebase change f ′ : X ′ → Y ′ is SS F ′ -transversal for the pull-back F ′ of F on X ′ = X × Y Y ′ .Then, f : X → Y is universally locally acyclic relatively to F . The following conditions are equivalent: (1) f : X → Y is SS F -transversal (resp. properly SS F -transversal). (2) For every integer q and for every constituant G of the perverse sheaf p H q F , themorphism f : X → Y is SS G -transversal (resp. properly SS G -transversal). Let k ′ be an extension of k . Then f : X → Y is SS F -transversal (resp. properly SS F -transversal) if and only if the base change f ′ : X ′ → Y ′ by Spec k ′ → Spec k is SS F ′ -transversal (resp. properly SS F ′ -transversal) for the pull-back F ′ on X ′ of F .Proof.
1. If F is locally constant, then the singular support SS F is a subset of the0-section T ∗ X X . Hence the assertion follows.Since the remaining assertions 2-4 are local on X , we may take a closed immersion i : X → P to a smooth scheme P over k such that f is the composition of i with amorphism P → Y over k . Replacing X and F by P and i ∗ F , we may assume that X issmooth over k . Set C = SS F .2. If f : X → Y is C -transversal, the morphism f : X → Y is universally locally acyclicrelatively to F by the definition of singular support. Thus under the weaker assumption,the morphism f ′ : X ′ → Y ′ is universally locally acyclic with respect to the pull-back F ′ . Since Y ′ → Y is quasi-finite and faithfully flat, the morphism f : X → Y itself isuniversally locally acyclic with respect to F .3. By [2, Theorem 1.4 (ii)], the singular support SS F equals the union of SS G for theconstituants G of the perverse sheaves p H q F for integers q . Hence the assertion follows.10. By [2, Theorem 1.4 (iii)], the construction of the singular support commutes withchange of base fields. Hence the assertion follows. Lemma 1.3.5.
Let f : X → Y be a morphism of schemes of finite type over a field k .Assume that Y is smooth over k . Let F be a constructible complex of Λ -modules on X .Assume that f : X → Y is SS F -transversal. Assume that F is a perverse sheaf. Let V ⊂ Y be a dense open subscheme and j : X V = X × Y V → V be the open immersion. Then, there is a unique isomorphism F → j ! ∗ j ∗ F . There exists a dense open subscheme V ⊂ Y such that the base change f : X → V is properly SS F -transversal on V .Proof.
1. By [3, Corollaire 1.4.25], it suffices to show that for every constituant of F , itsrestriction on X V is non-trivial. Let G be a constituant of F . By Lemma 1.3.4.3 and 2,the morphism f : X → Y is locally acyclic relatively to G . Let x be a geometric point of X such that G x = 0 and let y → f ( x ) be a specialization for a geometric point y of V .Then, since the canonical morphism G x → R Γ( X ( x ) × Y ( f ( x )) y, G ) is an isomorphism, therestriction of G on X V is non-trivial. Thus the assertion is proved.2. As in the proof of Lemma 1.3.4, we may assume that X is smooth over k . Set C = SS F . There exists a dense open subset V ⊂ Y such that for every irreducible component C a with the reduced scheme structure of C = S a C a , the base change C a × Y V → V isflat. Lemma 1.3.6.
Let f : X → Y be a morphism of schemes of finite type over a field k .Assume that Y is smooth over k . Let F be a constructible complex of Λ -modules on X .Let Y ′ → Y be a morphism of smooth schemes over k and let X h ←−−− X ′ f y (cid:3) y f ′ Y ←−−− Y ′ , be a cartesian diagram. Let F ′ denote the pull-back of F on X ′ . We consider the following conditions: (1) f : X → Y is SS F -transversal (resp. properly SS F -transversal). (2) f ′ : X ′ → Y ′ is SS F ′ -transversal (resp. properly SS F ′ -transversal).Then, we have (1) ⇒ (2) . Conversely, if Y ′ → Y is ´etale surjective, we have (2) ⇒ (1) . Assume that f : X → Y is SS F -transversal, that F is a perverse sheaf on X andthat dim Y ′ = dim Y + d . Then F ′ [ d ] is a perverse sheaf on X ′ . Assume that f : X → Y is smooth and is properly SS F -transversal. Then, we have SS F ′ = h ◦ SS F . Further if k is perfect, we have CC F ′ = h ! CC F .Proof. Since the assertions are local on X , we may take a closed immersion i : X → P to a smooth scheme P over Y . As in the proof of Lemma 1.3.4, we may assume that f : X → Y is smooth. Set C = SS F .1. Assume that f : X → Y is C -transversal. The pair ( h, f ′ ) of morphisms is C -transversal by Lemma 1.2.6.1. Hence, F ′ = h ∗ F is micro-supported on h ◦ C by [16,Lemma 4.2.4] and we have an inclusion SS F ′ ⊂ h ◦ C and f ′ is SS F ′ -transversal. Thusthe implication (1) ⇒ (2) is proved for the C -transversality. The assertion on the proper C -transversality follows from this and Lemma 1.2.6.1.11ince the formation of singular support is ´etale local, we have (2) ⇒ (1) if Y ′ → Y is´etale surjective.2. Since h : X ′ → X is C -transversal by Lemma 1.2.6.1, the assertion follows from [16,Lemma 8.6.5].3. Since h : X ′ → X is properly C -transversal by Lemma 1.2.6.1, the assertion for SS F ′ (resp. for CC F ′ ) follows from Lemma 1.3.1 (resp. [16, Theorem 7.6]).Lemma 1.3.6.3 is closely related to the subject studied in [9]. Lemma 1.3.7.
Let f : X → Y be a morphism of schemes of finite type over a field k .Assume that Y is smooth over k . Let F be a constructible complex of Λ -modules on X . Let g : Y → Z be a smooth morphism of smooth schemes over k . If f : X → Y is SS F -transversal (resp. properly SS F -transversal), then the composition gf : X → Z is SS F -transversal (resp. properly SS F -transversal). Let h : W → X be a smooth morphism of schemes of finite type over k . If f : X → Y is SS F -transversal (resp. properly SS F -transversal), then the composition f h : W → Y is SSh ∗ F -transversal (resp. properly SSh ∗ F -transversal). Let X r / / f ❇❇❇❇❇❇❇❇ X ′ f ′ (cid:15) (cid:15) Y be a commutative diagram of morphisms of schemes of finite type over k . Assume that r : X → X ′ is proper on the support of F and that f : X → Y is quasi-projective. If f : X → Y is SS F -transversal, then f ′ : X ′ → Y is SS Rr ∗ F -transversal.Proof.
1. As in the proof of Lemma 1.3.4, we may assume that X is smooth over k . Set C = SS F . Since g : Y → Z is smooth, the C -transversality of f implies that of gf by[16, Lemma 3.6.3]. The assertion on the proper C -transversality follows from this and thesmoothness of g .2. Since the question is ´etale local on W , we may assume that there exists a cartesiandiagram W h / / (cid:3) (cid:15) (cid:15) X f / / i (cid:15) (cid:15) YQ / / P ? ? ⑦⑦⑦⑦⑦⑦⑦⑦ of morphisms of schemes over k such that the vertical arrows are closed immersions and thehorizontal arrow Q → P is a smooth morphism of smooth schemes over k . By replacing X and F by P and i ∗ F , we may assume that X is smooth. Since W × X T ∗ X → T ∗ W isan injection and SSh ∗ F = h ◦ SS F by Lemma 1.3.1, the assertion follows.3. Since the assertion is local on X ′ , we may assume that X ′ and Y are affine andhence X is quasi-projective. By taking a closed immersion i ′ : X ′ → P ′ to an affine spaceand by factorizing X ′ → Y as the composition of the immersion ( i ′ , f ′ ) : X ′ → P ′ × Y and the projection P ′ × Y → Y , we may assume that X ′ is smooth. Similarly, we takean open subscheme P of a projective space and a closed immersion i : X → P . Then,by factorizing X → X ′ as the composition of the immersion ( i, r ) : X → P × X ′ andthe projection P × X ′ → X ′ , we may also assume that X is smooth, by [16, Lemma 3.8122) ⇒ (1)]. By [2, Lemma 2.2 (ii)], we have SS Rr ∗ F ⊂ r ◦ SS F . Hence the assertion followsfrom [16, Lemma 3.8 (2) ⇒ (1)].We give two methods to establish SS F -transversality. Lemma 1.3.8.
Let Y → S be a smooth morphism of smooth schemes of finite type overa field k and let f : X → Y be a morphism of schemes of finite type over a field k . Let F be a constructible complex of Λ -modules on X . Assume that the composition X → S is properly SS F -transversal. Assume that k is perfect. Then, the following conditions are equivalent: (1) f : X → Y is SS F -transversal (resp. properly SS F -transversal). (2) For every closed point s ∈ S , the fiber f s : X s → Y s is SS F s -transversal (resp.properly SS F s -transversal) for the pull-back F s of F on X s = X × S s . Assume that F is a perverse sheaf on X and that f : X → Y is locally acyclicrelatively to F . If there exists a closed subset Z ⊂ X quasi-finite over S such that f : X → Y is SS F -transversal (resp. properly SS F -transversal) on the complement of Z , then f : X → Y is SS F -transversal (resp. properly SS F -transversal) on X .Proof.
1. The implication (1) ⇒ (2) is a special case of Lemma 1.3.6.1. We show (2) ⇒ (1).Since the question is local on X , we may assume that f : X → Y is smooth. Let T ∗ X/S and T ∗ Y /S denote the relative cotangent bundles and let C = SS F . By the assumptionthat X → S is C -transversal, the canonical surjection T ∗ X → T ∗ X/S is finite on C by[2, Lemma 1.2 (ii)]. Hence its image ¯ C ⊂ T ∗ X/S is a closed conical subset and C → ¯ C isfinite.The morphism X → Y is C -transversal if and only if the inverse image of ¯ C ⊂ T ∗ X/S by the canonical injection X × Y T ∗ Y /S → T ∗ X/S is a subset of the 0-section. This isequivalent to that for every closed point s ∈ S and the closed immersion i s : X s → X , themorphism f s : X s → Y s is i ◦ s C -transversal. Further, under the assumption that f : X → Y is C -transversal, this is properly C -transversal if and only if f s : X s → Y s is properly i ◦ s C -transversal for every closed point s ∈ S .By the assumption that X → S is properly SS F -transversal and by Lemma 1.3.6.3,we have SS F s = i ◦ s SS F = i ◦ s C for every closed point s ∈ S . Hence the assertion is proved.2. By Lemma 1.3.4.4, we may assume that k is perfect. By 1 and Lemma 1.3.6.2, wemay assume that S is a point and further that S = Spec k . As in the proof of Lemma 1.3.6,we may assume that f : X → Y is smooth of relative dimension d . Let u ∈ Z . By replacing X by a neighborhood of u , we may assume Z = { u } . Set C = SS F , v = f ( u ) ∈ Y andregard X × Y T ∗ Y as a closed subscheme of T ∗ X .We show that f : X → Y is C -transversal, assuming that f : X → Y is locally acyclicrelatively to F . Namely, we show that the intersection C ′ = C ∩ ( X × Y T ∗ Y ) is a subset ofthe 0-section X × Y T ∗ Y Y . By the assumption that f : X → Y is C -transversal outside u , theintersection C ′ = C ∩ ( X × Y T ∗ Y ) is a subset of the union ( X × Y T ∗ Y Y ) ∪ ( u × Y T ∗ Y ) withthe fiber at u . Let ω ∈ u × Y T ∗ Y = v × Y T ∗ Y be a non-zero element. After shrinking Y toa neighborhood of v = f ( u ) if necessary, we take a smooth morphism Y → A = Spec k [ t ]such that dt ( v ) = ω . Then, by [16, Lemma 3.6.3], the point u is at most an isolated C -characteristic point of the composition g : X → Y → A .Since F is a perverse sheaf, the characteristic cycle CC F is an effective cycle andits support equals C = SS F by [16, Proposition 5.14]. Let dg denote the section of X × Y T ∗ Y ⊂ T ∗ X defined by the function g ∗ ( t ). Since the composition X → Y → A is13ocally acyclic relatively to F by [10, Corollaire 5.2.7], we have ( CC F , dg ) T ∗ X,u = 0 by theMilnor formula (1.5). Therefore by the positivity [7, Proposition 7.1 (a)], the intersection SS F ∩ dg = C ′ ∩ dg is empty and hence ω / ∈ C ′ . Since ω is any non-zero element of u × Y T ∗ Y , we conclude that C ′ ∩ ( u × Y T ∗ Y ) ⊂ f : X → Y is C -transversal.Assume further that f : X → Y is properly C -transversal outside u . Since f : X → Y is C -transversal, the morphism T ∗ X → T ∗ X/Y to the relative cotangent bundle is finiteon C by [2, Lemma 1.2 (ii)] and the image ¯ C ⊂ T ∗ X/Y of C is a closed conical subset. Itis sufficient to show that for every point y ∈ Y , the fiber ¯ C × Y y is of dimension ≦ d . For y = f ( u ), this follows from the assumption. Assume y = f ( u ). Then, every irreduciblecomponent of ¯ C × Y y is either a closure of an irreducible component of ¯ C × Y y ∩ ( X × Y y { u } ) or a subset of the fiber T ∗ u ( X × Y y ). Since dim T ∗ u ( X × Y y ) = d , the assertion isproved. Lemma 1.3.9.
Let W h −−−→ X f −−−→ Y j ′ x (cid:3) x j U ′ h U −−−→ U be a cartesian diagram of schemes of finite type over a field k . Assume that Y is smoothover k and that j is an open immersion. Let F U be a perverse sheaf of Λ -modules on U .Let F = j ! ∗ F U and let F ′ be a perverse sheaf on W such that the restriction F ′ U ′ on U ′ isisomorphic to the pull-back h ∗ U F U . If one of the following conditions (1) and (2) below issatisfied and if f ◦ h : W → Y is SS F ′ -transversal, then f : X → Y is SS F -transversal. (1) The morphism h : W → X is proper, surjective and generically finite and thecomposition W → Y is quasi-projective. The scheme U is smooth of dimension d over k and there exists a locally constant sheaf G of Λ -modules on U such that F U = G [ d ] . (2) The morphism h is quasi-finite and faithfully flat. For every constituant G of theperverse sheaf F U , the pull-back h ∗ U G is also a perverse sheaf on U ′ .Proof. Assume that (1) is satisfied. Since h is surjective, the canonical morphism G → h U ∗ h ∗ U G is an injection. Hence the constituants of the perverse sheaf F U = G [ d ] areconstituants of p H Rh U ∗ F ′ U ′ = j ∗ p H Rh ∗ F ′ . Consequently, the constituants of F = j ! ∗ F U are constituants of p H Rh ∗ F ′ . Since W → Y is quasi-projecitive, the assertion followsfrom Lemma 1.3.7.3 and Lemma 1.3.4.3.Assume that (2) is satisfied. By Lemma 1.3.4.3 and the perversity assumption on thepull-backs, we may assume that F U is a simple perverse sheaf. Then, by [3, Th´eor`eme 4.3.1(ii)], there exists a locally closed immersion i : V → U of a scheme V smooth of dimension d over k and a locally constant sheaf G on V such that F U = i ! ∗ G [ d ]. By replacing X and U by the closure of the image of V → X and V , we may assume that V = U . Since theassertion is ´etale local on X by Lemma 1.3.6.1, we may assume that h is finite and that X, W and Y are affine. Then the assertion follows from the case (1). Let f : X → Y be a morphism of smooth schemes over a field k and let D ⊂ Y be a divisorsmooth over k . In this article, we say that f : X → Y is semi-stable relatively to D if ´etale14ocally on X and on Y , there exists a cartesian diagram X f −−−→ Y ←−−− D y (cid:3) y (cid:3) y A n −−−→ A ←−−− A n = Spec k [ t , . . . , t n ] → A = Spec k [ t ] is definedby t t · · · t n and the lower right horizontal arrow is the inclusion of the origin 0 ∈ A .A semi-stable morphism f : X → Y is flat and the base change f V : X × Y V → V = Y D is smooth. We recall statements on the existence of alteration.
Lemma 1.4.1.
Let k be a perfect field and let f : X → Y be a dominant separated mor-phism of integral schemes of finite type over k . There exists a commutative diagram (1.6) X ←−−− W f y y g Y ←−−− Y ′ of integral schemes of finite type over k satisfying the following condition: The bottomhorizontal arrow Y ′ → Y is dominant and is the composition gh of an ´etale morphism g and a finite flat radicial morphism h . The schemes W and Y ′ are smooth over k andthe morphism g : W → Y ′ is quasi-projective and smooth. The induced morphism W → X × Y Y ′ is proper surjective and generically finite. Let ξ ∈ Y be a point such that the local ring O Y,ξ is a discrete valuation ring. Then,there exists a commutative diagram (1.6) of integral schemes of finite type over k satisfyingthe following condition: The bottom horizontal arrow Y ′ → Y is quasi-finite and flat andits image is an open neighborhood of ξ . The schemes W and Y ′ are smooth over k , theclosure D ′ ⊂ Y ′ of the inverse image of ξ is a divisor smooth over k and the morphism g : W → Y ′ is quasi-projective and is semi-stable relatively to D ′ . The induced morphism W → X × Y Y ′ is proper surjective and generically finite.Proof.
1. Let η be the generic point of Y . Then, it suffices to apply [5, Theorem 4.1] tothe generic fiber X × Y η .2. Let S = Spec O Y,ξ be the localization at ξ . Then, it suffices to apply [5, Theorem8.2] to the base change X × Y S → S .We prove an analogue of the generic local acyclicity theorem [6, Th´eor`eme 2.13]. Proposition 1.4.2.
Let f : X → Y be a morphism of schemes of finite type over a perfect field k . Let F be a constructible complex of Λ -modules on X . Then, there exists a cartesiandiagram (1.7) X ←−−− X ′ f y (cid:3) y f ′ Y ←−−− Y ′ of schemes of finite type over k satisfying the following conditions: The scheme Y ′ issmooth over k and the morphism Y ′ → Y is dominant and is the composition gh of an etale morphism g and a finite flat radicial morphism h . The morphism f ′ : X ′ → Y ′ isproperly SS F ′ -transversal for the pull-back F ′ of F on X ′ .Proof. We may assume that F is a simple perverse sheaf by Lemma 1.3.4.3 and Lemma1.3.6. Hence, we may assume that there exist a locally closed immersion j : Z → X ofa smooth irreducible scheme of dimension d and a simple locally constant sheaf G of Λ-modules such that j ! ∗ G [ d ] = F by [3, Th´eor`eme 4.3.1 (ii)]. By replacing X by the closureof j ( Z ), we may assume that j : Z → X is an open immersion. It suffices to consider thecase where Z → Y is dominant since the assertion is clear if otherwise.Let Z → Z be a finite ´etale covering such that the pull-back of G is constant and let X be the normalization of X in Z . Applying Lemma 1.4.1.1 to X → Y , we obtain acommutative diagram(1.8) X r ←−−− W f y y Y ←−−− Y ′ of schemes over k satisfying the following conditions: The scheme Y ′ is smooth and themorphism Y ′ → Y is dominant and is the composition gh of an ´etale morphism g anda finite flat radicial surjective morphism h . The morphism W → Y ′ is quasi-projectiveand smooth. The induced morphism r ′ : W → X ′ = X × Y Y ′ is proper surjective andgenerically finite. The pull-back G ′ W of G on W × X Z is a constant sheaf.We consider the cartesian diagram Z ←−−− Z ′ ←−−− W × X Z j y (cid:3) j ′ y (cid:3) y j W X ←−−− X ′ r ′ ←−−− W and let G ′ be the pull-back of G on Z ′ . Since the finite radicial surjective morphism h isuniversally a homeomorphism, we have F ′ = j ′ ! ∗ G ′ [ d ].Since G ′ W is a constant sheaf on W × X Z and W is smooth over k , the intermediateextension j W ! ∗ G ′ W [ d ] is constant. The smooth morphism W → Y ′ is properly SSj W ! ∗ G ′ W [ d ]-transversal by Lemma 1.3.4.1. Since W → X ′ is proper and W → Y ′ is quasi-projective,the morphism X ′ → Y ′ is SS F ′ -transversal by the case (1) in Lemma 1.3.9. After shrinking Y ′ , the morphism X ′ → Y ′ is properly SS F ′ -transversal by Lemma 1.3.5.2. Corollary 1.4.3.
Let f : X → Y and F be as in Proposition and assume that k is of characteristic p > . Then, there exist a dense open subscheme V ⊂ Y smoothover k and an iteration ˜ V → V of Frobenius such that the base change X × Y ˜ V → ˜ V is SS ˜ F -transversal for the pull-back ˜ F on ˜ X V = X × Y ˜ V .Proof. After shrinking Y ′ in the conclusion of Proposition 1.4.2, we may assume that Y ′ → Y is the composition jgh of an open immersion j : V → Y , a finite surjective radicalmorphism g and an ´etale surjective morphism h . By Lemma 1.3.6.1, we may assume that Y ′ → Y is jg . Thus, the assertion follows.We show an analogue of the stable reduction theorem.16 roposition 1.4.4. Let (1.9) X ⊃ ←−−− U f y (cid:3) y f V Y ⊃ ←−−− V be a cartesian diagram of schemes of finite type over a perfect field k . Assume that Y isnormal and that V is a dense open subset of Y smooth over k . Let F U be a perverse sheafof Λ -modules on U such that f V : U → V is SS F U -transversal.Then, there exists a cartesian diagram (1.10) X ←−−− X ′ j ′ ←−−− U ′ = U × X X ′ f y (cid:3) y f ′ (cid:3) y Y g ←−−− Y ′ ⊃ ←−−− V ′ = V × Y Y ′ of schemes of finite type over k satisfying the following conditions: The scheme Y ′ issmooth over k and V ′ ⊂ Y ′ is the complement of a divisor D ′ ⊂ Y ′ smooth over k . Themorphism g : Y ′ → Y is quasi-finite flat and the complement Y g ( Y ′ ) is of codimension ≧ in Y . The pull-back F ′ U ′ of F U is a perverse sheaf on U ′ and for F ′ = j ′ ! ∗ F ′ U ′ on X ′ ,the morphism f ′ : X ′ → Y ′ is SS F ′ -transversal. First, we prove a basic case.
Lemma 1.4.5.
Let X and Y be smooth schemes over a field k . Let D ⊂ Y be a divisorsmooth over k and V = Y D be the complement. Let f : X → Y be a morphism over k semi-stable relatively to D . Assume that dim X = n . For a cartesian diagram (1.10) such that Y ′ → Y is a quasi-finite flat morphism of smooth schemes over k , let F ′ be theperverse sheaf F ′ = j ′ ! ∗ Λ U ′ [ n ] on X ′ . Assume that dim Y = 1 . Let Y ′ → Y be a flat morphism of smooth curves over k such that for every y ′ ∈ Y ′ V ′ , the action of the inertia group I y ′ on R Ψ y ′ Λ U ′ is trivial.Then the morphism X ′ → Y ′ is properly SS F ′ -transversal. There exists a quasi-finite faithfully flat morphism Y ′ → Y of smooth schemes over k satisfying the following conditions: The open subscheme V ′ ⊂ Y ′ is the complement ofa divisor D ′ smooth over k and the morphism X ′ → Y ′ is properly SS F ′ -transversal.Proof.
1. Since the question is ´etale local, we may assume that Y = A k = Spec k [ t ],that X = X n = A nk = Spec k [ t , . . . , t n ] and that the morphism f : X → Y is defined by t t · · · t n . We prove the assertion by induction on n . If n = 1, then f : X → Y is ´etaleand F ′ is constant. Hence the assertion follows in this case by Lemma 1.3.4.1.Assume n >
1. Outside the closed point u ∈ X defined by t = · · · = t n = 0, locallythere exists a smooth morphism X = X n → X n − over Y . Hence, the induction hypothesisimplies the assertion on the complement X { u } by Lemma 1.3.7.2. Thus, the morphism f ′ : X ′ → Y ′ is properly SS F ′ -transversal outside the inverse image of u . By Proposition1.1.2.2 (3) ⇒ (1), the morphism f ′ : X ′ → Y ′ is locally acyclic relatively to F ′ . Hence byLemma 1.3.8.2, the morphism f ′ : X ′ → Y ′ is properly SS F ′ -transversal on X ′ .2. It follows from 1 by Lemma 1.3.6.1 and Lemma 1.3.5.1.17 roof of Proposition . The proof is similar to that of Proposition 1.4.2. By Lemma1.3.6 and Lemma 1.3.5, it suffices to show the assertion on a neighborhood of each point ξ ∈ Y of codimension 1 not contained in V . Thus, we may assume that the closure D of ξ is a divisor smooth over k and that V = Y D .We may assume that F U is a simple perverse sheaf by Lemma 1.3.4.3 and Lemma 1.3.6.Hence, similarly as in the proof of Proposition 1.4.2, we may assume that there exist adense open immersion j : Z → U of a smooth irreducible scheme of dimension d and asimple locally constant sheaf G of Λ-modules such that F U = j ! ∗ G [ d ]. Further, we mayassume that Z → Y is dominant.Taking a finite ´etale covering trivializing G and applying Lemma 1.4.1.2 as in the proofof Proposition 1.4.2, we obtain a commutative diagram(1.11) X r ←−−− W f y y Y g ←−−− Y of schemes over k satisfying the following conditions: The scheme Y is smooth over k ,the morphism g : Y → Y is quasi-finite and flat and Y g ( Y ) is of codimension ≧ Y . The inverse image V × Y Y is the complement of a divisor D smooth over k and the morphism W → Y is quasi-projective and is semi-stable relatively to D . Theinduced morphism r : W → X = X × Y Y is proper surjective and generically finite.The pull-back G ′ of G on W × X Z is a constant sheaf.By Lemma 1.4.5.2 applied to the semi-stable morphism W → Y , we obtain a quasi-finite faithfully flat morphism Y ′ → Y of smooth schemes satisfying the condition loc. cit.We consider the cartesian diagram Z ←−−− Z ′ ←−−− W ′ × X Z j Z y (cid:3) j Z ′ y (cid:3) y j W X ←−−− X ′ r ′ ←−−− W ′ = X × Y Y ′ = W × Y Y ′ and let G ′ and G ′ W ′ denote the pull-backs of G on Z ′ and on W ′ × X Z respectively. Since G ′ W ′ is a constant sheaf on W ′ × X Z , the morphism W ′ → Y ′ is SSj W ! ∗ G ′ W ′ [ d ]-transversalby Lemma 1.4.5.2.The pull-back F ′ U ′ is a perverse sheaf by Lemma 1.3.6. The perverse sheaf F ′ = j ′ ! ∗ F ′ U ′ is canonically identified with j Z ′ ! ∗ G ′ [ d ]. Since r ′ : W ′ → X ′ is proper surjective andgenerically finite and since r ′ : W ′ → Y ′ is quasi-projective, the morphism X ′ → Y ′ is SS F ′ -transversal by the case (1) in Lemma 1.3.9. Corollary 1.4.6.
Let the cartesian diagram (1.9) and a perverse sheaf F U on U = X × Y V be as in Proposition . Assume further that Y is smooth over k and that V is thecomplement of a divisor D ⊂ Y smooth over k . Then, there exist a cartesian diagram (1.10) satisfying the following conditions: The scheme Y ′ is smooth over k and V ′ ⊂ Y ′ isthe complement of a divisor D ′ ⊂ Y ′ smooth over k . The morphism g : Y ′ → Y is quasi-finite flat, the morphism D ′ → D is dominant and the morphism V ′ → V is ´etale . Thepull-back F ′ U ′ of F U is a perverse sheaf on U ′ and the morphism f ′ : X ′ → Y ′ is universallylocally acyclic relatively to F ′ = j ′ ! ∗ F ′ . roof. Let V ′ ⊂ Y ′ be as in the conclusion of Proposition 1.4.4. Let ¯ Y ′′ be the normal-ization of Y in the separable closure of k ( Y ) in k ( Y ′ ). Then, there exists a dense opensubset Y ′′ ⊂ ¯ Y ′′ smooth over k of the image of Y ′ → ¯ Y ′′ such that g ′′ : Y ′′ → Y is flat, that V ′′ = V × Y Y ′′ is the complement of a divisor D ′′ smooth over k , that D ′′ → D is dominant,and that V ′′ → V is ´etale . Since Y ′ × ¯ Y ′′ Y ′′ → Y ′′ is finite surjective radicial, the cartesiandiagram (1.10) defined by Y ′′ → Y in place of Y ′ → Y satisfies the conditions. Corollary 1.4.7.
Let X → Y be a morphism of schemes of finite type over a field k and assume that Y is smooth of dimension . Then, for a constructible complex F of Λ -modules on X , the following conditions are equivalent: (1) X → Y is locally acyclic relatively to F . (2) X → Y is universally locally acyclic relatively to F . (3) There exists a finite faithfully flat morphism Y ′ → Y of smooth curves over k suchthat the base change X ′ → Y ′ is SS F ′ -transversal for the base change F ′ of F on X ′ . The equivalence (1) ⇔ (2) is proved in [15]. Proof.
We show (1) ⇒ (3). Since the nearby cycles functor is t -exact, we may assumethat F is a perverse sheaf. Then, the assertion follows from Propositions 1.4.2, 1.4.4 and1.1.2. The implication (3) ⇒ (2) is proved in Lemma 1.3.4.2. The implication (2) ⇒ (1) istrivial.The following example shows that taking a covering Y ′ → Y in condition (3) is neces-sary. Example 1.4.8.
Let k be a field of characteristic p >
2. Let X = A × P and j : U = A × A = Spec k [ x, y ] → X be the open immersion. Let G be the locally constant sheafof Λ-modules of rank 1 on U defined by the Artin-Schreier covering t p − t = xy and by anon-trivial character F p → Λ × . Then, the second projection pr : X → Y = P is locallyacyclic relatively to F = j ! G [14, Th´eor`eme 2.4.4].On the other hand, the singular support C = SS F is the union of the zero-section T ∗ X X and the conormal bundles T ∗ X ∞ X of the fiber X ∞ = pr − ( ∞ ) and T ∗ (0 , ∞ ) X of thepoint (0 , ∞ ). Hence the projection pr : X → Y = P is not C -transversal.Let Y ′ = P → Y = P be the Frobenius. Then, for the pull-back F ′ of F on X ′ = X × Y Y ′ , the singular support C ′ = SS F ′ is the union of the zero-section T ∗ X ′ X ′ andthe image of the pull-back X ′∞ × A T ∗ A → T ∗ X ′ with respect to the first projection onthe fiber X ′∞ = pr − ( ∞ ) at infinity. Consequently, the projection pr : X ′ → Y ′ = P is C ′ -transversal.Let Y ′′ → Y ′ be a flat morphism of smooth curves over k and let F ′′ be the pull-backof F ′ on X ′′ = X ′ × Y ′ Y ′′ . Then, the morphism h : X ′′ → X ′ is properly C ′ -transversaland hence SS F ′′ = h ◦ SS F ′ is the union of T ∗ X ′′ X ′′ and X ′′∞ × A T ∗ A ⊂ T ∗ X ′′ by Lemma1.3.1. We prove a refinement of the analogue of the stable reduction theorem, using the followingconsequence of the stable reduction theorem for curves.19 emma 1.5.1.
Let U ⊂ −−−→ X f V y (cid:3) y f V ⊂ −−−→ Y g −−−→ S be a cartesian diagram of morphisms of smooth schemes of finite type over a perfect field k satisfying the following conditions: The morphism f : X → Y is flat and the morphisms g : Y → S and f V : U → V are smooth of relative dimension . The horizontal arrowsare open immersions and the open subset V ⊂ Y is the complement of a divisor D ⊂ Y smooth over k and quasi-finite and flat over S .Then, there exists a commutative diagram X ′ f ′ −−−→ Y ′ g ′ −−−→ S ′ y y y X f −−−→ Y g −−−→ S of smooth schemes over k satisfying the following conditions: The morphisms S ′ → S , Y ′ → Y × S S ′ and X ′ → X × Y Y ′ are quasi-finite flat and dominant. The morphisms g ′ : Y ′ → S ′ and f ′ : X ′ → Y ′ are smooth of relative dimension and that V ′ = V × Y Y ′ ⊂ Y ′ is the complement of a divisor D ′ ⊂ Y ′ smooth over k and quasi-finite and flat over S ′ . The morphism V ′ → V × S S ′ is ´etale and the morphism U ′ = X ′ × Y ′ V ′ → U × V V ′ is an isomorphism. The quasi-finite morphisms D ′ → D and X ′ × Y ′ D ′ → X × Y D ′ aredominant.Proof. Let ¯ η be a geometric point of S defined by an algebraic closure of the functionfield of an irreducible component. Then, it suffices to apply [18, Theorem 1.5] to the basechange of X → Y → S by ¯ η → S . Theorem 1.5.2.
Let U ⊂ −−−→ X y (cid:3) y f V ⊂ −−−→ Y −−−→ S be a cartesian diagram of morphisms of schemes of finite type over a perfect field k .Assume that Y and S are smooth over k , that Y → S is smooth of relative dimension and that V ⊂ Y is the complement of a divisor D smooth over k and quasi-finite and flatover S . Let F U be a perverse sheaf of Λ -modules on U = X × Y V such that U → V is SS F U -transversal.Then, there exists a commutative diagram (1.12) V ′ ⊂ −−−→ Y ′ −−−→ S ′ y (cid:3) y y V ⊂ −−−→ Y −−−→ S of smooth schemes over k satisfying the following conditions (1) and (2):(1) The morphisms S ′ → S and Y ′ → Y × S S ′ are quasi-finite flat and dominant. Thehorizontal arrow Y ′ → S ′ is smooth of relative dimension . The left square is cartesian nd V ′ ⊂ Y ′ is the complement of a divisor D ′ ⊂ Y ′ smooth over k and quasi-finite andflat over S ′ . The induced morphism V ′ → V × S S ′ is ´etale and D ′ → D is dominant . (2) Let (1.13) U ′ j ′ −−−→ X ′ f ′ −−−→ Y ′ y (cid:3) y (cid:3) y U −−−→ X f −−−→ Y ′ be a cartesian diagram and let F ′ U ′ denote the pull-back of F U on U ′ . Then the morphism f ′ : X ′ → Y ′ is SS F ′ -transversal for F ′ = j ′ ! ∗ F ′ U ′ .Proof. Since the assertion is local on X , we may assume that there exists a closed immer-sion i : X → A nY for an integer n ≧
0. By replacing X and F U by A nY and i | U ∗ F U on A nV ,we may assume that X is an open subscheme of A nY . We prove the assertion by inductionon n .Assume n = 0 and hence X → Y is an open immersion. Since the open immersion U → V is SS F U -transversal, the singular support SS F U is a subset of the 0-section T ∗ U U by Lemma [16, Lemma 3.6.3]. Hence F U is locally constant by [2, Lemma 2.1(iii)]. Let U → U be a finite ´etale covering such that the pull-back of F U is constant. Let Y bethe normalization of Y in U . There exists a quasi-finite flat and dominant morphism S ′ → S of smooth scheme such that the normalization Y ′ of Y × S S ′ is smooth over S ′ and that V ′ ⊂ Y ′ is the complement of a divisor D ′ ´etale over S ′ . After shrinking S ′ , wemay assume that Y ′ → Y × S S ′ is flat. After shrinking Y ′ keeping D ′ dominant over D ,we may assume that V ′ → V × S S ′ ´etale. Then, the condition (1) is satisfied. Since F ′ on X ′ ⊂ Y ′ is constant, the condition (2) is also satisfied by Lemma 1.3.4.1.Assume that n ≧ n −
1. For the proof of the inductionstep, we first show the following weaker assertion.
Claim.
Let X ⊂ A nY → A Y be a projection and assume that its restriction U ⊂ A nV → A V is SS F U -transversal. Then, there exist a commutative diagram (1.12) satisfying thecondition (1) and an open subset W ′ ⊂ A Y ′ satisfying the following condition: (2 ′ ) The intersection W ′ ∩ A D ′ is dense in A D ′ . For the cartesian diagram (1.13) andfor the pull-back F ′ U ′ of F on U ′ and F ′ = j ′ ! ∗ F ′ U ′ on X ′ , the morphism f ′ : X ′ → Y ′ is SS F ′ -transversal on the inverse image X ′ × A Y ′ W ′ ⊂ X ′ .Proof of Claim. By the induction hypothesis applied to X ⊂ A nY → A Y → A S , thereexists a commutative diagram Y −−−→ S y y A Y −−−→ A S satisfying the conditions (1) and (2) in Theorem 1.5.2. We consider the cartesian diagram U j −−−→ X −−−→ Y y (cid:3) y (cid:3) y U −−−→ X −−−→ A Y F U be the pull-back of F U . Then, for F = j ∗ F U on X , the morphism X → Y is SS F -transversal. The inverse image V = V × Y Y ⊂ Y is the complement of a divisor D ⊂ Y smooth over k and quasi-finite and flat over S . The quasi-finite morphism V → V × S S is ´etale and the quasi-finite morphism D → A D is dominant.Since the morphism S → A S is quasi-finite and flat, there exists a quasi-finite, flatand dominant morphism S ′ → S of smooth schemes over k such that the normalization S ′ of S × S S ′ is smooth over S ′ and that the induced morphism S ′ → S is also quasi-finite,flat and dominant. After shrinking S ′ if necessary, we may assume that the morphism Y × S S ′ → A Y × A S S ′ of smooth curves over S ′ is flat. Hence, by replacing S, Y, S and Y by S ′ , Y × S S ′ , S ′ and Y × S S ′ , we may assume that S → S is smooth of relativedimension 1.We consider the commutative diagram(1.14) V −−−→ Y −−−→ S y (cid:3) y y V −−−→ Y −−−→ S where the left square is cartesian. Since V → V × S S is ´etale, the left vertical arrow V → V is also smooth of relative dimension 1. The middle vertical arrow Y → Y is flat.Hence, by Lemma 1.5.1 applied to (1.14), there exists a commutative diagram Y ′ −−−→ Y ′ −−−→ S ′ y y y Y −−−→ Y −−−→ S of smooth schemes over k satisfying the following conditions: The morphisms S ′ → S , Y ′ → Y × S S ′ and Y ′ → Y × Y Y ′ are quasi-finite flat and dominant. The morphisms Y ′ → S ′ and Y ′ → Y ′ are smooth of relative dimension 1. The inverse image V ′ = V × Y Y ′ is the complement Y ′ D ′ of a divisor D ′ ⊂ Y ′ smooth over k and quasi-finite and flatover S ′ . The morphism V ′ → V × S S ′ is ´etale and the morphism Y ′ × Y ′ V ′ → V × V V ′ is an isomorphism. The quasi-finite morphisms D ′ → D and Y ′ × Y ′ D ′ → Y × Y D ′ aredominant. Thus the condition (1) in Theorem 1.5.2 is satisfied.The composition Y ′ → Y × Y Y ′ → A Y ′ is quasi-finite and flat. We consider thecartesian diagram X ′′ −−−→ Y ′ y (cid:3) y X ′ −−−→ A Y ′ −−−→ Y ′ and the pull-back F ′′ of F on X ′′ . Then, since X → Y is SS F -transversal, the mor-phism X ′′ → Y ′ is SS F ′′ -transversal by Lemma 1.3.6.1. Since Y ′ → Y ′ is smooth, thecomposition X ′′ → Y ′ is also SS F ′′ -transversal by Lemma 1.3.7.1. By Lemma 1.3.6.2and Lemma 1.3.4.3, the pull-back on U ′′ = U ′ × X ′ X ′′ of every constituant G of F ′ U ′ isa perverse sheaf. Since Y ′ → A Y ′ is quasi-finite and flat, the morphism f ′ : X ′ → Y ′ is SS F ′ -transversal on the image of X ′′ by the case (2) in Lemma 1.3.9.Let W ′ ⊂ A Y ′ be the image of Y ′ . The image of X ′′ → X ′ equals the inverse image X ′ × A Y ′ W ′ . Hence, f ′ : X ′ → Y ′ is SS F ′ -transversal on X ′ × A Y ′ W ′ ⊂ X ′ . Since22 ′ × Y ′ D ′ → Y × Y D ′ and D → A D are dominant, the intersection W ′ ∩ A D ′ is dense in A D ′ . Thus the condition (2 ′ ) in Claim is also satisfied.To complete the proof of the induction step, we use the following elementary lemma. Lemma 1.5.3.
Let X be an open subset of a vector space V of dimension n over an infinitefield k regarded as a smooth scheme over k . Let C ⊂ T ∗ X be a closed conical subset ofdimension ≦ n . Then, there exists an isomorphism V → A n of vector spaces over k suchthat the compositions X → V → A n → A with the projections pr i , i = 1 , . . . , n have atmost isolated C -characteristic points.Proof. Identify the cotangent bundle T ∗ X with the product X × V ∨ with the dual andlet P ( C ) ⊂ P ( T ∗ X ) = X × P ( V ∨ ) be the projectivization. Then, by the assumptiondim C ≦ n , the projection P ( C ) → P ( V ∨ ) is generically finite. By the assumption that k is infinite, there exists a basis p , . . . , p n of V ∨ such that the fibers of P ( C ) → P ( V ∨ )at ¯ p , . . . , ¯ p n ∈ P ( V ∨ ) are finite. Then, the product of p , . . . , p n : V → A satisfies thecondition.Set C U = SS F U ⊂ T ∗ U . By the assumption that U ⊂ A nV → V is SS F U -transversal,the morphism T ∗ U → T ∗ U/V to the relative cotangent bundle is finite on C U by [2,Lemma 1.2 (ii)]. The image ¯ C U ⊂ T ∗ U/V of C U and its closure ¯ C ⊂ T ∗ X/Y are closedconical subsets. Since every irreducible component of C U is of dimension dim X , everyirreducible component of ¯ C U is also of dimension dim X . Hence, for the generic point ofeach irreducible component of Y , the fiber of ¯ C U is of dimension ≦ n = dim X − dim Y .Consequently, for the generic point of each irreducible component of D ⊂ Y , the fiber of¯ C is also of dimension ≦ n .By Lemma 1.5.3 applied to the fibers of the generic points of irreducible components of D , after replacing S by a dense open subset, there exists a coordinate of A nY ⊃ X such that,for each i = 1 , . . . , n , there exist a dense open subset W i ⊂ A D and an open neighborhood X i ⊂ X of the inverse image W i × A Y X by the i -th projection pr i satisfying the followingcondition: The inverse image of ¯ C ⊂ T ∗ X/Y by the morphism X × A Y T ∗ A Y /Y → T ∗ X/Y of the relative cotangent bundles induced by pr i is a subset of the 0-section on X i .Then, the restriction U → A V of pr i is SS F U -transversal on X i ∩ U . By Claim appliedto the restriction X i → A Y of pr i , there exist a commutative diagram (1.12) satisfyingthe condition (1) and for each i = 1 , . . . , n a dense open subset W ′ i ⊂ A Y ′ satisfying thecondition (2 ′ ). Hence X ′ → Y ′ is SS F ′ -transversal on the union W ′ = S ni =1 pr − i W ′ i ⊂ X ′ ⊂ A nY ′ of the inverse images by the projections. Since X ′ → Y ′ is SS F ′ -transversal on U ′ , it is SS F ′ -transversal on W ′ ∪ U ′ . By shrinking S ′ if necessary, we may assume that Z ′ = X ′ ( W ′ ∪ U ′ ) = Q ni =1 ( A D ′ ( A D ′ ∩ W ′ i )) ⊂ A nD ′ is quasi-finite over S ′ .By Corollary 1.4.6, there exists a cartesian diagram U ′′ j ′′ −−−→ X ′′ f ′′ −−−→ Y ′′ ←−−− V ′′ y (cid:3) y (cid:3) y (cid:3) y U ′ j ′ −−−→ X ′ f ′ −−−→ Y ′ ←−−− V ′ of smooth schemes over k satisfying the following condition: The morphism V ′′ → V ′ is´etale and V ′′ ⊂ Y ′′ is the complement of a divisor D ′′ ⊂ Y ′′ smooth over k . The morphism D ′′ → D ′ is dominant. For the pull-back F ′′ U ′′ of F ′ U ′ on U ′′ and F ′′ = j ′′ ! ∗ F ′′ , the morphism f ′′ : X ′′ → Y ′′ is universally locally acyclic relatively to F ′′ .23y Lemma 1.3.6.1 and Lemma 1.3.5.1, F ′′ is the pull-back of F ′ outside the inverseimage Z ′′ of Z ′ and f ′′ : X ′′ → Y ′′ is SS F ′′ -transversal outside the inverse image Z ′′ .Let S ′′ → S ′ be a quasi-finite flat dominant morphism of smooth schemes over k suchthat the normalization Y ′′′ of Y ′′ × S ′ S ′′ is smooth over S ′′ of relative dimension 1 and that V ′′′ = V ′′ × Y ′′ Y ′′′ is the complement of a divisor D ′′′ smooth over k . Since V ′′ → V ′ is´etale, the morphism V ′′ → S ′ is smooth and V ′′′ → V ′′ × S ′ S ′′ is an isomorphism. Hencethe morphism V ′′′ → V ′ × S ′ S ′′ is ´etale. The morphism D ′′′ → D ′ is dominant. We considerthe commutative diagram U ′′′ j ′′′ −−−→ X ′′′ f ′′′ −−−→ Y ′′′ −−−→ S ′′ y (cid:3) y (cid:3) y y U ′ j ′ −−−→ X ′ f ′ −−−→ Y ′ −−−→ S ′ where the left and middle squares are cartesian. Then for the pull-back F ′′′ of F ′′ on X ′′′ , the morphism f ′′′ : X ′′′ → Y ′′′ is universally locally acyclic and is SS F ′′′ -transversaloutside the inverse image Z ′′′ of Z ′′ quasi-finite over S ′′ .Shrinking S ′′ , we may further assume that X ′′′ → S ′′ is properly SS F ′′′ -transversalby Lemma 1.3.5.2. Then, the morphism f ′′′ : X ′′′ → Y ′′′ is SS F ′′′ -transversal by Lemma1.3.8.2. Further we have an isomorphism F ′′′ = j ′′′ ! ∗ j ′′′∗ F ′′′ by Lemma 1.3.5.1. Thus, thecommutative diagram V ′′′ ⊂ −−−→ Y ′′′ −−−→ S ′′ y (cid:3) y y V ⊂ −−−→ Y −−−→ S satisfies the required conditions. Corollary 1.5.4.
Let f : X → Y be a morphism of scheme of finite type over a perfectfield k . Assume that Y is smooth of dimension . Let F be a constructible complex of Λ -modules on X . Assume that f : X → Y is locally acyclic relatively to F and that thereexists a dense open subset V ⊂ Y such that f : X → Y is SS F -transversal on V . Then,there exists a cartesian diagram X ←−−− X ′ f y (cid:3) y f ′ Y ←−−− Y ′ of morphisms of schemes of finite type over k satisfying the following condition: Themorphism Y ′ → Y is a finite generically ´etale morphism of smooth curves. For the pull-back F ′ of F on X ′ , the morphism f ′ : X ′ → Y ′ is SS F ′ -transversal.Proof. Since the shifted vanishing cycles functor R Φ[ −
1] is t -exact, we may assume that F is a perverse sheaf. Then the assertion follows from the case where S = Spec k inTheorem 1.5.2, Proposition 1.1.2.1 and weak approximation.24 Characteristic cycles and the direct image
To state the compatibility with push-forward, we fix some terminology and notations.Recall that a morphism f : X → Y of noetherian schemes is said to be proper on a closedsubset Z ⊂ X if its restriction Z → Y is proper with respect to a closed subschemestructure of Z ⊂ X .Let f : X → Y be a morphism of smooth schemes over a field k and we consider thediagram(2.1) T ∗ X ←−−− X × Y T ∗ Y −−−→ T ∗ Y as an algebraic correspondence from T ∗ X to T ∗ Y . Assume that every irreducible com-ponent of X (resp. of Y ) is of dimension n (resp. m ). Let B ⊂ X be a closed subset onwhich f : X → Y is proper and let C ⊂ T ∗ X be a closed subset of B × X T ∗ X . Then, theclosed subset f ◦ C ⊂ T ∗ Y is defined as the image by the right arrow in (2.1) of the inverseimage of C by the left arrow. It is a closed subset by the assumption that f is proper on B . The composition of the Gysin map [7, 6.6] for the first arrow and the push-forwardmap for the second arrow defines a morphism(2.2) f ! : CH n ( C ) −−−→ CH m ( f ◦ C )since dim T ∗ X − dim X × Y T ∗ Y = n − m . If every irreducible component of C (resp. f ◦ C )is of dimension ≦ n (resp. ≦ m ), the morphism (2.2) defines a morphism(2.3) f ! : Z n ( C ) −−−→ Z m ( f ◦ C )of free abelian groups of cycles. Lemma 2.1.1.
Let X g / / f ❇❇❇❇❇❇❇❇ X ′ f ′ (cid:15) (cid:15) Y be a commutative diagram of morphisms of smooth schemes over k . Assume that everyirreducible component of X (resp. of X ′ and Y ) is of dimension n (resp. n ′ and m ). Let B ⊂ X be a closed subset on which f : X → Y is proper and let C ⊂ T ∗ X be a closedsubset of B × X T ∗ X . Then, the diagram CH n ( C ) g ! / / f ! & & ◆◆◆◆◆◆◆◆◆◆◆ CH n ′ ( g ◦ C ) f ′ ! (cid:15) (cid:15) CH m ( f ◦ C ) is commutative.Proof. We consider the diagram T ∗ X ←−−− X × X ′ T ∗ X ′ −−−→ T ∗ X ′ x (cid:3) x X × Y T ∗ Y −−−→ X ′ × Y T ∗ Y −−−→ T ∗ Y Lemma 2.1.2.
Let f : X → Y be a smooth morphism of smooth irreducible schemesover a perfect field k . Assume that X (resp. of Y ) is of dimension n (resp. m ). Let C = S a C a ⊂ T ∗ X be a closed conical subset such that every irreducible component C a isof dimension n and that f : X → Y is properly C -transversal and is proper on the base B = C ∩ T ∗ X X ⊂ X .Let A = P a m a C a be a linear combination. Let y ∈ Y be a closed point, let A y = i ! y A be the pull-back [16, Definition 7.1] by the closed immersion i y : X y → X of the fiber andlet ( A y , T ∗ X y X y ) T ∗ X y denote the intersection number. Then, we have (2.4) f ! A = ( − m ( A y , T ∗ X y X y ) T ∗ X y · [ T ∗ Y Y ] in Z m ( T ∗ Y Y ) .Proof. Since the closed immersion i y : X y → X is properly C -transversal by Lemma1.2.6.1, the pull-back A y = i ! y A is defined. Further by the assumption that f : X → Y is C -transversal, we have an inclusion f ◦ C ⊂ T ∗ Y Y and f ! A is defined as an element ofCH m ( f ◦ C ) = Z m ( f ◦ C ) ⊂ Z m ( T ∗ Y Y ). Hence it suffices to show that the coefficient of T ∗ Y Y in f ! A equals the intersection number ( − m ( A y , T ∗ X y X y ) T ∗ X y .We consider the cartesian diagram T ∗ X ←−−− X × Y T ∗ Y −−−→ T ∗ Y x (cid:3) x (cid:3) x X y × X T ∗ X ←−−− X y × Y T ∗ Y −−−→ y × Y T ∗ Y y (cid:3) y (cid:3) y T ∗ X y ←−−− X y −−−→ y. We regard the four sides of the exterior square of the diagram as algebraic correspondences.The coefficient of f ! A is the image of A by the composition via the upper right corner. Itequals the composition via the upper right corner by [7, Theorem 6.2 (a)] applied to theupper right and the lower left squares. Since the definition of i ! y A in [16, Definition 7.1]involves the sign ( − dim X − dim X y = ( − m , the assertion follows.We study the case where Y is a smooth curve and dim f ◦ C = 1. Let f : X → Y bea morphism of smooth schemes over k . Assume that every irreducible component of X (resp. of Y ) is of dimension n (resp. 1). Let C ⊂ T ∗ X be a closed conical subset suchthat every irreducible component C a of C = S a C a is of dimension n and that f : X → Y is proper on the base B = C ∩ T ∗ X X ⊂ X . Let V ⊂ Y be a dense open subscheme suchthat the base change f V : X V → V is properly C V -transversal for the restriction C V of C on X V .Let y ∈ Y V be a closed point on the boundary and let t be a uniformizer at y and let df denote the section of T ∗ X defined on a neighborhood of the fiber X y by the pull-back f ∗ dt . Then, on a neighborhood of X y , the intersection C ∩ df ⊂ T ∗ X is supported on theinverse image of the intersection B ∩ X y . Hence for a linear combination A = P a m a C a ,the intersection product(2.5) ( A, df ) T ∗ X,X y X y is defined as an element of CH ( B ∩ X y ). Since C is conical, theintersection product ( A, df ) T ∗ X,X y does not depend on the choice of t . Thus the intersectionnumber also denoted ( A, df ) T ∗ X,X y is defined as its image by the degree mapping CH ( B ∩ X y ) → CH ( y ) = Z . Lemma 2.1.3.
Let f : X → Y be a morphism of smooth irreducible schemes over a perfectfield k . Assume that X (resp. of Y ) is of dimension n (resp. ). Let C = S a C a ⊂ T ∗ X be a closed conical subset as in Lemma . The following conditions are equivalent: (1) dim f ◦ C ≦ . (2) There exists a dense open subscheme V ⊂ Y such that the base change f V : X V → V is C V -transversal for the restriction C V of C on X V . (3) There exists a dense open subscheme V ⊂ Y such that the base change f V : X V → V is properly C V -transversal for the restriction C V of C on X V . Let V ⊂ Y be a dense open subscheme satisfying the condition (3) above. Let A = P a m a C a be a linear combination, let v ∈ V be a closed point and define the intersectionnumber ( A v , T ∗ X v X v ) T ∗ X v as in Lemma . Then, we have (2.6) f ! A = − ( A v , T ∗ X v X v ) T ∗ X v · [ T ∗ Y Y ] + X y ∈ Y V ( A, df ) T ∗ X,X y · [ T ∗ y Y ] in Z ( f ◦ C ) .Proof.
1. Since f ◦ C is a closed conical subset of the line bundle T ∗ Y , the condition (1)is equivalent to the existence of a dense open subset V ⊂ Y such that f ◦ C ⊂ T ∗ Y Y ∪ S y ∈ Y V T ∗ y Y . This is equivalent to the condition (2). The equivalence (2) ⇔ (3) followsfrom Lemma 1.3.5.2.2. It suffices to compare the coefficients of the 0-section T ∗ Y Y and of the fibers T ∗ y Y respectively. For those of T ∗ Y Y , it is proved in Lemma 2.1.2. For those of T ∗ y Y , it followsfrom the projection formula [7, Theorem 6.2 (a)] applied to the cartesian square in thediagram T ∗ X ←−−− X × Y T ∗ Y −−−→ T ∗ Y df x (cid:3) x dt X −−−→ Y. Lemma 2.1.4.
Let X be a scheme of finite type of dimension d over a field k and let E bea vector bundle on X associated to a locally free O X -module E of rank n . Let s : X → E be a section, X → E be the zero section and Z = Z ( s ) = 0( X ) ∩ s ( X ) ⊂ X be the zerolocus of s . Let K = [ O X s → E ] be the complex of O X -modules where E is put on degree and let c nXZ ( K ) be the localized Chern class defined in [4, Section 1] . Then, we have (0( X ) , s ( X )) E = c nXZ ( K ) ∩ [ X ] in CH d − n ( Z ) .Proof. We may assume that X is integral and Z $ X . By taking the blow-up at Z andby [13, Proposition 2.3.1.6], we may assume that Z is a Cartier divisor D ⊂ X . Then, we27ave an exact sequence of 0 → L → E → F → O X -modules where L and F are of rank 1 and n − s ∈ Γ( X, E ) is defined by s ∈ Γ( X, L ). Then,the right hand side equals c n − ( F ) ∩ [ D ] by [4, Proposition (1.1) (iii)]. The left hand sidealso equals c n − ( F ) ∩ [ D ] by the excess intersection formula [7, Theorem 6.3].We define the specialization of a cycle. Let f : X → Y be a smooth morphism ofsmooth schemes over a perfect field k and assume that X (resp. Y ) is equidimensional ofdimension n +1 (resp. 1). Let y ∈ Y be a closed point, V = Y { y } be the complement and U = X × Y V be the inverse image. Let C ⊂ T ∗ U be a closed conical subset equidimensionalof dimension n + 1 and assume that U → V is properly C -transversal. We define itsspecialization sp y C ⊂ T ∗ X y as follows. By the assumption that U → V is properly C -transversal and [16, Lemma3.1], the morphism T ∗ U → T ∗ U/V to the relative cotangent bundle is finite on C . Henceits image C ′ ⊂ T ∗ U/V is a closed conical subset. Let ¯ C ′ ⊂ T ∗ X/Y be the closure anddefine sp y C ⊂ T ∗ X y to be the fiber ¯ C ′ × Y y ⊂ T ∗ X/Y × Y y = T ∗ X y . The specializationsp y C ⊂ T ∗ X y is a closed conical subset equidimensional of dimension n .For a linear combination A = P a m a C a of irreducible components of C = S a C a , wedefine its specialization sp y A ∈ Z n (sp y C )as follows. First, we define A ′ ∈ Z n +1 ( C ′ ) as the push-forward of A by the morphism T ∗ U → T ∗ U/V finite on C . Let ¯ A ′ ∈ Z n +1 ( ¯ C ′ ) be the unique element extending A ′ ∈ Z n +1 ( C ′ ). Then, we define sp y A ∈ Z n (sp y C ) to be the minus of the pull-back of A ′ by theGysin map for the immersion i y : X y → X . If X → Y is proper, for a closed point v ∈ V and the closed immersion i v : X v → X , we have(2.7) (sp y A, T ∗ X y X y ) T ∗ X y = ( i ! v A, T ∗ X v X v ) T ∗ X v since the definition of i ! v A in [16, Definition 7.1] involves the sign ( − dim X − dim X v = − Lemma 2.1.5.
Let f : X → Y be a smooth morphism of smooth schemes over a perfectfield k and assume that X (resp. Y ) is equidimensional of dimension n + 1 (resp. ). Let y ∈ Y be a closed point, i y : X y → X be the closed immersion of the fiber, V = Y { y } bethe complement and U = X × Y V be the inverse image. Let C ⊂ T ∗ X be a closed conicalsubset equidimensional of dimension n + 1 such that f : X → Y is properly C -transversal. For the restriction C U of C on U , we have (2.8) sp y C U = i ◦ y C. For a linear combination A = P a m a C a of irreducible components of C = S a C a and its restriction A U on U , we have (2.9) sp y A U = i ! y A. Proof.
1. By the assumption that f : X → Y is properly C -transversal and [16, Lemma3.1], the morphism T ∗ X → T ∗ X/Y to the relative cotangent bundle is finite on C andhence its image C ′ ⊂ T ∗ X/Y is a closed conical subset. Further C ′ with reduced schemestructure is flat over Y . Hence it equals the closure of the restriction C ′ U and we obtain(2.8). 28. We consider the cartesian diagram T ∗ X −−−→ T ∗ X/Y x (cid:3) x X y × X T ∗ X −−−→ T ∗ X y . The right hand side is the minus of the image of A by the push-forward and the pull-backvia upper right. The left hand side is the minus of the image of A by the pull-back andthe push forward via lower left. Hence the assertion follows from the projection formula[7, Theorem 6.2 (a)]. Let k be a field and let Λ be a finite field of characteristic ℓ invertible in k . Let X be asmooth scheme over k such that every irreducible component is of dimension n . Let F bea constructible complex of Λ-modules on X and C = SS F be the singular support. Then,every irreducible component C a of C = S a C a has the same dimension as X [2, Theorem1.3 (ii)] and the base B = C ∩ T ∗ X X ⊂ T ∗ X X = X defined as the intersection with the0-sections equals the support of F [2, Lemma 2.1 (i)]. Let f : X → Y be a morphism ofsmooth schemes over k , proper on the support of F . Then, we have an inclusion(2.10) SSRf ∗ F ⊂ f ◦ SS F by [2, Lemma 2.2 (ii)].We restate a conjecture from [17, Conjecture 1]. Conjecture 2.2.1.
Let f : X → Y be a morphism of smooth schemes over a perfect field k . Assume that every irreducible component of X (resp. of Y ) is of dimension n (resp. m ). Let F be a constructible complex of Λ -modules on X and C = SS F be the singularsupport. Assume that f is proper on the support of F . Then, we have (2.11) CCRf ∗ F = f ! CC F in CH m ( f ◦ SS F ) . If dim f ◦ SS F ≦ m , the equality (2.11) is an equality as cycles in CH m ( f ◦ SS F ) = Z m ( f ◦ SS F ) without rational equivalence.A weaker version of Conjecture 2.2.1 is proved in the case k is finite and X and Y areprojective in [19] using ε -factors.If Y = Spec k , the equality (2.11) means the index formula(2.12) χ ( X ¯ k , F ) = ( CC F , T ∗ X X ) T ∗ X where the right hand side denotes the intersection number. Further if X is projective, theequality (2.12) is proved in [16, Theorem 7.13]. Lemma 2.2.2.
Let f : X → Y be a morphism of smooth schemes over k and let F be aconstructible complex of Λ -modules. Assume that f : X → Y is proper on the support of F . Assume that every irreducible component of X (resp. of Y ) is of dimension n (resp. m ). Let X g / / f ❇❇❇❇❇❇❇❇ X ′ f ′ (cid:15) (cid:15) Y be a commutative diagram of morphisms of smooth schemes over k . Then, we have (2.13) f ! CC F = f ′ ! ( g ! CC F ) in CH m ( f ◦ SS F ) . Assume that one of the following conditions (1) and (2) is satisfied: (1) f : X → Y is an immersion. (2) f : X → Y is quasi-projective and SS F -transversal.Then, we have dim( f ◦ SS F ) ≦ m = dim Y and (2 . CCRf ∗ F = f ! CC F in Z m ( f ◦ SS F ) .Proof.
1. It follows from Lemma 2.1.1.2. The case (1) is proved in [16, Lemma 5.13.2]. We show the case (2). Since f ◦ SS F isa subset of the 0-section T ∗ Y Y , we have dim( f ◦ SS F ) ≦ m = dim Y . We may assume that Y is connected and affine and hence X is quasi-projective. Let X → P be an immersion toa projective space and factorize X → Y as the composition of an immersion X → Y × P and the projection Y × P → Y . Then, by 1 and the case (1), we may assume that f : X → Y is projective and smooth.By the assumption that f : X → Y is SS F -transversal, it is locally acyclic relatively to F by Lemma 1.3.4.1. Since f : X → Y is proper, the direct image Rf ∗ F is locally constantby [10, 5.2.4]. By Lemma 1.3.5.2, there exists a dense open subscheme V ⊂ Y such that f : X → Y is properly SS F -transversal on V . By [16, Lemma 5.11.1] and Lemma 2.1.2,it suffices to show the equality(2.14) rank Rf ∗ F = ( i ! y CC F , T ∗ X y X y ) T ∗ X y for a closed point y ∈ V . Since rank Rf ∗ F = χ ( X ¯ y , i ∗ y F ), the equality (2.14) follows fromthe compatibility CCi ∗ y F = i ! y CC F with the pull-back [16, Theorem 7.6] and the indexformula [16, Theorem 7.13].We consider the case where Y is a smooth curve and dim f ◦ SS F ≦
1. We recallthe definition of the Artin conductor and the description of the characteristic cycle of asheaf on a curve. Let Y be a smooth irreducible curve over a perfect field k and let G be a constructible complex of Λ-modules on Y . Let V ⊂ Y be a dense open subschemesuch that the restriction G V is locally constant i. e. the cohomology sheaf H q G V is locallyconstant for every integer q . For a closed point y ∈ Y , the Artin conductor a y G is definedby(2.15) a y G = rank G V − rank G ¯ y + Sw y G . Here ¯ y denotes a geometric point above y and Sw y denotes the alternating sum of theSwan conductor. The characteristic cycle is given by(2.16) CC G = − (cid:16) rank G V · [ T ∗ Y Y ] + X y ∈ Y V a y G · [ T ∗ y Y ] (cid:17)
30y [16, Lemma 5.11.3]. Here T ∗ y Y is the fiber of y .Let f : X → Y be a morphism of smooth schemes over a perfect field k and y ∈ Y bea closed point. Assume that dim Y = 1. Let F be a constructible complex of Λ-moduleson X . Assume that f : X → Y is proper on the support of F . Under the assumptiondim f ◦ SS F ≦
1, the equality
CCRf ∗ F = f ! CC F (2.11) in Z ( f ◦ SS F ) is equivalent tothe equality(2.17) − a y Rf ∗ F = ( CC F , df ) T ∗ X,X y . for every closed point y ⊂ Y V by (2.16), Lemma 2.2.2.2 (2) and Lemma 2.1.3.2, wherethe right hand side is defined as in (2.5).
Theorem 2.2.3.
Let f : X → Y be a quasi-projective morphism of smooth schemes overa perfect field k and y ∈ Y be a closed point. Assume that dim Y = 1 . Let F be aconstructible complex of Λ -modules on X . Assume that f : X → Y is proper on thesupport of F and is properly SS F -transversal on a dense open subscheme V ⊂ Y . Then,we have (2 . − a y Rf ∗ F = ( CC F , df ) T ∗ X,X y . Proof.
We may assume that k is algebraically closed. By the same argument as in theproof of Lemma 2.2.2.2, we may assume that f : X = Y × P → Y is the projection for aprojective space P . By Lemma 2.1.3.1 and by replacing Y by a projective smooth curveover k containing Y as a dense open subscheme, we may assume that Y is projective andsmooth.By Lemma 2.2.2 applied to X → Y → Spec k , we obtain( f ! CC F , T ∗ Y Y ) T ∗ Y = ( CC F , T ∗ X X ) T ∗ X . By the index formula [16, Theorem 7.13], we have(
CCRf ∗ F , T ∗ Y Y ) T ∗ Y = χ ( Y ¯ k , Rf ∗ F ) = χ ( X ¯ k , F ) = ( CC F , T ∗ X X ) T ∗ X . Thus, we have (
CCRf ∗ F − f ! CC F , T ∗ Y Y ) T ∗ Y = 0 . Since the coefficients of T ∗ Y Y in CCRf ∗ F and f ! CC F are equal by (2.4), (2.16) and theindex formula [16, Theorem 7.13], we obtain(2.18) X y ∈ Y V − a y Rf ∗ F = X y ∈ Y V ( CC F , df ) T ∗ X,X y . By d´evissage using Lemma 1.3.4.3 and [16, Lemma 5.13.1], we may assume that F is aperverse sheaf. Set e V = V ∪ { y } and Z = Y e V . By Corollary 1.1.3, Corollary 1.5.4 andweak approximation, there exists a faithfully flat finite morphism Y ′ → Y of projectivesmooth curves ´etale at y satisfying the following condition: Let X ←−−− X ′ ˜ j ′ ←−−− X ′ e V ′ f y (cid:3) f ′ y (cid:3) y Y ←−−− Y ′ ←−−− e V ′ = Y ′ × Y e V
31e a cartesian diagram and set F ′ = ˜ j ′ ! ∗ F ′ e V ′ for the pull-back F ′ e V ′ of F on X ′ e V ′ . Then on Y ′ = Y ′ Y ′ × Y y , the morphism f ′ : X ′ → Y ′ is SS F ′ -transversal and hence is universallylocally acyclic relatively to F ′ .For each y ′ ∈ Z ′ = Z × Y Y ′ , we have a y ′ Rf ′∗ F ′ = ( CC F ′ , df ′ ) T ∗ X ′ ,y ′ = 0. Since Y ′ → Y is ´etale at y , for each y ′ ∈ Y ′ × Y y , we have a y Rf ∗ F = a y ′ Rf ′∗ F ′ and ( CC F , df ) T ∗ X,X y =( CC F ′ , df ′ ) T ∗ X ′ ,y ′ . Thus, by applying (2.18) to f ′ : X ′ → Y ′ and F ′ , we obtain − [ Y ′ : Y ] · a y Rf ∗ F = [ Y ′ : Y ] · ( CC F , df ) T ∗ X,y and hence (2.17).
Corollary 2.2.4 (cf. [4, Conjecture]) . Let f : X → Y be a projective flat morphism ofsmooth schemes over a perfect field k . Assume that dim X = n , dim Y = 1 and that thereexists a dense open subscheme V ⊂ Y such that the base change f V : X × Y V → V issmooth. Then, for a closed point y ∈ Y , we have (2.19) − a y Rf ∗ Λ = ( − n c nXX y (Ω X/Y ) ∩ [ X ] . Proof.
Applying Theorem 2.2.3 to the constant sheaf F = Λ and CC Λ = ( − n [ T ∗ X X ], weobtain − a y Rf ∗ Λ = ( − n ( T ∗ X X, df ) T ∗ X,X y . By applying Lemma 2.1.4 to the right handside and [ f ∗ Ω Y/k → Ω X/k ], we obtain (2.19).
Theorem 2.2.5.
Let f : X → Y be a morphism of smooth schemes over a perfect field k . Let F be a constructible complex of Λ -modules on X and C = SS F be the singularsupport. Assume that Y is projective, that f : X → Y is quasi-projective and is proper onthe support of F and that we have an inequalty (2.20) dim f ◦ C ≦ dim Y = m. Then, we have (2.21)
CCRf ∗ F = f ! CC F in Z m ( f ◦ SS F ) .Proof. We may assume that k is algebraically closed. Since X is quasi-projective, thereexists a locally closed immersion i : X → P to a projective space P . By decomposing f asthe composition of the immersion ( i, f ) : X → P × Y and the second projection P × Y → Y ,we may assume that f is projective and smooth by Lemma 2.2.2. Set C = f ◦ SS F ⊂ T ∗ Y .We have SSRf ∗ F ⊂ f ◦ SS F = C . By the assumption, we have dim C ≦ m . By theindex formula [16, Theorem 7.13] and Theorem 2.2.3, the equality (2.21) is proved for Y of dimension ≦
1. We show the general case by reducing to the case dim Y = 1.We take a closed immersion of Y to a projective space i : Y → P . We use the notations P p ← Q p ∨ → P ∨ in (1.4) and let p X : X × P Q → X be the projection. After replacingthe immersion i by the composition with a Veronese embedding if necessary, we mayassume that the restriction to P ( i ◦ C ) ⊂ Q = P ( T ∗ P ) of the projection p ∨ : Q → P ∨ isgenerically radicial by the assumption dim C ≦ m = dim Y and by [16, Corollary 3.21].Let C ∨ = p ∨◦ p ◦ Y C ⊂ T ∗ P ∨ and let D denote the image p ∨ ( P ( i ◦ C )) ⊂ P ∨ . By Lemma1.2.7, Lemma 1.2.3.3 and the Bertini theorem, there exists a line L ⊂ P ∨ satisfyingthe following conditions: The immersion h : L → P ∨ is properly C ∨ -transversal. The32orphism h : L → P ∨ meets p ◦ X SS F properly. The axis A L of L meets Y transverselyand L meets D transversely.Since A L meets Y transversely, the blow-up Y ′ of Y at Y ∩ A L is smooth. We considerthe cartesian diagram(2.22) X p X ←−−− X × P Q h X ←−−− X ′ f y (cid:3) ˜ f y (cid:3) y f ′ Y p Y ←−−− Y × P Q h Y ←−−− Y ′ p ∨ y (cid:3) y p L P ∨ h ←−−− L of projective smooth schemes over k . The equality (2.21) is equivalent to p ∨ ! p ! Y CCRf ∗ F = p ∨ ! p ! Y f ! CC F . It suffices to compare the coefficients of C ∨ a = p ∨◦ p ◦ Y C a for each irreducible component of C = S a C a of dimension m = dim Y . Hence, this is further equivalent to(2.23) h ! p ∨ ! p ! Y CCRf ∗ F = h ! p ∨ ! p ! Y f ! CC F since P ( i ◦ C ) → D is generically radicial, h : L → P ∨ is properly C ∨ -transversal and L meets D transversely. Let π X : X ′ → X denote the composition p X ◦ h X of the top line in(2.22). We show that the equality (2.23) is equivalent to(2.24) CCR ( p L f ′ ) ∗ π ∗ X F = ( p L f ′ ) ! CCπ ∗ X F . First, we compare the left hand sides. By [16, Corollary 7.12] applied to i ∗ Rf ∗ F on P ,the left hand side of (2.23) equals h ! CCRp ∨∗ p ∗ Y Rf ∗ F . Since SSRp ∨∗ p ∗ Y Rf ∗ F ⊂ C ∨ and since h : L → P ∨ is properly C ∨ -transversal, the left hand side further equals CCh ∗ Rp ∨∗ p ∗ Y Rf ∗ F by [16, Theorem 7.6]. By proper base change theorem, this is equal to the left hand side CCR ( p L f ′ ) ∗ π ∗ X F of (2.24).Next, we compare the right hand sides. The right hand side of (2.23) is equal to( p L f ′ ) ! π ! X CC F by the projection formula [7, Theorem 6.2 (a)]. Since h : L → P ∨ is C ∨ -transversal and C ∨ = p ∨◦ p ◦ Y C = ( p ∨ ˜ f ) ◦ p ◦ X SS F , the immersion h X : X ′ → X × P Q is p ◦ X SS F -transversal by Lemma 1.2.6.2. Further since h : L → P ∨ meets p ◦ X SS F properly,the immersion h X : X ′ → X × P Q is properly p ◦ X SS F -transversal. Since p X is smooth,the composition π X = p X ◦ h X is properly SS F -transversal. Thus by [16, Theorem 7.6],it further equals to the right hand side ( p L f ′ ) ! CCπ ∗ X F of (2.24).We show the equality (2.24) by applying Theorem 2.2.3 to complete the proof. Since P ( i ◦ C ) ⊂ Y × P Q = P ( Y × P T ∗ P ) is the complement of the largest open subset where p ∨ : Y × P Q → P ∨ is p ◦ Y C -transversal and since p ◦ Y C = p ◦ Y f ◦ SS F = ˜ f ◦ p ◦ X SS F =˜ f ◦ SSp ∗ X F , the composition p ∨ ˜ f : X × P Q → P ∨ is SSp ∗ X F -transversal on the comple-ment P ∨ D by [16, Lemma 3.8 (1) ⇒ (2)]. By Lemma 1.2.6, the morphism p L f ′ : X ′ → L is SSπ ∗ X F -transversal on the dense open subset L L ∩ D . Hence the equality (2.24)follows from Theorem 2.2.3 applied to π ∗ X F and the equality (2.21) is proved.In the case of characteristic 0, we recover the classical result as in [12, Proposition9.4.2], in a slightly weaker form. Let X be a smooth scheme equidimensional of dimension33 over a field k and let ω X ∈ Ω ( T ∗ X ) denote the canonical symplectic form on thecotangent bundle T ∗ X . Let C ⊂ T ∗ X be a closed conical subset. We say that C is isotropic if the restriction of ω X on C is 0. We say that C is Lagrangean if it is isotropicand if C is equidimensional of dimension n . Lemma 2.2.6.
Let k be a field of characteristic and let f : X → Y be a morphism ofsmooth schemes over k . Assume that X (resp. Y ) is equidimensional of dimension n (resp. m ). Let C ⊂ T ∗ X be a closed conical subset. If C ⊂ T ∗ X is isotropic, then f ◦ C ⊂ T ∗ Y is also isotropic. The author learned the following proof from Beilinson.
Proof.
Let T ∗ Γ ( X × Y ) ⊂ T ∗ ( X × Y ) be the normal bundle of the graph Γ ⊂ X × Y of f : X → Y and let p : T ∗ ( X × Y ) = T ∗ X × T ∗ Y → T ∗ Y be the projection. The directimage f ◦ C ⊂ T ∗ Y equals the image by p of the intersection C = T ∗ Γ ( X × Y ) ∩ ( C × T ∗ Y ).Since the normal bundle T ∗ Γ ( X × Y ) ⊂ T ∗ ( X × Y ) is isotropic and since ω X × Y equalsthe sum p ∗ ω X + p ∗ ω Y of the pull-backs by projections, the assumption that C ⊂ T ∗ X isisotropic implies that the restriction of p ∗ ω Y on C is 0. Since k is of characteristic 0,for each irreducible component C ′ of f ◦ C ⊂ T ∗ Y , there exists a closed subset C ′ ⊂ C generically ´etale over C ′ . Hence the assertion follows. Proposition 2.2.7.
Let k be a field of characteristic and let X be a smooth schemesover k . Let F be a constructible complex of Λ -modules on X . The singular support SS F is Lagrangean. Let f : X → Y be a morphism of smooth schemes over k . Assume that f is properon the support of F . Then, the inequality (2.20) holds. Further if f : X → Y is quasi-projective, the equality (2.21) holds.Proof.
1. We may assume that X is equidimensional of dimension n . Since the singularsupport SS F is equidimensional of dimension n [2, Theorem 1.3 (ii)], it suffices to showthat SS F is isotropic. By devissage, we may assume that there exist a locally closedimmersion i : V → X of smooth scheme, a locally constant sheaf G on V and F = i ! G .Since the resolution of singularity is known in characteristic 0, the immersion i isdecomposed by an open immersion j : V → W and a proper morphism h : W → X suchthat W is smooth and V is the complement of a divisor with simple normal crossings.Thus, by the inclusion SS F = SSRh ∗ j ! G ⊂ h ◦ SSj ! G and Lemma 2.2.6, it is reduced tothe case where i = j is an open immersion of the complement of a divisor with simplenormal crossings. Since k is of characteristic 0, this case is proved in [16, Proposition 4.11].2. By 1 and Lemma 2.2.6, the direct image f ◦ SS F is isotropic. Hence the inequalitydim f ◦ SS F ≦ dim Y (2.20) holds.We show the equality CCRf ∗ F = f ! CC F (2.21). Similarly as in the proof of Theorem2.2.5, we may assume that Y is affine and f : X = P × Y → Y is the projection for aprojective smooth scheme P over k . By resolution of singularity, we may assume that Y is projective and smooth. Then since the inequality (2.20) holds, we may apply Theorem2.2.5. We prepare some notation to formulate an index formula for vanishing cycle complex. Let f : X → Y be a smooth morphism of smooth schemes over a perfect field k . Assume that34 (resp. Y ) is equidimensional of dimension n + 1 (resp. 1). Let F be a constructiblecomplex of Λ-modules on X . Let y ∈ Y be a closed point and i y : X y → X be theclosed immersion of the fiber. Assume that f : X → Y is properly SS F -transversal onthe complement X X y of the fiber X y = f − ( y ). Then, the specialization(2.25) sp y SS F ⊂ T ∗ X y is defined as a closed conical subset equidimensional of dimension n . Further, the special-ization(2.26) sp y CC F ∈ Z n (sp y SS F )is defined as a cycle. Lemma 2.3.1.
Let f : X → Y be a smooth morphism of smooth schemes over a field k and assume dim Y = 1 . Let F be a constructible complex of Λ -modules on X and assumethat f : X → Y is properly SS F -transversal. Let y ∈ Y be a closed point. Then, we have (2.27) SSR Ψ y F = sp y SS F . Further if k is perfect, we have (2.28) CCR Ψ y F = sp y CC F . Proof.
Let i y : X y → X denote the closed immersion of the fiber. Then, by the assumptionthat f : X → Y is properly SS F -transversal, we have sp y SS F = i ◦ y SS F and sp y CC F = i ! y CC F . Recall that the definitions of sp y and i ! y both involve the minus sign.Since f : X → Y is locally acyclic relatively to F by Lemma 1.3.4.2, the canonicalmorphism i ∗ y F → R Ψ y F is an isomorphism. Hence the equalities (2.27) and (2.28) followfrom Lemma 1.3.1 and [16, Theorem 7.6] respectively.The following example shows that the inclusion SSR Ψ F ⊂ sp y SS F does not hold ingeneral. Example 2.3.2.
Let k be a field of characteristic p >
2. Let X = A × P and j : U = A × A = Spec k [ x, y ] → X be the open immersion. Let G be the locally constant sheafof Λ-modules of rank 1 on U defined by the Artin-Schreier covering t p − t = x p y and by anon-trivial character F p → Λ × . Then, the nearby cycles complex R Ψ ∞ F is acyclic exceptat the closed point (0 , ∞ ) or at degree 1 and dim R Ψ F (0 , ∞ ) = 1 . Hence, the singularsupport
SSR Ψ ∞ F equals the fiber T ∗ (0 , ∞ ) X ∞ at the closed point and is not a subset of thezero-section sp ∞ SS F = T ∗ X ∞ X ∞ .Let Z ⊂ X y be a closed subset. Assume that f : X → Y is properly SS F -transversalon the complement of Z . Then, on the complement X y Z , we have sp y CC F = i ! y CC F = CCi ∗ y F by Lemma 2.1.5 and the compatibility with the pull-back [16, Theorem 7.6]. Thus,the difference(2.29) δ y CC F = sp y CC F −
CCi ∗ y F is defined as a cycle in Z n (cid:0) Z × X (sp y SS F ∪
SSi ∗ y F ) (cid:1) supported on Z . If Z is proper over Y , the intersection number ( δ y SS F , T ∗ X y X y ) T ∗ X y is defined.35 roposition 2.3.3. Let f : X → Y be a smooth morphism of smooth schemes over aperfect field k . Assume that X (resp. Y ) is equidimensional of dimension n + 1 (resp. ).Let F be a constructible complex of Λ -modules on X . Let y ∈ Y be a closed point and let Z ⊂ X y be a closed subset. Assume that f : X → Y is properly SS F -transversal on thecomplement of Z and that either of the following conditions (1) and (2) is satisfied: (1) f : X → Y is projective. (2) dim Z = 0 .Then, for the vanishing cycles complex R Φ y F , we have (2.30) χ ( Z ¯ k , R Φ y F ) = ( δ y CC F , T ∗ X y X y ) T ∗ X y . Proof.
We may assume that k is algebraically closed.We show the case (1). Let v ∈ Y be a closed point different from y and let i v : X v → X be the closed immersion. Then, since the projective morphism f : X → Y is locally acyclicrelative to F outside Z by Lemma 1.3.4.2, the left hand side of (2.30) equals(2.31) χ ( Z, R Φ y F ) = χ ( X y , R Ψ y F ) − χ ( X y , i ∗ y F ) = χ ( X v , i ∗ v F ) − χ ( X y , i ∗ y F )The right hand side of (2.30)( δ y CC F , T ∗ X y X y ) T ∗ X y = (sp y CC F , T ∗ X y X y ) T ∗ X y − ( CCi ∗ y F , T ∗ X y X y ) T ∗ X y equals(2.32) ( i ! v CC F , T ∗ X v X v ) T ∗ X v − ( CCi ∗ y F , T ∗ X y X y ) T ∗ X y by (2.7). Since i v : X v → X is properly SS F -transversal by Lemma 1.2.6, the right handside of (2.31) equals (2.32) by the compatibility with the pull-back [16, Theorem 7.6] andthe index formula [16, Theorem 7.13]. Thus the equality (2.30) is proved.We show the case (2). Since the formation of nearby cycles complex commutes withbase change by [6, Proposition 3.7], we may assume that the action of the inertia group I y on R Ψ y F is trivial. Since the vanishing cycles functor is t -exact by [11, Corollaire 4.6],we may assume that F is a simple perverse sheaf.First, we consider the case F is supported on the closed fiber X y . By the assumptionthat f : X → Y is properly SS F -transversal on the complement of Z , the morphism f : X → Y is locally acyclic relatively to F on the complement of Z . Thus F is supportedon Z and the assertion follows in this case.We may assume that the restriction F | X η on the generic fiber is non-trivial. Then, byProposition 1.1.2.2, the morphism f : X → Y is locally acyclic relatively to F . Hence byLemma 1.3.8.2, the morphism f : X → Y is properly SS F -transversal and the assertionfollows from Lemma 2.3.1.In the case (2) dim Z = 0, Proposition 2.3.3 means CCR Φ y F = δ y CC F . However,Examples 1.4.8 and 2.3.2 show that one cannot expect to have CCR Ψ y F = sp y CC F orequivalently CCR Φ y F = δ y CC F in general. References [1] M. Artin,
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