Non-logarithmic case

following condition:

(R(n)) X is a regular and flat equidimensional scheme of finite type over OK of relative dimension n−1. The generic fiber XK is smooth.

Lemma 5.1.1. — Let X be a scheme over OK satisfying the condition (R(n)) and x be a point of Xin the closed fiber. Then there exist an open neighborhood U ofx and a regular immer-sion U → P of codimension 1 into a smooth scheme P of relative dimension n over O^K. Namely, Xis locally a hypersurface of virtual relative dimension n−1 over OK.

Proof. — Let t1, ...,tm ∈O^X,x be a minimal system of generators of the maximal ideal mx of the local ring O^X,x. Let tm+1, ...,tn ∈ O^X,x be a lifting of a transcendental basis of the residue field κ(x) over F such that κ(x) is a finite separable extension of F(tm+1, ...,tn). We take an open neighborhood U of x and define a map U→ A_Oⁿ

K = Spec OK[T1, ...,Tn] by sending Ti to ti. Then we have Ω¹U/Aⁿ_O

K,x =0. By shrinking U if necessary, we may assume Ω¹U/Aⁿ_O

K = 0, namely U → A_Oⁿ_K is unramified. By [15]

Corollaire (18.4.7), further shrinking U if necessary, there exist a closed immersion U → P and an etale morphism P → Aⁿ_OK such that the composition is the map U→Aⁿ_OK. The schemePis smooth over OK of relative dimension n. Hence it is regu-lar of dimension n+1. Therefore the immersion U→P is regular of codimension 1.

We give a local description of the sheaf Ω¹_X/S using an immersion as in Lem-ma 5.1.1.

Corollary 5.1.2. — Let X be a scheme over OK satisfying the condition (R(n)). Then 1. The canonical map LX/S →Ω¹X/S is an isomorphism.

2. Let U→P be an immersion as in Lemma 5.1.1. Then we have an exact sequence 0 −−−→ NU/P −−−→ Ω¹P/S⊗OP OU −−−→ Ω¹U/S −−−→ 0. (5.1.2.1)

TheO^U-module Ω¹P/S⊗OPO^U is locally free of rank n and the conormal sheaf NU/P is invertible.

3. The cotangent complex L_X/S satisfies the conditions (L(n)) and (G) in Section 2.4.

Proof. — 1. It follows immediately from Lemma 1.6.2.3 and from the assertion 2.

2. For the exact sequence (5.1.2.1), it is sufficient to show the injectivity of NU/P

→ Ω¹P/S ⊗OP OU. Since the generic fiber is smooth, it is injective there. Since X is normal, the map is injective. The rest of assertion is clear.

3. It follows from 2 and Lemma 2.1.1.

Lemma5.1.3. — LetXbe a scheme over OKsatisfying the condition (R(n)). Leti :Z→X be the closed immersion defined by the ideal Ann ΩⁿX/S and LZ be the invertible OZ-module L¹i^∗Ω¹X/S.

1. Let W be a normal scheme of finite type over s = Spec F and ϕ : W → Z be a morphism over S. Then, there exists a canonical isomorphism ϕ^∗L^Z = L¹(i ◦ϕ)^∗Ω¹X/S → Ns/S⊗O^W O^W of invertible O^W-modules.

2. The bivariant Chern class c1(LZ)∈CH¹(Z→Z) is 0.

3. For a scheme T of finite type over Z, the map ·LZ : G(T) → G(T) sending [F]

to [F ⊗OZ LZ] is the identity. The canonical map G(T) → G(T)/LZ = Coker(1− ·LZ : G(T)→ G(T)) is an isomorphism.

Proof. — 1. The OW-module ϕ^∗LZ = L¹(i ◦ ϕ)^∗Ω¹X/S is invertible by Corol-lary 5.1.2.2. Therefore, to define an isomorphism L¹(i◦ϕ)^∗Ω¹X/S→ Ns/S⊗OW of in-vertibleOW-modules, we may shrink Wto an open subset containing all the points of codimension 1. Shrinking W, we may assume W is smooth over s. The distinguished triangle (1.4.0.1) gives us distinguished triangles

→L(i◦ϕ)^∗Ω¹X/S −−−→ LW/S −−−→ LW/X −−−→

(5.1.3.1)

and → Ls/S ⊗OW → LW/S → LW/s →. Since Ls/S = Ns/S[1] and LW/s = Ω¹W/s, we have H0(LW/S)=Ω¹W/s andH1(LW/S)=Ns/S⊗FOW. Taking the cohomology sheaves H1 of the distinguished triangle (5.1.3.1), we obtain an exact sequence

0 −−−→ L¹(i◦ϕ)^∗Ω¹_X/S −−−→^a N_s/S⊗FOW

−−−→b H1(L_W/X).

We show that the mapa is an isomorphism. SinceWis locally of complete intersection over X, the OW-module H1(LW/X) is locally a subsheaf of a locally free OW-module and hence is torsion free. On the other hand, since a is injective, the cokernel of a is torsion. Hence the mapb is 0 and a is an isomorphism.

2. For a scheme T of finite type over Z, the Chow group CHi(T) is generated by π∗[W] where π : W → T runs through the normalization of integral closed sub-schemes of T of dimension i. By 1, we have c1(LZ)∩π∗[W] =π∗(c1(π^∗LZ)∩ [W])

=0 and the assertion follows.

3. For a scheme T of finite type over Z, the K-group G(T) is generated by π∗[O^W] where π : W → T runs through the normalization of integral closed sub-schemes of T. By 1, we have LZ ·π∗[OW] = π∗[LZ ⊗OZ OW] = π∗[OW] and the

assertion follows.

Proposition 5.1.4. — Let Xbe a scheme over S=Spec OK satisfying the condition (R(n)) and let Z⊂X be the closed subscheme defined by the ideal Ann ΩⁿX/S.

Then the spectral sequence

E¹_p,q =L^2p+qΛ^−pΩ¹X/S⇒ Tor_p^O₊^X×_q^{S X}(OX,OX)

(1.6.4.3) degenerates at E¹-terms. It defines an increasing filtration F_• on Torn^{OX×S X}(O^X,O^X) satisfying Fn = Torn^{OX×S X}(OX,OX) and F₋1 = 0 and isomorphisms L^pΛ^qΩ¹X/S → Gr^F_pTorn^{OX×S X}(O^X,O^X) for p+q=n. The O^X-modules L^pΛ^qΩ¹X/S areO^Z-modules for p > 0.

Proof. — We have an isomorphism MX/X×SX → Ω¹X/S by Corollaries 5.1.2 and 3.4.5. By applying Proposition 1.6.7 to the diagonal embeddingX→X×SX, we see that the spectral sequence (1.6.4.3) degenerates at E¹-terms. It defines a filtration F_• satisfying the condition up to decalage. TheOX-modulesL^pΛ^qΩ¹X/S areOZ-modules

forp > 0 by Lemma 2.4.2.1.

We define the non-logarithmic localized intersection product. LetXbe a scheme over OK satisfying the condition (R(n)) as above. Let i :Z→X be the closed immer-sion defined by the ideal Ann Ωⁿ_X/S and LZ be the invertible OZ-module L¹i^∗Ω¹_X/S as in Lemma 5.1.3. Then, by Lemmas 5.1.1 and 3.2.4, the projectionpr2:X×^SX→X is locally a hypersurface of virtual relative dimensionn−1over X and the closed sub-scheme of X×^SX defined by the ideal AnnΩⁿX×SX/X is the pull-back Z×^X(X×^SX) of Z ⊂ X by the first projection. Let W be a noetherian scheme over X×^S X and let V be a closed subscheme of X×SX. We put T =V×X×SXW and ZT =Z×XT be the pull-back by the composition T → X×SX→ X with the first projection. By Lemma 5.1.3.3, we have G(ZT)/LZ =G(ZT). Thus, the localized intersection product (3.2.2.1) defines a map[[ , ]]X×SX :G(V)×G(W)→G(ZT). Since the generic fiber is smooth, the subschemeZ is supported on the closed fiberXs and we have a natural map G(ZT)→G(Ts).

Definition 5.1.5. — Let Xbe a scheme over S=Spec OK satisfying the condition (R(n)) and Z → X be the closed subscheme defined by the ideal Ann ΩⁿX/S. For a closed subscheme V of X×S X and a noetherian scheme W over X×S X, we put T = V×X×SXW and call the composition

G(V)×G(W) −−−−−→^{[[ , ]]X×S X} G(ZT)/LZ =G(ZT) −−−→ G(Ts) (5.1.5.1)

the localized intersection product. We also define

[[ ,W]]X×SX:G(X×SX) −−−→ G(Ws) (5.1.5.2)

as the localized intersection product with the class [OW] ∈ G(W) by taking V = X×S X. If V=X→X×SX is the diagonal map, we call the localized intersection product

[[X, ]]^X×SX:G(W) −−−→ G(Ts) (5.1.5.3)

with the class [OX] ∈G(X) the localized intersection product with the diagonal.

By Theorem 3.4.3.1, the map [[X, ]]X×SX :G(W)→G(Ts)induces FpG(W) −−−→ Fp−nG(Ts),

Gr^F_pG(W) −−−→ Gr^F_p−nG(Ts).

(5.1.5.4)

By abuse of notation, we use the same notation [[X, ]]X×SX for them. For W = X×SX, we have

[[X, ]]X×SX:G(X×SX) −−−→ G(Xs).

(5.1.5.5)

For the self-intersection, we have an equality [[X,X]]X×SX=(−1)ⁿcn

X Z

Ω¹_X/S

∩ [X] =(∆X,∆X)S

(5.1.5.6)

in Gr^F₀G(Xs) by Corollaries 5.1.2.1 and 3.4.5.

The localized Chern class cnX

XF(Ω¹_X/S)∩ [X] ∈ CH0(XF) is computed explicitly as follows.

Lemma 5.1.6. — Let Xbe a scheme over OK satisfying the condition (R(n)) and let Z be the closed subscheme defined by the ideal Ann ΩⁿX/S as in Lemma 5.1.3. Let π :X →X be the blow-up at Z and D=Z×XX be the exceptional divisor.

Then the pull-back π^∗Ω¹X/S is an extension of a locally free OX-module E of rank n−1 by an invertible OD-module and we have

cnX Z

Ω¹_X/S

∩ [X] =π∗(c_n−1(E)∩ [D]).

Another computation of deg(∆X,∆X)S in terms of the torsion parts of Ω^qX/S is given in [39].

Example. — Let the notation be as in Lemma 5.1.6. Assume x∈Xis an isolated non-degenerate quadratic singularity of the mapX→ Sand assumeX−{x} is smooth over S. Then Z = {x} with reduced scheme structure, D Pⁿ_x⁻¹ is the exceptional divisor and E⊗OX OD is a quotient of ODⁿ by OD(−1). Hence c_n−1(E)∩ [D] is the class [x] of a κ(x)-rational point x ofD and cnX

Z(Ω¹X/S)∩ [X] = π∗[x] = [x].

Proof. — By Corollary 5.1.2.3, we may apply Corollary 2.4.5. The assertion

fol-lows by Lemma 5.1.3.2.

We prove a K-theoretic version of the projection formula conjectured in [1] Sec-tion 6 formula (20).

Lemma 5.1.7. — Let X and Y be schemes over OK satifying the condition (R(n)) and f :X→Y be a morphism over OK. Then, for a closed subscheme Γ of X×SXof dimension n, we have an equality

[[Γ, (f ×f)^∗∆Y]]X×SX= [[Y,Γ]]Y×SY

in F0G((X×^YX)^s).

Proof. — We apply Corollary 3.3.4.3 by taking Y ← Y ×S Y ← X×S X = X×S X→ X, [∆Y] ∈ G(Y×SY) and Γ ⊂ X×S X as S ← X← W → X → S, Γ∈G(X) andV⊂X. Then, since the map X×SX→ Y×SYis locally of complete intersection, it is of finite tor-dimension. Thus the assumption of Corollary 3.3.4.3 is satisfied and we obtain the equality in G((X×^YX)^s).

We show the right hand side is in F0G((X×^YX)^s). Since dimΓ = n, we have [OΓ] ∈FnG(X×^SX). Thus the assertion follows from Theorem 3.4.3.1.

5.2. Logarithmic self-products. — We keep the notation that K is a discrete