<Emphasis Type="Italic">p</Emphasis>-adic deformation of algebraic cycle classes

Spencer Bloch; Hélène Esnault; Moritz Kerz

doi:10.1007/s00222-013-0461-4

Inventiones mathematicae

March 2014, Volume 195, Issue 3, pp 673–722 | Cite as

p-adic deformation of algebraic cycle classes

Authors
Authors and affiliations

Spencer Bloch
Hélène EsnaultEmail author
Moritz Kerz

Article

First Online: 26 March 2013

Received: 14 March 2012

Accepted: 05 March 2013

650 Downloads
2 Citations

Abstract

We study the p-adic deformation properties of algebraic cycle classes modulo rational equivalence. We show that the crystalline Chern character of a vector bundle on the closed fibre lies in a certain part of the Hodge filtration if and only if, rationally, the class of the vector bundle lifts to a formal pro-class in K-theory on the p-adic scheme.

Keywords

Chern Class Chern Character Discrete Valuation Ring Cycle Class Hodge Conjecture

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

Acknowledgements

It is our pleasure to thank Lars Hesselholt for explaining to us topological cyclic homology and Marc Levine for many important comments. We also thank Markus Spitzweck and Chuck Weibel for helpful discussions. We are grateful to the mathematicians from the Feza Gürsey Institute in Istanbul for giving us the opportunity to present a preliminary version of our results in March 2011. After our work was completed, Alexander Beilinson proposed to us an alternative definition of our motivic complex (see Sect. 3). We thank him for his interest in our work and for his contribution to it. We also thank the various referees who sent comments to us.

Appendix A: Homological algebra

In this section we collect some standard facts from homological algebra that we use. Let $\mathcal{T}$ be a triangulated category with t-structure, see [2, Sect. 1.3].

Lemma A.1

For an integer r and for an exact triangle

$$A\to B \to C \xrightarrow{[1]} A[1] $$

in $\mathcal{T}$ with $A \in \mathcal{T}^{\le r}$ the triangle

$$A \to \tau_{\le r} B \to \tau_{\le r} C \xrightarrow{[1]} A[1] $$

is exact.

Lemma A.2

For $A,B \in \mathcal{T}$ with $A\in \mathcal{T}^{\le r}$ and $B\in \mathcal{T}^{\le r} \cap \mathcal{T}^{\ge r} $ assume given an epimorphism $\mathcal{H}^{r} (A) \to \mathcal{H}^{r}(B)$. Then this epimorphism lifts uniquely to a morphism A→B in $\mathcal{T}$, sitting inside an exact triangle

$$A \to B \to C \to A[1] $$

which is unique up to unique isomorphism.

Proof

The existence of such an exact triangle is clear from the axioms of triangulated categories. Note that $C\in \mathcal{T}^{<r} $. Uniqueness means that there exists a unique dotted isomorphism α in a commutative diagram with exact triangles as rows

Open image in new window

Existence and uniqueness follow from the exact sequence

$$0= \mathrm {Hom}(C,B) \to \mathrm {Hom}\bigl(C,C'\bigr) \to \mathrm {Hom}\bigl(C, A[1]\bigr) \to \mathrm {Hom}\bigl(C, B[1]\bigr) . $$

□

Now we discuss pro-sheaves on sites. Let ${\mathbb{N}}$ be the category with the objects {1,2,3,…} and morphisms n ₁→n ₂ for n ₁≥n ₂. By the category of pro-systems C_pro, for a category C, we mean the category of diagrams in C with index category ${\mathbb{N}}$ and with morphisms

$$\mathrm{Mor}_{\mathrm{C}_\mathrm {pro}} (Y_\centerdot , Z_\centerdot ) = \varprojlim_{n} \varinjlim_{m} \mathrm{Mor}_\mathrm{C}(Y_m, Z_n). $$

Definition A.3

Let ${\mathbb{S}}$ be a small site.

(a)

By $\mathrm {Sh}({\mathbb{S}})$ we denote the category of sheaves of abelian groups on ${\mathbb{S}}$. By $\mathrm {C}({\mathbb{S}})$ we denote the category of unbounded complexes in $\mathrm {Sh}({\mathbb{S}})$.
(b)

By $\mathrm {Sh}_{\mathrm {pro}}(\mathbb{S})$ we denote the category of pro-systems in $\mathrm {Sh}({\mathbb{S}})$.
(c)

By $\mathrm {C}_{\mathrm {pro}}( {\mathbb{S}})$ we denote the category of pro-systems in $\mathrm {C}({\mathbb{S}})$.
(d)

By $\mathrm {D}_{\mathrm {pro}}(\mathbb{S})$ we denote the Verdier localization of the homotopy category of $\mathrm {C}_{\mathrm {pro}}({\mathbb{S}})$, where we kill objects which are represented by systems of complexes which have level-wise vanishing cohomology sheaves.

For the construction of Verdier localization in (d) see [53, Sect. 2.1].

Lemma A.4

The triangulated category $\mathrm {D}_{\mathrm {pro}}(\mathbb{S})$ has a natural t-structure $(\mathrm {D}^{\le 0}({\mathbb{S}}), \mathrm {D}^{\ge 0}({\mathbb{S}}))$ with ${\mathcal{F}}_{\centerdot }\in \mathrm {D}_{\mathrm {pro}}^{\le 0}$ resp. ${\mathcal{F}}_{\centerdot }\in \mathrm {D}_{\mathrm {pro}}^{\ge 0}$ if ${\mathcal{F}}_{\centerdot }$ is isomorphic in $\mathrm {D}_{\mathrm {pro}}(\mathbb{S})$ to ${\mathcal{F}}'_{\centerdot }$ with $\mathcal{H}^{i}({\mathcal{F}}'_{n})=0$ for all $n\in {\mathbb{N}}$ and i>0 resp. for i<0. The t-structure has heart $\mathrm {Sh}_{\mathrm {pro}}(\mathbb{S})$.

We write $\mathrm {D}^{+}_{\mathrm {pro}}({\mathbb{S}})$, $\mathrm {D}^{-}_{\mathrm {pro}}({\mathbb{S}})$ and $\mathrm {D}^{b}_{\mathrm {pro}}({\mathbb{S}})$ for the bounded above, bounded below and bounded objects in $D({\mathbb{S}})$ with respect to the t-structure.

Appendix B: Homotopical algebra

In this section we introduce certain standard model categories of pro-systems over a small site $\mathbb{S}$. We uniquely specify our model structures by explaining what are the cofibrations and weak equivalences. The fibrations are then defined to be the maps which have the right lifting property with respect to all trivial cofibrations. Our definition of closed model category is as in [55].

Definition B.1

(a)

Let $\mathrm {S}({\mathbb{S}})$ be the proper closed simplicial model category of simplicial presheaves on ${\mathbb{S}}$, where cofibrations are injective morphisms of presheaves and weak equivalences are those maps which induce isomorphisms on homotopy sheaves, cf. [40, Sect. 2].
(b)

We endow the category of unbounded complexes of abelian sheaves $\mathrm {C}({\mathbb{S}})$ with the proper closed simplicial model structure where cofibrations are injective morphisms and weak equivalences are those maps which induce isomorphisms on cohomology sheaves, see App. C in [12] and Thm. 2.3.13 in [34].

Explicit characterizations of the classes of fibrations for the two model categories are given in the references. For the crucial notion of level representation in the following definitions see [37, Sect. 2.1].

Definition B.2

(a)

By $\mathrm {S}_{\mathrm {pro}}({\mathbb{S}})$ we denote the proper closed simplicial model category of pro-systems of simplicial presheaves on ${\mathbb{S}}$, where cofibrations are those maps which have a level representation by levelwise injective morphisms and where weak equivalences are those maps which have a level representation which induces a levelwise isomorphism on homotopy sheaves.
(b)

We endow $\mathrm {C}_{\mathrm {pro}}({\mathbb{S}})$ with the proper closed simplicial model structure, where cofibrations are those maps which have a level representation by levelwise injective morphisms and where weak equivalences are those maps which have a level representation which induces a levelwise isomorphism on cohomology sheaves.

Notation B.3

For a model category M we write hM for the associated homotopy category.

The pro-model structures in Definition B.2 are due to Isaksen [37]. He uses all pro-systems indexed by small cofiltering categories, whereas we allow only ${\mathbb{N}}$ as index category. In fact all his definitions and proofs work in a simpler way in this setting, except for the following points: In our model categories only countable inverse limits and finite direct limits exist, cf. [36, Sect. 11]. Also for our categories the simplicial functors K⊗− resp. (−)^K exist only for a finite resp. countable simplicial set K. This is why we use Quillen’s original notion of a closed simplicial model category [55]. Note that Isaksen calls his pro-category strict model category.

Isaksen gives the following concrete description of fibrations.

Proposition B.4

(Trivial) fibrations in $\mathrm {S}_{\mathrm {pro}}({\mathbb{S}})$ resp. $\mathrm {C}_{\mathrm {pro}}({\mathbb{S}})$ are precisely those maps, which are retracts of maps having a level representation f:X _·→Y _· such that

$$f_n:X_n \to X_{n-1} \times_{Y_{n-1}} Y_n $$

are (trivial) fibrations in $\mathrm {S}({\mathbb{S}})$ resp. $\mathrm {C}({\mathbb{S}})$ for n≥1. Here we let X ₀=Y ₀ be the final object.

Sketch of Isaksen’s construction (Definition B.2)

In a first step one shows the two out of three property for weak equivalences. The key lemma in this step is [37, Lem. 3.2], which is the only part of the construction where Isaksen constructs a new non-trivial index category. For index category ${\mathbb{N}}$ the argument simplifies. In a second step one shows the various left and right lifting properties of a model category. Here one takes the description of fibrations given in Proposition B.4 as a definition and thereby also obtains a proof of this proposition.

Proposition B.5

(a)

There are Quillen adjoint functors
Open image in new window
where the right adjoint K is the composition of the good truncation τ _≤0 and the Eilenberg-MacLane space construction.
(b)

There is a canonical isomorphism of categories
$$\mathrm {D}_\mathrm {pro}({\mathbb{S}}) \xrightarrow{\simeq} \mathrm{hC}_\mathrm {pro}({\mathbb{S}}). $$
(c)

There are Quillen adjoint functors
Open image in new window
where the left adjoint is the constant pro-system functor and the right adjoint is the inverse limit functor.

Notation B.6

We write
$$\mathrm{K}: \mathrm{hC}_\mathrm {pro}({\mathbb{S}}) \to \mathrm{hS}_\mathrm {pro}({\mathbb{S}}) $$
for the functor induced by $\mathrm{K}: \mathrm {C}_{\mathrm {pro}}({\mathbb{S}}) \to \mathrm {S}_{\mathrm {pro}}( {\mathbb{S}})$.
We write [Y ₁,Y ₂] for the set of morphisms from Y ₁ to Y ₂ in the homotopy category.
The right derived functor $\mathrm{holim}: \mathrm{hS}_{\mathrm {pro}}({\mathbb{S}}) \to \mathrm{hS}_{\mathrm {pro}}({\mathbb{S}}) $ of $\varprojlim: \mathrm {S}_{\mathrm {pro}}({\mathbb{S}})\to \mathrm {S}({\mathbb{S}})$ is called homotopy inverse limit. By $R \varprojlim : \mathrm {D}_{\mathrm {pro}}({\mathbb{S}}) \to \mathrm {D}({\mathbb{S}})$ we denote the right derived functor of $\varprojlim: \mathrm {C}_{\mathrm {pro}}({\mathbb{S}})\to \mathrm {C}({\mathbb{S}})$.

There is a standard method for calculating the derived inverse limit $R^{i} \varprojlim: \mathrm {Sh}_{\mathrm {pro}}({\mathbb{S}}) \to \mathrm {Sh}({\mathbb{S}})$ which shows in particular that $R^{i} \varprojlim =0$ for i>1, see [61, Sect. 3.5].

Definition B.7

We define continuous cohomology of ${\mathcal{F}}_{\centerdot }\in \mathrm {D}_{\mathrm {pro}}({\mathbb{S}})$ by

$$H^i_\mathrm {cont}({\mathbb{S}}, {\mathcal{F}}_\centerdot ) = \bigl[{\mathbb{Z}}[-i], {\mathcal{F}}_\centerdot \bigr] , $$

where ${\mathbb{Z}}$ denotes the constant sheaf of integers.

Continuous cohomology of sheaves was first studied in [39]. Note that we have a short exact sequence

$$ 0\to \varprojlim\limits_n{\!}^1 H^{i-1}(\mathbb{S} , {\mathcal{F}}_n ) \to H_\mathrm {cont}^i(\mathbb{S}, {\mathcal{F}}_\centerdot ) \to \varprojlim\limits_n H^i(\mathbb{S}, {\mathcal{F}}_n ) \to 0. $$

(B.1)

Lemma B.8

For ${\mathcal{F}}_{\centerdot }\in \mathrm {D}_{\mathrm {pro}}^{+}({\mathbb{S}})$ there is a convergent spectral sequence

$$E_2^{p,q} = H^p_\mathrm {cont}\bigl({\mathbb{S}}, {\mathcal{H}}^q({\mathcal{F}}_\centerdot )\bigr) \Longrightarrow H^{p+q}_\mathrm {cont}({\mathbb{S}}, {\mathcal{F}}_\centerdot ) $$

with differential $d_{r}:E_{r}^{p,q} \to E_{r}^{p+r,q - r+1}$.

Lemma B.9

Let C _· be a pointed object in $\mathrm {S}_{\mathrm {pro}}({\mathbb{S}})$ and assume that $\tilde{\pi}_{1}(\mathrm {C}_{n})$ is commutative for any n≥1. If there is N such that $H^{i}_{\mathrm {cont}}({\mathbb{S}}, \tilde{\pi}_{j}(C_{\centerdot })) =0$ for i>N and j>0, then there is a completely convergent Bousfield-Kan spectral sequence

$$E^{s,t}_2 = H^{s}_\mathrm {cont}\bigl({\mathbb{S}}, \tilde{\pi}_t (C_\centerdot ) \bigr) \Longrightarrow \bigl[S^{t-s}, C_\centerdot \bigr] \quad \mathit{with}\ t\ge s $$

and differential $d_{r}:E_{r}^{s,t} \to E_{r}^{s+r,t+r-1}$.

Here $\tilde{\pi}_{i}$ is the pro-system of sheaves of homotopy groups and $H^{0}_{\mathrm {cont}}$ of the sheaf of sets $\tilde{\pi}_{0}(C_{\centerdot })$ means simply global sections of the inverse limit. The indexing of the spectral sequence is as in [9, Sect. IX.4.2].

For $C_{\centerdot }= \mathrm{K} ({\mathcal{F}}_{\centerdot })$ with K as in Proposition B.5(a) and ${\mathcal{F}}_{\centerdot }$ as in Lemma B.8 there is a natural morphism

$$E^{s,t}_r({\mathcal{F}}_\centerdot )\to E^{s,t}_r \bigl( \mathrm{ K}({\mathcal{F}}_\centerdot ) \bigr) \quad (t\ge s, r\ge 2), $$

compatible with the differential d _r, where the left side is a Bousfield-Kan renumbering of the spectral sequence of Lemma B.9 and the right side is the spectral sequence of Lemma B.9. This morphism is injective for t=s and bijective for t>s.

Lemma B.9 implies in particular the following lemma.

Lemma B.10

Let $C_{\centerdot }, C'_{\centerdot }\in \mathrm {S}_{\mathrm {pro}}({\mathbb{S}}) $ satisfy the assumptions of Lemma B.9 and let $\varPsi: C_{\centerdot }\to C'_{\centerdot }$ be a morphism.

(a)

Assume that for an integer n≥1 the induced map
$$ H^{s}_\mathrm {cont}\bigl({\mathbb{S}}, \tilde{\pi}_t (C_\centerdot ) \bigr) \xrightarrow{\varPsi_*} H^{s}_\mathrm {cont}\bigl({\mathbb{S}}, \tilde{\pi}_t \bigl(C'_\centerdot \bigr) \bigr), $$

(B.2)
is injective for all t,s with t−s=n−1, bijective for t−s=n and surjective for t−s=n+1. Then $\varPsi_{*}:[S^{n}, C_{\centerdot }]\to [S^{n}, C'_{\centerdot }]$ is an isomorphism.
(b)

Assume that (B.2) is surjective for t−s=1 and injective for t=s. Then $\varPsi_{*}:[S^{0}, C_{\centerdot }]\to [S^{0}, C'_{\centerdot }]$ is injective.

Definition B.11

An object $C_{\centerdot }\in \mathrm {S}_{\mathrm {pro}}({\mathbb{S}})$ satisfies descent if for any object $U\in {\mathbb{S}}$

$$\varGamma (U,C_\centerdot ) \to \varGamma(U, \mathrm{F}C_\centerdot ) $$

is a an isomorphism in hS_pro({∗}). Here FC _· is a fibrant replacement in $\mathrm {S}_{\mathrm {pro}}({\mathbb{S}})$.

Appendix C: The motivic complex: a crystalline construction

In this appendix we continue to assume that r<p. We identify the motivic complex $\widetilde{{\mathbb{Z}}}_{X_{\centerdot }}(r)$ as constructed in Sect. 3 with the complex ${\mathbb{Z}}_{X_{\centerdot }}(r)$ given in Definition 7.1. The later is defined via a cone involving the Nisnevich syntomic complex $\frak{S}_{X_{\centerdot }}(r)$ (Definition 4.2). As a preliminary simplification, we may modify the cone (3.3) and define

Open image in new window

(C.1)

Here $\psi_{2,3}: \varOmega^{{\bullet }}_{X_{1},\log}[-r] \to q(r)W\varOmega^{{\bullet }}_{X_{1}}$ is the natural inclusion. We will exhibit a canonical quasi-isomorphism $\widetilde{\frak{S}}_{X_{\centerdot }}(r) \simeq \frak{S}_{X_{\centerdot }}(r)$. The desired result for motivic cohomology will follow by a further cone construction for the map $d\log: {\mathbb{Z}}_{X_{1}}(r) \to W\varOmega^{{\bullet }}_{X_{1},\log}[-r]$ (7.4).

Let us write $C^{{\bullet }}:= \widetilde{\frak{S}}_{X_{\centerdot }}(r)$.

Lemma C.1

${\mathcal{H}}^{j}(C^{{\bullet }}) = (0)$ for j≥r+1, i.e. $\tau_{\le r}C^{{\bullet }}\xrightarrow{\simeq} C^{{\bullet }}$.

Proof

It suffices to show the map

Open image in new window

(C.2)

is an isomorphism for j≥r+1 and is surjective for j=r. This follows from the assertion $I(r)\varOmega^{{\bullet }}_{D} \simeq q(r)W\varOmega^{{\bullet }}_{X_{1}}$ which is a consequence of formulas (2.4) and (2.5) in the paper. □

Lemma C.2

Let ε:X _ét→X _Nis be the map of sites. Then there is a canonical quasi-isomorphism ${\varepsilon }^{*}C^{{\bullet }}\simeq \frak{S}_{X_{\centerdot }}(r)_{\acute {\mathrm {e}}\mathrm {t}}$, the syntomic complex in the étale topology.

Proof

There is a natural inclusion of cones

$$\mathrm{Cone}\bigl(W\varOmega^r_{X_1,\log}[-r] \to q(r)W \varOmega^{\bullet }_{X_1}\bigr)[-1] \to C^{\bullet }. $$

In the étale site, the cone on the left is quasi-isomorphic to $W\varOmega^{{\bullet }}_{X_{1}}[-1]$ (Corollary 4.6). As a consequence, in the étale site we get

$$ C^{\bullet }[-1] \simeq \mathrm{Cone}\bigl( \varOmega_{X_\centerdot }^{\ge r} \oplus p(r)\varOmega^{\bullet }_D \to p(r)\varOmega^{\bullet }_{X_\centerdot } \oplus W\varOmega^{\bullet }_{X_1} \bigr)[-1]. $$

(C.3)

(The map $p(r)\varOmega^{{\bullet }}_{D} \to W\varOmega^{{\bullet }}_{X_{1}}$ is (1−F _r)∘μ, where $\mu: \varOmega^{{\bullet }}_{D} \to W\varOmega^{{\bullet }}_{X_{1}}$ is the composition of (2.4) and (2.7) in the paper.) Let $\xi: J(r)\varOmega^{{\bullet }}_{D_{\centerdot }} \to \varOmega_{X_{\centerdot }}^{\ge r}$ be as in (2.8) in the paper, and let $\iota: J(r)\varOmega^{{\bullet }}_{D_{\centerdot }} \subset \varOmega^{{\bullet }}_{D_{\centerdot }}$ be the natural inclusion. Construct a commutative diagram

Open image in new window

(C.4)

This diagram yields the desired quasi-isomorphism in the étale site. □

We have by Lemma C.2, $C^{{\bullet }}\to R{\varepsilon }_{*}{\varepsilon }^{*}C^{{\bullet }}\simeq R{\varepsilon }_{*}\frak{S}_{X_{\centerdot }}(r)_{\acute {\mathrm {e}}\mathrm {t}}$. Applying τ _≤r and using Lemma C.1 we get

$$ C^{\bullet }\simeq \tau_{\le r} C^{\bullet }\to \tau_{\le r}R{\varepsilon }_*\frak{S}_{X_\centerdot }(r)_{\acute {\mathrm {e}}\mathrm {t}}=: \frak{S}_{X_\centerdot }(r)_\mathrm{Nis}. $$

(C.5)

We must show the map (C.5) is a quasi-isomorphism. Consider the commutative diagram

Open image in new window

(C.6)

Here the bottom line is as in (C.1) and the top as in (C.4). The sheaves on the top are ε-acyclic, so the top complex represents $R{\varepsilon }_{*}\frak{S}_{X_{\centerdot }}(r)_{\acute {\mathrm {e}}\mathrm {t}}$ and the whole diagram represents $C^{{\bullet }}\to R{\varepsilon }_{*}{\varepsilon }^{*}C^{{\bullet }}\simeq R{\varepsilon }_{*}\frak{S}_{X_{\centerdot }}(r)_{\acute {\mathrm {e}}\mathrm {t}}$. It will suffice to check that this vertical map of Nisnevich complexes induces an isomorphism in cohomology in degrees ≤r.

In the Nisnevich topology, consider the double complex of complexes which we position so $W\varOmega^{r}_{X_{1},\log}[-r]$ is in position (0,0).

Open image in new window

(C.7)

Lemma C.3

The total complex of Nisnevich sheaves associated to (C.7) is acyclic away from degree r+2.

Proof

Writing T for the total complex, we have a triangle

$$T \to \bigl(q(r)W\varOmega^{\bullet }/W\varOmega^r_{\log}[-r] \bigr)[-1] \xrightarrow{1-F_r} W\varOmega^{\bullet }[-1] \xrightarrow{+1} . $$

The map 1−F _r induces isomorphisms in cohomology (Lemmas 4.4 and 4.5) except

Open image in new window

is not surjective so ${\mathcal{H}}^{i}(T) = (0), i\neq r+2$. □

Let U be the corresponding total complex for the diagram (C.6). The inclusion T↪U is a quasi-isomorphism so ${\mathcal{H}}^{i}(U) = (0), i\neq r+2$. It follows that ${\mathcal{H}}^{i}(C^{{\bullet }}) \to {\mathcal{H}}^{i}(R{\varepsilon }_{*}\frak{S}_{X_{\centerdot }}(r)_{\acute {\mathrm {e}}\mathrm {t}})$ is an isomorphism except possibly for i=r+1,r+2. This implies ${\mathcal{H}}^{i}(\tau_{\le r}C^{{\bullet }}) \to {\mathcal{H}}^{i}(\tau_{\le r}R{\varepsilon }_{*}\frak{S}_{X_{\centerdot }}(r)_{\acute {\mathrm {e}}\mathrm {t}})$ is a quasi-isomorphism for all i. From Lemma C.1 we conclude

$$\widetilde{\frak{S}}_{X_\centerdot }(r)= C^{\bullet }\to \tau_{\le r}R {\varepsilon }_*\frak{S}_{X_\centerdot }(r)_{\acute {\mathrm {e}}\mathrm {t}}=\frak{S}_{X_\centerdot }(r)_{\mathrm {Nis}} $$

is a quasi-isomorphism as desired.

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Authors and Affiliations

Spencer Bloch
- 1
Hélène Esnault
- 2
Email author
Moritz Kerz
- 3

1.ChicagoUSA
2.FB Mathematik und InformatikFU BerlinBerlinGermany
3.Fakultät für MathematikUniversität RegensburgRegensburgGermany

p-adic deformation of algebraic cycle classes

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