Weil-étale cohomology for arbitrary arithmetic schemes and n < 0. Part II: The special value conjecture
aa r X i v : . [ m a t h . AG ] F e b Weil-´etale cohomology for arbitrary arithmetic schemes and n <
Alexey BeshenovFebruary 25, 2021
Abstract
Following the ideas of Flach and Morin [FM2018], we state a conjecture in terms of Weil-´etale coho-mology for the vanishing order and special value of the zeta function ζ ( X, s ) at s = n <
0, where X is aseparated scheme of finite type over Spec Z . We prove that the conjecture is compatible with closed-opendecompositions of schemes and affine bundles, and as a consequence, that it holds for cellular schemesover certain 1-dimensional bases.This is a continuation of author’s preprint [Bes2020], which gives a construction of Weil-´etale coho-mology for n < X . Contents
Let X be an arithmetic scheme , by which we will mean throughout this paper that it is a separatedscheme of finite type over Spec Z . Then the corresponding zeta function is defined by ζ ( X, s ) = Y x ∈ X closed pt. − κ ( x ) − s , where κ ( x ) = O X,x / m X,x denotes the residue field of a point. The above product converges for Re s > dim X ,and conjecturally admits a meromorphic continuation to the whole complex plane. For basic facts andconjectures about zeta functions of schemes, see Serre’s survey [Ser1965].1f particular interest are the so-called special values of ζ ( X, s ) at integers s = n ∈ Z . To define these,assume that ζ ( X, s ) admits a meromorphic continuation around s = n . We will denote by d n = ord s = n ζ ( X, s )the vanishing order of ζ ( X, s ) at s = n . That is, d n > d n <
0) if ζ ( X, s ) has a zero (resp. pole) oforder d n at s = n . The corresponding special value of ζ ( X, s ) at s = n is defined to be the leading nonzerocoefficient of the Taylor expansion: ζ ∗ ( X, n ) = lim s → n ( s − n ) − d n ζ ( X, s ) . Early on Stephen Lichtenbaum conjectured that both numbers ord s = n ζ ( X, s ) and ζ ∗ ( X, n ) should havea cohomological interpretation, related to ´etale motivic cohomology of X (see e.g. [Lic1984] for varieties overfinite fields). This is made more precise in Lichtenbaum’s Weil-´etale program . In particular it suggests theexistence of a cohomology theory H i W,c ( X, Z ( n )), Weil-´etale cohomology with compact support, whichencodes the information about the vanishing order and special value of ζ ( X, n ) at s = n .For Lichtenbaum’s recent work on the topic, we refer to his papers [Lic2005, Lic2009b, Lic2009a, Lic2021].The case of varieties over finite fields X/ F q was further studied by Thomas Geisser [Gei2004b, Gei2006,Gei2010a], and it is rather well understood now.Matthias Flach and Baptiste Morin considered the case of proper, regular arithmetic schemes X . In[FM2012] they defined and studied the corresponding Weil-´etale topos. Later, Morin gave in [Mor2014] anexplicit construction of Weil-´etale cohomology groups H i W,c ( X, Z ( n )) for n = 0, and X a proper, regulararithmetic scheme, under assumptions on finite generation of suitable ´etale motivic cohomology groups. Thisconstruction was further generalized by Flach and Morin in [FM2018] to any n ∈ Z , again for proper andregular X .Motivated by the work of Flach and Morin, the author constructed in [Bes2020] the Weil-´etale cohomologygroups H i W,c ( X, Z ( n )) for any arithmetic scheme X (thus removing the assumptions that X is proper orregular) and n < §
8] for further details and known cases).
Conjecture. L c ( X ´et , n ): the cohomology groups H i ( X ´et , Z c ( n )) are finitely generated for all i ∈ Z .Namely, assuming L c ( X ´et , n ), we defined in [Bes2020] perfect complexes of abelian groups R Γ W,c ( X, Z ( n )).This text is a continuation of [Bes2020] and it explores the conjectural relation of our Weil-´etale cohomologyto the special value of ζ ( X, s ) at s = n <
0. Specifically, we make the following conjectures.1)
Conjecture VO ( X, n ): the vanishing order is given by ord s = n ζ ( X, s ) = X i ∈ Z ( − i · i · rk Z H i W,c ( X, Z ( n )) .
2) A consequence of
Conjecture B ( X, n ) (see § after tensoring with R , one obtains a long exactsequence of finite dimensional real vector spaces · · · → H i − W,c ( X, R ( n )) ⌣θ −−→ H i W,c ( X, R ( n )) ⌣θ −−→ H i +1 W,c ( X, R ( n )) → · · · Here H i W,c ( X, R ( n )) = H i W,c ( X, Z ( n )) ⊗ Z R = H i ( R Γ W,c ( X, Z ( n )) ⊗ Z R ).It follows that there is a canonical isomorphism λ : R ∼ = −→ (det Z R Γ W,c ( X, Z ( n ))) ⊗ R . Here det Z R Γ W,c ( X, Z ( n )) is the determinant of the perfect complex of abelian groups R Γ W,c ( X, Z ( n )),in the sense of Knudsen and Mumford [KM1976]. In particular, det Z R Γ W,c ( X, Z ( n )) is a free Z -moduleof rank 1. For the reader’s convenience, we include a brief overview of determinants in appendix A.3) Conjecture C ( X, n ): the special value is determined up to sign by λ ( ζ ∗ ( X, n ) − ) · Z = det Z R Γ W,c ( X, Z ( n )) . X is proper and regular, then our construction of R Γ W,c ( X, Z ( n )) and the above conjectures agreewith those of Flach and Morin from [FM2018]. Apart from removing the assumption that X is proper andregular, one novelty of this work is that we prove the compatibility of the above conjectures with operationson schemes, in particular closed-open decompositions Z ֒ → X ← ֓ U , where Z ⊂ X is a closed subschemeand U = X \ Z is the open complement, and affine bundles A rX = A r Z × X . (See proposition 6.3 andtheorem 6.8.) This gives a machinery that allows one to start from the particular instances of schemes forwhich the conjectures are known, and construct new schemes for which the conjectures hold as well. As anapplication, we prove in § Main theorem.
Let B be a -dimensional arithmetic scheme, such that each of the generic points η ∈ B satisfies one of the following properties:a) char κ ( η ) = p > ;b) char κ ( η ) = 0 , and κ ( η ) / Q is an abelian number field.If X is a B -cellular arithmetic scheme with smooth quasi-projective fiber X red , C , then the conjectures VO ( X, n ) and C ( X, n ) hold unconditionally for any n < . In fact, this result will be established for a bigger class of arithmetic schemes C ( Z ); we refer to § Outline of the paper In § B ( X, n ) related to it.Then § VO ( X, n ). We also explain why it is consistentwith a conjecture of Soul´e and the vanishing orders that come from the expected functional equationIn § C ( X, n ).We explain in § X/ F q a variety over a finite field, the special value conjecture C ( X, n ) isconsistent with the conjectures considered by Geisser in [Gei2004b, Gei2006, Gei2010a]. In particular, itholds for any smooth projective X/ F q , assuming that the groups H i ( X ´et , Z ( n )) are finite for all i . This alsogeneralizes to all varieties X/ F q if we assume the resolution of singularities over F q .Then we prove in § VO ( X, n ) and C ( X, n ) are compatible with basic operations onschemes: disjoint unions, closed-open decompositions, and affine bundles. Using these results, we concludein § Notation
In this paper, X will always denote an arithmetic scheme (separated, of finite type over Spec Z ), and n will always be a strictly negative integer.We denote by R Γ fg ( X, Z ( n )) and R Γ W,c ( X, Z ( n )) the complexes constructed in [Bes2020], assuming theconjecture L c ( X ´et , n ) stated above. For Q -coefficients, we put R Γ fg ( X, Q ( n )) := R Γ fg ( X, Z ( n )) ⊗ L Z Q = R Γ fg ( X, Z ( n )) ⊗ Z Q ,R Γ W,c ( X, Q ( n )) := R Γ W,c ( X, Z ( n )) ⊗ L Z Q = R Γ W,c ( X, Z ( n )) ⊗ Z Q . Accordingly, H i fg ( X, Q ( n )) := H i ( R Γ fg ( X, Q ( n ))) = H i fg ( X, Z ( n )) ⊗ Z Q ,H i W,c ( X, Q ( n )) := H i ( R Γ W,c ( X, Q ( n ))) = H i W,c ( X, Z ( n )) ⊗ Z Q . R -coefficients.By X ( C ) we denote the space of complex points of X with the usual analytic topology. It carries anatural action of G R = Gal( C / R ) via complex conjugation. For a subring A ⊆ R we denote by A ( n ) the G R -module (2 πi ) n A , and also the corresponding constant G R -equivariant sheaf on X ( C ).We denote by R Γ c ( X ( C ) , A ( n )) (resp. R Γ c ( G R , X ( C ) , A ( n ))) the cohomology with compact support(resp. G R -equivariant cohomology with compact support) of X with A ( n )-coefficients. For more details on G R -equivariant cohomology, see [Bes2020]. With real coefficients, H ic ( G R , X ( C ) , R ( n )) = H ic ( X ( C ) , R ( n )) G R ,where the G R -action on H ic ( X ( C ) , R ( n )) naturally comes from the corresponding action on X ( C ) and R ( n ).The Borel–Moore homology is defined as dual to the cohomology with compact support. In particular,we will be interested in R Γ BM ( X ( C ) , R ( n )) := R Hom( R Γ c ( X ( C ) , R ( n )) , R ) ,R Γ BM ( G R , X ( C ) , R ( n )) := R Hom( R Γ c ( G R , X ( C ) , R ( n )) , R ) . Acknowledgments
Parts of this work are based on the results of my PhD thesis, prepared in Universit´e de Bordeaux andUniversiteit Leiden under supervision of Baptiste Morin and Bas Edixhoven. I am deeply grateful for theirsupport during working on this project. I am also indebted to Matthias Flach, since the ideas of this workcome from [FM2018]. I thank Stephen Lichtenbaum and Niranjan Ramachandran who kindly accepted tobe the referees for my thesis and provided many useful comments and suggestions. Finally, I thank PedroLuis del ´Angel, Jos´e Jaime Hern´andez Castillo, Diosel L´opez Cruz, and Maxim Mornev for various fruitfulconversations.This paper was edited while I was visiting Center for Research in Mathematics (CIMAT), Guanajuato.I am grateful personally to Pedro Luis del ´Angel and Xavier G´omez Mont for their hospitality.
To state the special value conjecture, we need to define the regulator morphism from motivic cohomology toDeligne-Be˘ılinson (co)homology. It was introduced by Bloch in [Blo1986b], and here we are going to use theconstruction of Kerr, Lewis, and M¨uller-Stach from [KLMS2006], which works on the level of complexes. Wewill simply call it “the KLM morphism”. It works under assumption that X red , C is a smooth quasi-projectivevariety.For simplicity we will assume in this section that X is reduced (the motivic cohomology does not dis-tinguish between X and X red ), and that X C is connected of dimension d C (otherwise, the arguments belowmay be applied to each connected component). We fix a compactification by a normal crossing divisor X C X C D j The KLM regulator takes form of a morphism in the derived category z p ( X C , −• ) ⊗ Q → ′ C p − d C + •D ( X C , D, Q ( p − d C )) . (2.1)Here z p ( X C , −• ) denotes the Bloch’s cycle complex [Blo1986a]. We recall that by the definition, z p ( X C , i )is freely generated by algebraic cycles Z ⊂ X C × Spec C ∆ i C of codimension p that intersect properly the facesof the algebraic simplex ∆ i C = Spec C [ t , . . . , t i ] / (1 − P j t j ). For us it will be more convenient to pass to z d C − p ( X C , i ) = z p ( X C , i ) , generated by cycles Z ⊂ X C × Spec C ∆ i C of dimension p + i .The complex ′ C •D ( X C , D, Q ( k )) on the right hand side of (2.1) computes Deligne(–Be˘ılinson) homology,as defined by Jannsen [Jan1988]. In particular, taking p = d C + 1 − n , tensoring with R , and shifting by 2 n ,we obtain z n − ( X C , −• ) ⊗ R [2 n ] → ′ C •D ( X C , D, R (1 − n )) . (2.2)4 .1. Remark. Some comments are in order.1. Originally, the KLM morphism is defined using a cubical version of cycle complexes, but these are quasi-isomorphic to the usual simplicial cycle complexes (see [Lev1994]), so we do not make a distinctionhere. For a simplicial version, see also [KLL2018].2. The KLM morphism is defined as a genuine morphism of complexes (not just a morphism in the derivedcategory) on a certain subcomplex z r R ( X C , • ) ⊂ z r ( X C , • ). This inclusion is a quasi-isomorphism, if wepass rational coefficients. This is stated without tensoring with Q in the original paper [KLMS2006],but the omission is acknowledged later in [KL2007]. For our particular purposes, it will be enough tohave a regulator with coefficients in Q , or in fact in R .3. The case of smooth quasi-projective X C , where one has to consider a compactification by a normalcrossing divisor as above, is treated in [KLMS2006, § n <
0, the Deligne homology corresponds to Borel–Moore homology.
For any n < there is a quasi-isomorphism ′ C •D ( X C , D, R (1 − n )) ∼ = R Γ BM ( X ( C ) , R ( n ))[ −
1] := R Hom( R Γ c ( X ( C ) , R ( n )) , R )[ − . Further, this respects the natural actions of G R on both complexes.Proof. From the proof of [Jan1988, Theorem 1.15], for any k ∈ Z we have a quasi-isomorphism ′ C •D ( X C , D, R ( k )) ∼ = R Γ( X ( C ) , R ( k + d C ) D - Б , ( X C ,X C ) )[2 d C ] , (2.3)where R ( k + d C ) D - Б , ( X C ,X C ) = Cone (cid:16) Rj ∗ R ( k + d C ) ⊕ Ω > k + d C X ( C ) (log D ) ǫ − ι −−→ Rj ∗ Ω • X ( C ) (cid:17) [ − X ( C ) gives Deligne-Be˘ılinson cohomology (see [EV1988] for furtherdetails).Here Ω • X ( C ) denotes the usual de Rham complex of holomorphic differential forms, and Ω • X ( C ) (log D ) is thecomplex of forms with at most logarithmic poles along D ( C ). The latter complex is filtered by subcomplexesΩ > • X ( C ) (log D ). The morphism ǫ : Rj ∗ R ( k ) → Rj ∗ Ω • X ( C ) is induced by the canonical morphism of sheaves R ( k ) → O X ( C ) , and ι is induced by the natural inclusion Ω • X ( C ) (log D ) ∼ = −→ j ∗ Ω • X ( C ) = Rj ∗ Ω • X ( C ) , which is aquasi-isomorphism of filtered complexes.We will be interested in the case of k >
0, when the part Ω > k + d C X ( C ) (log D ) disappears, and we obtain R ( k + d C ) D - Б , ( X C ,X C ) ∼ = Rj ∗ Cone (cid:16) R ( k + d C ) ǫ −→ Ω • X ( C ) (cid:17) [ − ∼ = Rj ∗ (cid:16) R ( k + d C ) ǫ −→ Ω • X ( C ) [ − (cid:17) ∼ = Rj ∗ (cid:16) R ( k + d C ) → C [ − (cid:17) (2.4) ∼ = Rj ∗ R ( k + d C − −
1] (2.5)Here (2.4) comes from the Poincar´e lemma C ∼ = Ω • X ( C ) , and (2.5) comes from the short exact sequence of G R -modules R ( k + d C ) C ։ R ( k + d C − k = 1 − n , we conclude that ′ C •D ( X C , D, R (1 − n )) ∼ = R Γ( X ( C ) , R ( d C − n ))[2 d C − ∼ = R Hom( R Γ c ( X ( C ) , R ( n )) , R )[ − . Here the last isomorphism is Poincar´e duality. 5ow coming back to (2.2), the above lemma allows us to reinterpret the KLM morphism as z n − ( X C , −• ) ⊗ R [2 n ] → R Γ BM ( X ( C ) , R ( n )) , R )[1] . (2.6)By the definition, we have z n − ( X C , −• ) ⊗ R [2 n ] = z n − ( X C , −• ) ⊗ R [2 n − X C , ´et , R c ( n − , (2.7)where the complex of sheaves R c ( p ) is defined by U z p ( U, −• ) ⊗ R [2 p ]. By ´etale cohomological descent[Gei2010b, Theorem 3.1], we have * Γ( X C , ´et , R c ( n − ∼ = R Γ( X C , ´et , R c ( n − . (2.8)Finally, the base change from X to X C naturally maps cycles Z ⊂ X × ∆ i Z of dimension n to cycles in X C × Spec C ∆ i C of dimension n −
1, so that there is a morphism R Γ( X ´et , R c ( n )) → R Γ( X C , ´et , R c ( n − . (2.9) Assuming that X is flat of pure Krull dimension d , we have R c ( n ) X = R ( d − n ) X [2 d ], where R ( • ) is the usual cycle complex defined by z n ( , −• )[ − n ]. Similarly, R c ( n ) X C = R ( d C − n ) X C [2 d C ], with d C = d −
1. With this renumbering, the morphism (2.9) becomes R Γ( X ´et , R ( d − n ))[2 d ] → R Γ( X C , ´et , R ( d − n ))[2 d ] . This probably looks more natural, but we do not impose extra assumptions on X and work exclusivelywith complexes A c ( • ) defined in terms of dimension of algebraic cycles, instead of A ( • ) defined in terms ofcodimension. Given an arithmetic scheme X with smooth quasi-projective X C , and n <
0, consider thecomposition of morphisms R Γ( X ´et , R c ( n )) (2.9) −−−→ R Γ( X C , ´et , R c ( n − (2.8) ∼ = Γ( X C , ´et , R c ( n − (2.7) = z n − ( X C , −• ) R [2 n ] (2.6) −−−→ R Γ BM ( X ( C ) , R ( n )) , R )[1] . Further, we take G R -invariants, which gives us the (´etale) regulator Reg
X,n : R Γ( X ´et , R c ( n )) → R Γ BM ( G R , X ( C ) , R ( n ))[1] . Now we state our conjecture about the regulator, which will play an important role in everything thatfollows. ( X, n ): given an arithmetic scheme X with smooth quasi-projective X C and n <
0, theregulator morphism
Reg
X,n induces a quasi-isomorphism of complexes of real vector spaces
Reg ∨ X,n : R Γ c ( G R , X ( C ) , R ( n ))[ − → R Hom( R Γ( X ´et , Z c ( n )) , R ) . If X/ F q is a variety over a finite field, then X ( C ) = ∅ , so the regulator map in not interesting. Indeed, itspurpose is to take care of the archimedian places of X . It will be more convenient to formulate the followingconjecture. c fin ( X/ F q , n ): given a variety X/ F q and n <
0, the cohomology groups H i ( X ´et , Z c ( n ))are finite for all i ∈ Z . * We note that [Gei2010b, Theorem 3.1] holds unconditionally, since the Be˘ılinson–Lichtenbaum conjecture follows fromthe Bloch–Kato conjecture, which is now a theorem. See also [Gei2004a] where the consequences of Bloch–Kato for motiviccohomology are deduced.
6s explained in [Bes2020, § For a variety over a finite field X/ F q one has L c ( X, n ) and B ( X, n ) = ⇒ L c fin ( X/ F q , n ) . Proof.
Since X ( C ) = ∅ , we note that B ( X, n ) implies that H i ( X ´et , Z c ( n )) are torsion for all i , hence finite,if we further assume L c ( X ´et , n ). We reiterate that our construction of
Reg
X,n works for X red , C smooth quasi-projective. Ineverything that follows, whenever the regulator morphism or the conjecture B ( X, n ) is brought, we will tacitlyassume this. This is quite unfortunate, because the Weil-´etale complexes R Γ W,c ( X, Z ( n )) were constructedin [Bes2020] for any arithmetic scheme, assuming only L c ( X ´et , n ). Defining the regulator for singular X red , C is an interesting project for further work. Assuming that ζ ( X, s ) admits a meromorphic continuation around s = n <
0, we state the followingconjecture for the vanishing order at s = n . ( X, n ): one hasord s = n ζ ( X, s ) = χ ′ ( R Γ W,c ( X, Z ( n ))) := X i ∈ Z ( − i · i · rk Z H i W,c ( X, Z ( n )) . We note that the right hand side makes sense assuming the conjecture L c ( X ´et , n ), under which H i W,c ( X, Z ( n ))are finitely generated groups, trivial for | i | ≫ The alternating sum in the formula is the so-called secondary Euler characteristic of theWeil-´etale complex R Γ W,c ( X, Z ( n )). The easy calculations below show that the usual Euler characteristicof R Γ W,c ( X, Z ( n )) vanishes, assuming conjectures L c ( X ´et , n ) and B ( X, n ). See [Ram2016] for more detailsabout secondary Euler characteristic and its appearances in nature.Under the regulator conjecture, our conjectural vanishing order formula takes form of the usual Eulercharacteristic of the equivariant cohomology R Γ c ( G R , X ( C ) , R ( n )) or motivic cohomology R Γ( X ´et , Z c ( n ))[1]. Assuming L c ( X ´et , n ) and B ( X, n ) , the conjecture VO ( X, n ) is equivalent to ord s = n ζ ( X, s ) = χ ( R Γ c ( G R , X ( C ) , R ( n )) = X i ∈ Z ( − i dim R H ic ( X ( C ) , R ( n )) G R = − χ ( R Γ( X ´et , Z c ( n ))) = X i ∈ Z ( − i +1 rk Z H i ( X ´et , Z c ( n )) . Further, we have χ ( R Γ W,c ( X, Z ( n ))) = 0 . Proof.
Thanks to [Bes2020, Proposition 7.8], the Weil-´etale complex tensored with R splits as R Γ W,c ( X, R ( n )) ∼ = R Hom( R Γ( X ´et , Z c ( n )) , R )[ − ⊕ R Γ c ( G R , X ( C ) , R ( n ))[ − . Assuming the conjecture B ( X, n ), we also have a quasi-isomorphism R Γ c ( G R , X ( C ) , R ( n ))[ − ∼ = R Hom( R Γ( X ´et , Z c ( n )) , R ) ,
7o that dim R H i W,c ( X, R ( n )) = dim R H i − c ( X ( C ) , R ( n )) G R + dim R H i − c ( X ( C ) , R ( n )) G R . Using this, we may rewrite the sum X i ∈ Z ( − i · i · rk Z H i W,c ( X, Z ( n )) = X i ∈ Z ( − i · i · dim R H i W,c ( X, R ( n ))= X i ∈ Z ( − i · i · dim R H i − c ( X ( C ) , R ( n )) G R + X i ∈ Z ( − i · i · dim R H i − c ( X ( C ) , R ( n )) G R = − X i ∈ Z ( − i dim R H i − c ( X ( C ) , R ( n )) G R = χ ( R Γ c ( G R , X ( C ) , R ( n )) . Similarly, X i ∈ Z ( − i · i · rk Z H i W,c ( X, Z ( n )) = χ ( R Hom( R Γ( X ´et , Z c ( n )) , R )[1]) = − χ ( R Γ( X ´et , Z c ( n ))) . These considerations also show that the usual Euler characteristic of R Γ W,c ( X, Z ( n )) vanishes. The conjecture VO ( X, n ) is related to a conjecture of Soul´e [Sou1984, Conjecture 2.2], whichoriginally reads in terms of K ′ -theoryord s = n ζ ( X, s ) = X i ∈ Z ( − i +1 dim Q K ′ i ( X ) ( n ) . Then, as explained in [Kah2005, Remark 43], this may be rewritten in terms of Borel–Moore motivic homol-ogy as X i ∈ Z ( − i +1 dim Q H BMi ( X, Q ( n )) . In our setting, R Γ( X ´et , Z c ( n )) plays the role of Borel–Moore homology, which explains the formulaord s = n ζ ( X, s ) = X i ∈ Z ( − i +1 rk Z H i ( X ´et , Z c ( n )) . As for the formulaord s = n ζ ( X, s ) = X i ∈ Z ( − i dim R H ic ( X ( C ) , R ( n )) G R , it basically means that the the vanishing order at s = n < §§ § X C is a smooth projective variety, we consider the Hodge decomposition H i ( X ( C ) , C ) = M p + q = i H p,q , which carries an action of G R = { id, σ } such that σ ( H p,q ) = H q,p . We set h p,q = dim C H p,q . For p = i/ H p,p = H p, + ⊕ H p, − , where H p, + = { x ∈ H p,p | σ ( x ) = ( − p x } ,H p, − = { x ∈ H p,p | σ ( x ) = ( − p +1 x } , and set accordingly h p, ± = dim C H p, ± . It is expected that the completed zeta function ζ ( X, s ) = ζ ( X, s ) ζ ( X ∞ , s ) , A d − s ζ ( X, d − s ) = A s ζ ( X, s ) . Here ζ ( X ∞ , s ) = Y i ∈ Z L ∞ ( H i ( X ) , s ) ( − i ,L ∞ ( H i ( X ) , s ) = Y p = i/ Γ R ( s − p ) h p, + Γ R ( s − p + 1) h p, − Y p + q = ip 0. We havedim R H i ( X ( C ) , R ( n )) G R = dim R H i ( X ( C ) , R ) σ =( − n = dim C H i ( X ( C ) , C ) σ =( − n = X p = i/ h p, ( − n − p + X p + q = ip Suppose that X = Spec O F is the spectrum of the ring of integers of a number field F/ Q . Let r be the number of real embeddings F ֒ → R and r the number of conjugate pairs of complexembeddings F ֒ → C . The space X ( C ) with the action of complex conjugation may be pictured as follows: • • · · · •• • · · · • • • · · · • r points 2 r pointsThe complex R Γ c ( X ( C ) , R ( n )) consists of a single G R -module in degree 0 given by R ( n ) ⊕ r ⊕ ( R ( n ) ⊕ R ( n )) r , with the action of G R on the first summand R ( n ) ⊕ r by complex conjugation and the action on the secondsummand ( R ( n ) ⊕ R ( n )) r via ( x, y ) ( y, x ). The corresponding real space of fixed points has dimensiondim R H c ( G R , X ( C ) , R ( n )) = ( r , n odd ,r + r , n even , ζ ( X, s ) = ζ F ( s ) at s = n < n < 0, the groups H i ´et ( X, Z c ( n )) = H i +2 ´et ( X, Z (1 − n )) are finite, except for i = − 1, whererk Z H − ´et ( X, Z c ( n )) = rk Z H ´et ( X, Z (1 − n )) = ( r , n odd ,r + r , n even . Suppose that X is a variety over a finite field F q . Then the vanishing order conjecture isnot very interesting, as the formula givesord s = n ζ ( X, s ) = X i ∈ Z ( − i dim R H ic ( X ( C ) , R ( n )) G R = X i ∈ Z ( − i +1 rk Z H i ( X ´et , Z c ( n )) = 0 , since X ( C ) = ∅ , and also because B ( X, n ) implies rk Z H i ( X ´et , Z c ( n )) = 0. Therefore, the conjecture simplyasserts that ζ ( X, s ) does not have zeros or poles at s = n < ζ ( X, s ) = Z ( X, q − s ), where Z ( X, t ) = exp (cid:16)X k ≥ X ( F q k ) k t k (cid:17) is the Hasse–Weil zeta function. By Deligne’s work on Weil’s conjectures [Del1980], the zeros and poles of Z ( X, s ) satisfy | s | = q − w/ , where 0 ≤ w ≤ X (see e.g. [Kat1994, pp. 26–27]). In particular, q − s for s = n < Z ( X, s ).We also note that our definition of H i W,c ( X, Z ( n )), and pretty much all the above, makes sense only for n < 0. Already for n = 0, for instance, the zeta function of a smooth projective curve X/ F q has a simplepole at s = 0. Let X = E be an integral model of an elliptic curve over Q . Then as a consequence of themodularity theorem (Wiles–Breuil–Conrad–Diamond–Taylor) it is known that ζ ( E, s ) admits a meromorphiccontinuation, which satisfies the functional equation with the Γ-factors discussed in 3.5. In this particularcase ord s = n ζ ( E, s ) = 0 for all n < 0. This is consistent with the fact that χ ( R Γ c ( G R , E ( C ) , R ( n ))) = 0.Indeed, the equivariant cohomology groups H ic ( E ( C ) , R ( n )) G R are the following: i = 0 i = 1 i = 2 n even: R R n odd: 0 R R —see for instance the calculation in [Sie2019, Lemma A.6]. Define a morphism of complexes ⌣ θ : R Γ W,c ( X, Z ( n )) ⊗ R → R Γ W,c ( X, Z ( n ))[1] ⊗ R using the splitting R Γ W,c ( X, R ( n )) ∼ = R Hom( R Γ( X ´et , Z c ( n )) , R )[ − ⊕ R Γ c ( G R , X ( C ) , R ( n ))[ − R Γ W,c ( X, R ( n )) R Γ W,c ( X, R ( n ))[1] R Hom( R Γ( X ´et , Z c ( n )) , R )[ − R Hom( R Γ( X ´et , Z c ( n )) , R ) ⊕ ⊕ R Γ c ( G R , X ( C ) , R ( n ))[ − R Γ c ( G R , X ( C ) , R ( n )) ∼ = ⌣θ ∼ = Reg ∨ X,n .2. Lemma. Assuming L c ( X ´et , n ) and B ( X, n ) , the morphism ⌣ θ induces a long exact sequence of finitedimensional real vector spaces · · · → H i − W,c ( X, R ( n )) ⌣θ −−→ H i W,c ( X, R ( n )) ⌣θ −−→ H i +1 W,c ( X, R ( n )) → · · · Proof. We obtain a sequence · · · H i − W,c ( X, R ( n )) H i W,c ( X, R ( n )) H i +1 W,c ( X, R ( n )) · · · Hom( H − i ( X ´et , Z c ( n )) , R ) Hom( H − i − ( X ´et , Z c ( n )) , R ) Hom( H − i − ( X ´et , Z c ( n )) , R ) · · · ⊕ ⊕ ⊕ · · · H i − c ( G R , X ( C ) , R ( n )) H i − c ( G R , X ( C ) , R ( n )) H ic ( G R , X ( C ) , R ( n )) ∼ = ⌣θ ∼ = ⌣θ ∼ = ∼ = ∼ = Here the diagonal arrows are isomorphisms according to B ( X, n ), so the sequence is exact.We recall that the Weil-´etale complex R Γ W,c ( X, Z ( n )) was defined in [Bes2020] up to a non-unique isomorphism in the derived category D ( Z ) via a distinguished triangle R Γ W,c ( X, Z ( n )) → R Γ fg ( X, Z ( n )) i ∞ −−→ R Γ c ( G R , X ( C ) , Z ( n )) → R Γ W,c ( X, Z ( n ))[1] (4.1)This is quite unpleasant, and there ought to be a better, more canonical construction of R Γ W,c ( X, Z ( n )).However, this is not a big issue for the moment, since the special value conjecture will be formulated not interms of R Γ W,c ( X, Z ( n )), but in terms of its determinant det Z R Γ W,c ( X, Z ( n )) (see appendix A), which iswell-defined. The determinant det Z R Γ W,c ( X, Z ( n )) is well-defined up to a canonical isomorphism.Proof. Two different choices of the mapping fiber in (4.1) give us an isomorphism of distinguished triangles R Γ W,c ( X, Z ( n )) R Γ fg ( X, Z ( n )) R Γ c ( G R , X ( C ) , Z ( n )) R Γ W,c ( X, Z ( n ))[1] R Γ W,c ( X, Z ( n )) ′ R Γ fg ( X, Z ( n )) R Γ c ( G R , X ( C ) , Z ( n )) R Γ W,c ( X, Z ( n )) ′ [1] f ∼ = i ∞ id id f ∼ = i ∞ Now the idea is to use the functoriality of determinants with respect to isomorphisms of distinguishedtriangles (see A.4). The only technical issue is that whenever X ( R ) = ∅ , the complexes R Γ fg ( X, Z ( n )) and R Γ c ( G R , X ( C ) , Z ( n )) are not perfect, but possibly have finite 2-torsion in H i ( − ) for arbitrarily big i (in[Bes2020] we called such complexes almost perfect ). On the other hand, the determinants are only definedfor perfect complexes. Luckily, H i ( i ∗∞ ) is an isomorphism for i ≫ 0, so that for m big enough we may takethe corresponding canonical truncations τ ≤ m : τ ≤ m R Γ W,c ( X, Z ( n )) τ ≤ m R Γ fg ( X, Z ( n )) τ ≤ m R Γ c ( G R , X ( C ) , Z ( n )) τ ≤ m R Γ W,c ( X, Z ( n ))[1] R Γ W,c ( X, Z ( n )) R Γ fg ( X, Z ( n )) R Γ c ( G R , X ( C ) , Z ( n )) R Γ W,c ( X, Z ( n ))[1]0 τ ≥ m +1 R Γ fg ( X, Z ( n )) τ ≥ m +1 R Γ c ( G R , X ( C ) , Z ( n )) 0 τ ≤ m R Γ W,c ( X, Z ( n ))[1] τ ≤ m R Γ fg ( X, Z ( n ))[1] τ ≤ m R Γ c ( G R , X ( C ) , Z ( n ))[1] τ ≤ m R Γ W,c ( X, Z ( n ))[2] ∼ = ∼ = i ∞ ∼ = τ ≤ m R Γ c ( G R , X ( C ) , Z ( n ))[ − R Γ W,c ( X, Z ( n )) τ ≤ m R Γ fg ( X, Z ( n )) τ ≤ m R Γ c ( G R , X ( C ) , Z ( n )) τ ≤ m R Γ c ( G R , X ( C ) , Z ( n ))[ − R Γ W,c ( X, Z ( n )) ′ τ ≤ m R Γ fg ( X, Z ( n )) τ ≤ m R Γ c ( G R , X ( C ) , Z ( n )) id f ∼ = id id Now according to A.4, we have a commutative diagramdet Z τ ≤ m R Γ c ( G R , X ( C ) , Z ( n ))[ − ⊗ Z det Z τ ≤ m R Γ fg ( X, Z ( n )) det Z R Γ W,c ( X, Z ( n ))det Z τ ≤ m R Γ c ( G R , X ( C ) , Z ( n ))[ − ⊗ Z det Z τ ≤ m R Γ fg ( X, Z ( n )) det Z R Γ W,c ( X, Z ( n )) i ∼ = id det Z ( f ) ∼ = i ′ ∼ = so that det Z ( f ) = i ′ ◦ i − . The non-canonical splitting R Γ W,c ( X, Q ( n )) ∼ = R Hom( R Γ( X ´et , Z c ( n )) , Q )[ − ⊕ R Γ c ( G R , X ( C ) , Q ( n ))[ − . Gives a canonical isomorphism of determinants det Q R Γ W,c ( X, Q ( n )) ∼ = det Q R Hom( R Γ( X ´et , Z c ( n )) , Q )[ − ⊗ Q det Q R Γ c ( G R , X ( C ) , Q ( n ))[ − Proof. This is similar to the previous lemma; in fact, after tensoring with Q , we obtain perfect complexes of Q -vector spaces, so that the truncations are not needed anymore.We recall from [Bes2020, Proposition 7.4] that i ∗∞ ⊗ Q = 0, and thanks to this there is an isomorphismof triangles R Γ c ( G R , X ( C ) , Q ( n ))[ − R Γ c ( G R , X ( C ) , Q ( n ))[ − R Γ W,c ( X, Q ( n )) R Hom( R Γ( X ´et , Z c ( n )) , Q )[ − ⊕ R Γ c ( G R , X ( C ) , Q ( n ))[ − R Γ fg ( X, Q ( n )) R Hom( R Γ( X ´et , Z c ( n )) , Q )[ − R Γ c ( G R , X ( C ) , Q ( n )) R Γ c ( G R , X ( C ) , Q ( n )) idf ∼ = g ⊗ Q ∼ = id (4.2)Here the third horizontal arrow comes from the triangle defining R Γ fg ( X, Z ( n )) R Hom( R Γ( X ´et , Z c ( n )) , Q )[ − α X,n −−−→ R Γ c ( X ´et , Z ( n )) → R Γ fg ( X, Z ( n )) g −→ R Hom( R Γ( X ´et , Z c ( n )) , Q )[ − Q (see [Bes2020, § R Γ c ( G R , X ( C ) , Q ( n ))[ − 1] 0 R Γ c ( G R , X ( C ) , Q ( n ))[ − ⊕ R Hom( R Γ( X ´et , Z c ( n )) , Q )[ − R Hom( R Γ( X ´et , Z c ( n )) , Q )[ − R Γ c ( G R , X ( C ) , Q ( n )) 0 id id f in (4.2) is by no means canonical. However, after taking the determinants, weobtain a commutative diagram (see A.4)det Q R Γ c ( G R , X ( C ) , Q ( n ))[ − ⊗ Q det Q R Γ fg ( X, Q ( n )) det Q R Γ W,c ( X, Q ( n ))det Q R Γ c ( G R , X ( C ) , Q ( n ))[ − ⊗ Q det Q R Hom( R Γ( X ´et , Z c ( n )) , Q )[ − 1] det Q R Hom( R Γ( X ´et , Z c ( n )) , Q )[ − ⊕ R Γ c ( G R , X ( C ) , Q ( n ))[ − i ∼ = id ⊗ det Z ( g ⊗ Q ) ∼ = det Z ( f ) ∼ = ∼ = i ′ ∼ = Here the dashed diagonal arrow is the desired canonical isomorphism. Given an arithmetic scheme X and n < 0, assume the conjectures L c ( X ´et , n ) and B ( X, n ).Consider the quasi-isomorphism R Γ c ( G R , X ( C ) , R ( n ))[ − ⊕ R Γ c ( G R , X ( C ) , R ( n ))[ − R Hom( R Γ( X ´et , Z c ( n )) , R )[ − ⊕ R Γ c ( G R , X ( C ) , R ( n ))[ − R Γ W,c ( X, R ( n )) Reg ∨ X,n [ − ⊕ id ∼ = split ∼ = (4.3)Note that the first complex has determinantdet R R Γ c ( G R , X ( C ) , R ( n ))[ − ⊕ R Γ c ( G R , X ( C ) , R ( n ))[ − ∼ = det R R Γ c ( G R , X ( C ) , R ( n )) ⊗ R (det R R Γ c ( G R , X ( C ) , R ( n ))) − ∼ = R , and for the last complex, by compatibility with base change, we have a canonical isomorphismdet R R Γ W,c ( X, R ( n )) ∼ = (det Z R Γ W,c ( X, Z ( n ))) ⊗ Z R . Therefore, after taking determinants, the quasi-isomorphism (4.3) induces a canonical isomorphism λ = λ X,n : R ∼ = −→ (det Z R Γ W,c ( X, Z ( n ))) ⊗ Z R . (4.4) An equivalent way to define λ is λ : R ∼ = −→ O i ∈ Z (det R H i W,c ( X, R ( n ))) ( − i ∼ = −→ (cid:16)O i ∈ Z (det Z H i W,c ( X, Z ( n ))) ( − i (cid:17) ⊗ Z R ∼ = −→ (det Z R Γ W,c ( X, Z ( n ))) ⊗ Z R . Here the first isomorphism comes from lemma 4.2.We are ready to state the main conjecture of this paper. The determinant det Z R Γ W,c ( X, Z ( n ))) is a free Z -module of rank 1 (which does not sound very interesting), but the whole point of the determinant businessis that the isomorphism (4.4) embeds it canonically into R . We conjecture that this embedding gives thespecial value of ζ ( X, s ) at s = n in the following sense. ( X, n ): let X be an arithmetic scheme and n < L c ( X ´et , n ), B ( X, n ), and meromorphic continuation of ζ ( X, s ) around s = n < 0, the corresponding specialvalue is determined up to sign by λ ( ζ ∗ ( X, n ) − ) · Z = det Z R Γ W,c ( X, Z ( n )) , where λ is the canonical isomorphism (4.4). 13 .8. Remark. This conjecture is similar to [FM2018, Conjecture 5.11]. When X is proper and regular,the above conjecture is the same as the special value conjecture of Flach and Morin. They prove thattheir conjecture is consistent with the Tamagawa number conjecture of Bloch–Kato–Fontaine–Perrin-Riou[FPR1994]. Some canonical isomorphisms of determinants involve multiplication by ± 1, so there is nosurprise that the resulting conjecture is stated up to sign ± 1. However, this is not a big issue, since the signmay be recovered from the (conjectural) functional equation. For varieties over finite fields, our special value conjecture corresponds to the conjectures studied by Geisserin [Gei2004b], [Gei2006], [Gei2010a]. If X/ F q is a variety over a finite field, then assuming L c fin ( X/ F q , n ) , the special valueconjecture C ( X, n ) is equivalent to ζ ∗ ( X, n ) = ± Y i ∈ Z | H i W,c ( X, Z ( n )) | ( − i = ± Y i ∈ Z | H i ( X ´et , Z c ( n )) | ( − i = ± Y i ∈ Z | H ci ( X ar , Z ( n )) | ( − i +1 , (5.1) where H ci ( X ar , Z ( n )) are Geisser’s arithmetic homology groups defined in [Gei2010a].Proof. Assuming L c fin ( X/ F q , n ), thanks to [Bes2020, Proposition 7.9] we have H i W,c ( X, Z ( n )) ∼ = Hom( H − i ( X ´et , Z c ( n )) , Q / Z ) ∼ = Hom( H ci − ( X ar , Z ( n )) , Q / Z ) . The involved cohomology groups are finite, and therefore by A.6, the determinant is given bydet Z R Γ W,c ( X, Z ( n )) ⊂ det Z R Γ W,c ( X, Z ( n )) ⊗ Z Q m Z ⊂ Q where m = Y i ∈ Z | H i W,c ( X, Z ( n )) | ( − i . Formulas similar to (5.1) were suggested long time ago by Lichtenbaum in [Lic1984].To deal with singular varieties, we recall the following strong conjecture on resolution of singularities. ( k, d ). For a field k and d ∈ N , for varieties X/k of dimension ≤ d the followingconditions hold. • For any integral variety X/k of dimension ≤ d there is a proper, birational map f : Y → X with Y smooth. • For every smooth variety Y /k of dimension ≤ d and every proper birational map f : Y → X , there isa sequence of blowups along smooth centers X n → X n − → · · · → X → X such that the composition X n → X factors through f . Let X be a smooth projective variety X/ F q . Assuming L c fin ( X/ F q , n ) , the special valueconjecture C ( X, n ) holds.Moreover, assuming the resolution of singularities R ( F q , d ) and L c fin ( X/ F q , n ) for any smooth projectivevariety X/ F q of dimension ≤ d , the conjecture C ( X, n ) holds for any variety X/ F q of dimension ≤ d . roof. Proposition 5.1 gives ζ ∗ ( X, n ) = ± Y i ∈ Z | H ci ( X ar , Z ( n )) | ( − i +1 . Under L c ( X ´et , n ), the groups H ci ( X ar , Z ( n )) ∼ = H − i ( X ´et , Z c ( n )), are finitely generated, hence the con-jecture P ( X ) from [Gei2010a, § 4] holds, using [Gei2010a, Proposition 4.1]. Then the statement is precisely[Gei2010a, Theorem 4.5]. It is probably worth noting that Geisser’s proof of the special value conjecture is via reductionto Milne’s work [Mil1986].To generalize to all varieties of dimension ≤ d under assumption of R ( F q , d ), one uses the d´evissage lemma[Gei2006, Lemma 2.7] and compatibility of the special value conjecture with closed-open decompositions ofschemes. In the next section this will be verified in full generality (for any arithmetic scheme X , notnecessarily X/ F q ) for the conjecture C ( X, n ).Let us consider a couple of particular examples to see how the special value conjecture works. It is tobe noted that for a general arithmetic scheme X , calculating the motivic cohomology H i ( X ´et , Z c ( n )) (andtherefore our Weil-´etale cohomology H i W,c ( X, Z ( n ))) is by no means a trivial task. The finite generation of H i ( X ´et , Z c ( n )) is only known for particular cases (see [Bes2020, § Reg X,n is highly nontrivial. Therefore, for the moment we give a couple of toy examples over finitefields. If X = Spec F q , then ζ ( X, s ) = − q − s . In this case for n < H i (Spec F q, ´et , Z c ( n )) = H i (Spec F q, ´et , Z ( − n )) = ( Z / ( q − n − , i = 1 , , i = 1 (5.2)(see for instance [Gei2017, Example 4.2]). Therefore, the formula (5.1) indeed recovers ζ ( X, n ) up to sign.Similarly, replacing Spec F q with Spec F q m , viewed as a variety over F q , we have ζ (Spec F q m , s ) = ζ (Spec F q , ms ), and (5.2) also changes accordingly. Consider X = P F q / (0 ∼ ζ ( X, s ) = − q − s . We may calculate the groups H i ( X ´et , Z c ( n )) using the blowup squareSpec F q ⊔ Spec F q P F q Spec F q X y This is similar to [Gei2006, § 8, Example 2]. Geisser uses eh-topology and long exact sequences associated toabstract blowup squares [Gei2006, Proposition 3.2]. In our case, the same works, since according to [Bes2020,Theorem I], one has H i ( X ´et , Z c ( n )) ∼ = Hom( H − ic ( X ´et , Z ( n )) , Q / Z ), where Z ( n ) = lim −→ p ∤ m µ ⊗ nm [ − H i ( P F q , ´et , Z c ( n )) = Z / ( q − n − , i = − , Z / ( q − n − , i = +1 , , i = ± . Following the same argument from [Gei2006, § 8, Example 2], the short exact sequences0 → H i ( P F q , ´et , Z c ( n )) → H i ( X ´et , Z c ( n )) → H i +1 ((Spec F q ) ´et , Z c ( n )) → H i ( X ´et , Z c ( n )) = Z / ( q − n − , i = − , Z / ( q − n − , i = 0 , , , otherwise . The formula (5.1) recovers the correct value ζ ( X, n ). From the definition of ζ ( X, s ), the following basic properties follow easily.1) Disjoint unions : if X = ` ≤ i ≤ r X i is a finite disjoint union of arithmetic schemes, then ζ ( X, s ) = Y ≤ i ≤ r ζ ( X i , s ) . (6.1)In particular, ord s = n ζ ( X, s ) = X ≤ i ≤ r ord s = n ζ ( X i , s ) ,ζ ∗ ( X, n ) = Y ≤ i ≤ r ζ ∗ ( X i , n ) . Closed-open decompositions : if Z ⊂ X is a closed subscheme and U = X \ Z is its open complement,then we will say that we have a closed-open decomposition and write Z ֒ → X ← ֓ U . In this case ζ ( X, s ) = ζ ( Z, s ) · ζ ( U, s ) . (6.2)In particular, ord s = n ζ ( X, s ) = ord s = n ζ ( Z, s ) + ord s = n ζ ( U, s ) ,ζ ∗ ( X, n ) = ζ ∗ ( Z, n ) · ζ ∗ ( U, n ) . Affine bundles : for any r ≥ A rX = A r Z × X satisfies ζ ( A rX , s ) = ζ ( X, s − r ) . (6.3)In particular, ord s = n ζ ( A rX , s ) = ord s = n − r ζ ( X, s ) ,ζ ∗ ( A rX , n ) = ζ ∗ ( X, n − r ) . This suggests that the conjectures VO ( X, n ) and C ( X, n ) should also satisfy the corresponding compat-ibilities. We verify in this section that this is indeed the case. Let n < .1) If X = ` ≤ i ≤ r X i is a finite disjoint union of arithmetic schemes, then L c ( X ´et , n ) ⇐⇒ L c ( X i, ´et , n ) for all i. ) For a closed-open decomposition Z ֒ → X ← ֓ U , if two out of three conjectures L c ( X ´et , n ) , L c ( Z ´et , n ) , L c ( U ´et , n ) hold, then the third holds as well.3) For an arithmetic scheme X and any r ≥ , one has L c ( A rX, ´et , n ) ⇐⇒ L c ( X ´et , n − r ) . Proof. We already verified this in [Bes2020, Lemma 8.9]. 1) If X = ` ≤ i ≤ r X i is a finite disjoint union of arithmetic schemes, then Reg X,n = M ≤ i ≤ r Reg X i ,n : M ≤ i ≤ r R Γ( X i, ´et , R c ( n )) → M i ≤ i ≤ r R Γ BM ( G R , X i ( C ) , R ( n ))[1] . In particular, B ( X, n ) ⇐⇒ B ( X i , n ) for all i. 2) For a closed-open decomposition of arithmetic schemes Z ֒ → X ← ֓ U , the corresponding regulators givea morphism of distinguished triangles R Γ( Z ´et , R c ( n )) R Γ( X ´et , R c ( n )) R Γ( U ´et , R c ( n )) · · · [1] R Γ BM ( G R , Z ( C ) , R ( n ))[1] R Γ BM ( G R , X ( C ) , R ( n ))[1] R Γ BM ( G R , U ( C ) , R ( n ))[1] · · · [2] Reg Z,n Reg X,n Reg U,n Reg Z,n [1] In particular, if two out of three conjectures B ( X, n ) , B ( Z, n ) , B ( U, n ) hold, then the third holds as well.3) For any r ≥ , the diagram R Γ( X ´et , R c ( n − r ))[2 r ] R Γ( A rX, ´et , R c ( n )) R Γ BM ( G R , X ( C ) , R ( n − r ))[2 r ] R Γ BM ( G R , A rX ( C ) , R ( n )) Reg X,n − r ∼ = Reg A rX,n ∼ = commutes. In particular, one has B ( A rX , n ) ⇐⇒ B ( X, n − r ) . Proof. Part 1) is clear, since all cohomologies involved in the definition of Reg X,n decompose into directsums over i = 1 , . . . r . Parts 2) and 3) boil down to the corresponding functoriality properties for the KLMmorphism (2.1), namely that it commutes with proper pushforwards and flat pullbacks. For this we refer to[Wei2017, Lemma 3, Lemma 4], and it may be also verified directly from the KLM formula. For closed-opendecompositions, the distinguished triangle R Γ( Z ´et , R c ( n )) → R Γ( X ´et , R c ( n )) → R Γ( U ´et , R c ( n )) → R Γ( Z ´et , R c ( n ))[1]comes precisely from proper pushforward along Z ֒ → X and flat pullback along U ֒ → X (see [Gei2010b, Corol-lary 7.2] and [Blo1986a, § R Γ( X ´et , R c ( n − r ))[2 r ] ∼ = R Γ( A rX, ´et , R c ( n ))comes from the flat pullback along p : A rX → X . 17 .3. Proposition. For each arithmetic scheme X below and n < , assume L c ( X ´et , n ) , B ( X, n ) , and themeromorphic continuation of ζ ( X, s ) around s = n .1) If X = ` ≤ i ≤ r X i is a finite disjoint union of arithmetic schemes, then VO ( X, n ) ⇐⇒ VO ( X i , n ) for all i. 2) For a closed-open decomposition Z ֒ → X ← ֓ U , if two out of three conjectures VO ( X, n ) , VO ( Z, n ) , VO ( U, n ) hold, then the third holds as well.3) For any r ≥ , one has VO ( A rX , n ) ⇐⇒ VO ( X, n − r ) . Proof. We already observed in proposition 3.3 above that under the conjecture B ( X, n ) we can rewrite VO ( X, n ) as ord s = n ζ ( X, s ) = χ ( R Γ c ( G R , X ( C ) , R ( n ))) . In part 1), we have ord s = n ζ ( X, s ) = X ≤ i ≤ r ord s = n ζ ( X i , s ) , and for the corresponding G R -equivariant cohomology, R Γ c ( G R , X ( C ) , R ( n )) = M ≤ i ≤ r R Γ c ( G R , X ( C ) , R ( n )) . The statement follows from the additivity of Euler characteristic:ord s = n ζ ( X, s ) χ ( R Γ c ( G R , X ( C ) , R ( n ))) P ≤ i ≤ r ord s = n ζ ( X i , s ) P ≤ i ≤ r χ ( R Γ c ( G R , X i ( C ) , R ( n ))) VO ( X,n ) ∀ i VO ( X i ,n ) Similarly in part 2), we may consider the distinguished triangle R Γ c ( G R , U ( C ) , R ( n )) → R Γ c ( G R , X ( C ) , R ( n )) → R Γ c ( G R , Z ( C ) , R ( n )) → R Γ c ( G R , U ( C ) , R ( n ))[1]and the additivity of Euler characteristic givesord s = n ζ ( X, s ) χ ( R Γ c ( G R , X ( C ) , R ( n )))ord s = n ζ ( Z, s ) χ ( R Γ c ( G R , Z ( C ) , R ( n )))+ +ord s = n ζ ( U, s ) χ ( R Γ c ( G R , U ( C ) , R ( n ))) VO ( X,n ) VO ( Z,n ) VO ( Z,n ) Finally, in part 3), assume for simplicity that X C is connected of dimension d C . Then the Poincar´e dualityand homotopy invariance of the usual cohomology without compact support give us R Γ c ( G R , A r ( C ) × X ( C ) , R ( n )) P.D. ∼ = R Hom( R Γ( G R , A r ( C ) × X ( C ) , R ( d C + r − n )) , R )[ − d C − r ] H.I. ∼ = R Hom( R Γ( G R , X ( C ) , R ( d C + r − n )) , R )[ − d C − r ] P.D. ∼ = R Γ c ( G R , X ( C ) , R ( n − r ))[ − r ] . − r ] is even, hence it does not affect the Euler characteristic, so that we obtainord s = n ζ ( A rX , s ) χ ( R Γ c ( G R , A r ( C ) × X ( C ) , R ( n )))ord s = n − r ζ ( X, s ) χ ( R Γ c ( G R , X ( C ) , R ( n − r ))) VO ( A rX ,n ) VO ( X,n − r ) Recall that the formula that appears in the original statement of VO ( X, n ) readsord s = n ζ ( X, s ) = χ ′ ( R Γ W,c ( X, Z ( n ))) := X i ∈ Z ( − i · i · rk Z H i W,c ( X, Z ( n )) . (6.4)The conjecture B ( X, n ) in the above argument is needed to rewrite this in terms of the usual Euler charac-teristic. We used χ ( R Γ c ( G R , X ( C ) , R ( n ))), but we could do the same with χ ( R Hom( R Γ( X ´et , Z c ( n )) , R )[1]).The least interesting part 1) of the previous proposition could be proved directly from (6.4), since H i W,c ( X, Z ( n )) = L j H i W,c ( X j , Z ( n )). Parts 2) and 3) would be problematic to prove directly from (6.4)without assuming B ( X, n ), since the secondary Euler characteristic χ ′ ( − ) does not behave as the usual Eulercharacteristic χ ( − ). In particular, it is not additive for distinguished triangles.Our next goal is to prove similar compatibilities for the special value conjecture C ( X, n ), the same wayit was done in proposition 6.3 for VO ( X, n ). We will split the proof into three technical lemmas 6.5, 6.6,6.7, each for the corresponding compatibility. We briefly recall the construction of our Weil-´etale complex.It fits in the following diagram in the derived category D ( Z ) with distinguished triangles: R Γ W,c ( X, Z ( n )) R Hom( R Γ( X ´et , Z c ( n )) , Q [ − R Γ c ( X ´et , Z ( n )) R Γ fg ( X, Z ( n )) · · · R Γ c ( G R , X ( C ) , Z ( n )) R Γ c ( G R , X ( C ) , Z ( n )) 0 R Γ W,c ( X, Z ( n ))[1] α X,n u ∗∞ i ∗∞ id For further details, the reader may consult [Bes2020]. Let n < and let X = ` ≤ i ≤ r X i be a finite disjoint union of arithmetic schemes. Assume L c ( X ´et , n ) and B ( X, n ) . Then there is a quasi-isomorphism of complexes M ≤ i ≤ r R Γ W,c ( X i , Z ( n )) ∼ = R Γ W,c ( X, Z ( n )) , which after passing to the determinants gives a commutative diagram R ⊗ R · · · ⊗ R R R N ≤ i ≤ r (det Z R Γ W,c ( X i , Z ( n ))) ⊗ Z R (det Z R Γ W,c ( X i , Z ( n ))) ⊗ Z R λ X ,n ⊗···⊗ λ Xr,n ∼ = x ⊗···⊗ x r x ··· x r ∼ = λ X,n ∼ = ∼ = (6.5)19 roof. From the construction of R Γ W,c ( X, Z ( n )) it is clear that for X = ` ≤ i ≤ r X i all involved cohomologiesdecompose into the corresponding direct sum over i = 1 , . . . , r , and at the end after tensoring with R oneobtains a commutative diagram L i R Γ c ( G R , X i ( C ) , R ( n ))[ − ⊕ R Γ c ( G R , X i ( C ) , R ( n ))[ − R Γ c ( G R , X ( C ) , R ( n ))[ − ⊕ R Γ c ( G R , X ( C ) , R ( n ))[ − L i R Hom( R Γ( X i, ´et , Z c ( n )) , R )[ − ⊕ R Γ c ( G R , X i ( C ) , R ( n ))[ − R Hom( R Γ( X ´et , Z c ( n )) , R )[ − ⊕ R Γ c ( G R , X ( C ) , R ( n ))[ − L i R Γ W,c ( X i , R ( n )) R Γ W,c ( X, R ( n )) L i Reg ∨ Xi,n [ − ⊕ id ∼ = ∼ = Reg ∨ X,n [ − ⊕ id ∼ =split ∼ = ∼ = split ∼ = ∼ = Taking the determinants, we obtain (6.5). Let n < and let Z ֒ → X ← ֓ U be a closed-open decomposition of arithmetic schemes, suchthat the conjectures L c ( U ´et , n ) , L c ( X ´et , n ) , L c ( Z ´et , n ) , B ( U, n ) , B ( X, n ) , B ( Z ´et , n ) hold (in each case, it is enough to assume two out of three thanks to lemmas 6.1 and 6.2). Then there is anisomorphism of determinants det Z R Γ W,c ( U, Z ( n )) ⊗ Z det Z R Γ W,c ( Z, Z ( n )) ∼ = det Z R Γ W,c ( X, Z ( n )) (6.6) making the following diagram commute up to signs: R ⊗ R R R (det Z R Γ W,c ( U, Z ( n ))) ⊗ Z R ⊗ R (det Z R Γ W,c ( Z, Z ( n ))) ⊗ Z R (det Z R Γ W,c ( X, Z ( n ))) ⊗ Z R x ⊗ y xyλ U,n ⊗ λ Z,n ∼ = λ X,n ∼ = ∼ = (6.7) Proof. Morally, we expect in this situation a distinguished triangle of the form R Γ W,c ( U, Z ( n )) → R Γ W,c ( X, Z ( n )) → R Γ W,c ( Z, Z ( n )) → R Γ W,c ( U, Z ( n ))[1] . (6.8)However, even the complex R Γ W,c ( X, Z ( n )) was constructed in [Bes2020] up to a non-canonical isomorphismin the derived category D ( Z ), so this is problematic. In the absence of a better definition, we will constructthe isomorphism (6.6) in an ad hoc manner.A closed-open decomposition Z ֒ → X ← ֓ U gives us distinguished triangles R Γ( Z ´et , Z c ( n )) R Γ( X ´et , Z c ( n )) R Γ( U ´et , Z c ( n )) R Γ( Z ´et , Z c ( n ))[1] R Γ c ( U ´et , Z ( n )) R Γ c ( X ´et , Z ( n )) R Γ c ( Z ´et , Z ( n )) R Γ c ( U ´et , Z ( n ))[1] R Γ c ( G R , U ( C ) , R ( n )) R Γ c ( G R , X ( C ) , R ( n )) R Γ c ( G R , Z ( C ) , R ( n )) R Γ c ( G R , U ( C ) , R ( n ))[1]The first triangle is [Gei2010b, Corollary 7.2], and it means that R Γ( − , Z c ( n )) behaves like Borel–Moorehomology, while the following two triangles are the usual ones for cohomology with compact support.20hese fit together in a commutative diagram displayed on figure 1 below (page 25). For brevity, we de-note R Hom( X, Y ) by [ X, Y ] in the diagram. Similarly, figure 2 displays the same diagram tensored with Q . In this diagram, we start from the morphism of triangles ( α U,n , α X,n , α Z,n ), and then take the respectivecones R Γ fg ( − , Z ( n )). In fact, by [Bes2020, Proposition 5.5], these cones are well-defined up to a unique isomorphism in the derived category D ( Z ), and the same argument shows that the induced morphisms ofcomplexes R Γ fg ( U, Z ( n )) → R Γ fg ( X, Z ( n )) → R Γ fg ( Z, Z ( n )) → R Γ fg ( U, Z ( n ))[1] (6.9)are also uniquely defined (see [Bes2020, Corollary A.3]). A priori, it does not have to be a distinguishedtriangle * , but we claim that it induces a long exact sequence in cohomology.For this note that tensoring the diagram with Z /m Z gives us an isomorphism R Γ c ( U ´et , Z /m Z ( n )) R Γ c ( X ´et , Z /m Z ( n )) R Γ c ( Z ´et , Z /m Z ( n )) R Γ c ( U ´et , Z /m Z ( n ))[1] R Γ fg ( U, Z ( n )) ⊗ L Z Z /m Z R Γ fg ( X, Z ( n )) ⊗ L Z Z /m Z R Γ fg ( Z, Z ( n )) ⊗ L Z Z /m Z R Γ fg ( U, Z ( n )) ⊗ L Z Z /m Z [1] ∼ = ∼ = ∼ = ∼ = More generally, for each prime p we may take the corresponding derived p -adic completions (see [BS2015]and [Stacks, Tag 091N]) R Γ fg ( − , Z ( n )) ∧ p := R lim ←− k ( R Γ fg ( − , Z ( n )) ⊗ L Z Z /p k Z ) , and these give us a distinguished triangle for each prime pR Γ fg ( U, Z ( n )) ∧ p → R Γ fg ( X, Z ( n )) ∧ p → R Γ fg ( Z, Z ( n )) ∧ p → R Γ fg ( U, Z ( n )) ∧ p [1] . On the level of cohomology, there are natural isomorphisms [Stacks, Tag 0A06] H i ( R Γ fg ( − , Z ( n )) ∧ p ) ∼ = H i fg ( − , Z ( n )) ⊗ Z Z p . In particular, for each p there is a long exact sequence of cohomology groups · · · → H i fg ( U, Z ( n )) ⊗ Z Z p → H i fg ( X, Z ( n )) ⊗ Z Z p → H i fg ( Z, Z ( n )) ⊗ Z Z p → H i +1 fg ( U, Z ( n )) ⊗ Z Z p → · · · induced by (6.9). But now since the groups H i fg ( − , Z ( n )) are finitely generated, by flatness of Z p this impliesthat the sequence · · · → H i fg ( U, Z ( n )) → H i fg ( X, Z ( n )) → H i fg ( Z, Z ( n )) → H i +1 fg ( U, Z ( n )) → · · · (6.10)is exact. * Taking naively a “cone of a morphism of distinguished triangles” X Y Z X [1] X ′ Y ′ Z ′ X ′ [1] X ′′ Y ′′ Z ′′ X ′′ [1] X [1] Y [1] Z [1] X [2] (ac) normally does not give a distinguished triangle X ′′ → Y ′′ → Z ′′ → X ′′ [1], as discussed in [Nee1991]. Here we are dealing withnotorious issues associated to working with the classical derived (1-)categories. τ ≤ m R Γ c ( G R , U ( C ) , Z ( n ))[ − R Γ W,c ( U, Z ( n )) τ ≤ m R Γ fg ( U, Z ( n )) τ ≤ m R Γ c ( G R , U ( C ) , Z ( n )) τ ≤ m R Γ c ( G R , X ( C ) , Z ( n ))[ − R Γ W,c ( X, Z ( n )) τ ≤ m R Γ fg ( X, Z ( n )) τ ≤ m R Γ c ( G R , X ( C ) , Z ( n )) τ ≤ m R Γ c ( G R , Z ( C ) , Z ( n ))[ − R Γ W,c ( Z, Z ( n )) τ ≤ m R Γ fg ( Z, Z ( n )) τ ≤ m R Γ c ( G R , Z ( C ) , Z ( n )) τ ≤ m R Γ c ( G R , U ( C ) , Z ( n )) R Γ W,c ( U, Z ( n ))[1] τ ≤ m R Γ fg ( U, Z ( n ))[1] τ ≤ m R Γ c ( G R , U ( C ) , Z ( n ))[1]Here we took truncations for m big enough similarly to the proof of lemma 4.3. There are canonicalisomorphisms det Z R Γ W,c ( U, Z ( n )) ∼ = det Z ( τ ≤ m R Γ c ( G R , U ( C ) , Z ( n ))[ − ⊗ Z det Z ( τ ≤ m R Γ fg ( U, Z ( n ))) , det Z R Γ W,c ( X, Z ( n )) ∼ = det Z ( τ ≤ m R Γ c ( G R , X ( C ) , Z ( n ))[ − ⊗ Z det Z ( τ ≤ m R Γ fg ( X, Z ( n ))) , det Z R Γ W,c ( Z, Z ( n )) ∼ = det Z ( τ ≤ m R Γ c ( G R , Z ( C ) , Z ( n ))[ − ⊗ Z det Z ( τ ≤ m R Γ fg ( Z, Z ( n ))) , det Z ( τ ≤ m R Γ c ( G R , X ( C ) , Z ( n ))) ∼ = det Z ( τ ≤ m R Γ c ( G R , U ( C ) , Z ( n ))) ⊗ Z det Z ( τ ≤ m R Γ c ( G R , Z ( C ) , Z ( n ))) , det Z ( τ ≤ m R Γ fg ( X, Z ( n ))) ∼ = det Z ( τ ≤ m R Γ fg ( U, Z ( n ))) ⊗ Z det Z ( τ ≤ m R Γ fg ( Z, Z ( n ))) . Here the first four isomorphisms come from true distinguished triangles, while the last isomorphism comesfrom the cohomology long exact sequence (6.10), which gives an isomorphism O i ≤ m (cid:16) det Z H i fg ( U, Z ( n )) ( − i ⊗ Z det Z H i fg ( X, Z ( n )) ( − i +1 ⊗ Z det Z H i fg ( Z, Z ( n )) ( − i (cid:17) ∼ = Z . We may rearrange the terms at the cost of introducing a ± sign , to obtaindet Z ( τ ≤ m R Γ fg ( X, Z ( n ))) ∼ = O i ≤ m det Z H i fg ( X, Z ( n )) ∼ = O i ≤ m det Z H i fg ( U, Z ( n )) ⊗ Z O i ≤ m det Z H i fg ( Z, Z ( n )) ∼ =det Z ( τ ≤ m R Γ fg ( U, Z ( n ))) ⊗ Z det Z ( τ ≤ m R Γ fg ( Z, Z ( n ))) . All the above gives us the desired isomorphism of integral determinants (6.6).Now we consider the following diagram with distinguished rows: R Γ c ( G R , U ( C ) , R ( n ))[ − ⊕ R Γ c ( G R , U ( C ) , R ( n ))[ − R Γ c ( G R , X ( C ) , R ( n ))[ − ⊕ R Γ c ( G R , X ( C ) , R ( n ))[ − R Γ c ( G R , Z ( C ) , R ( n ))[ − ⊕ R Γ c ( G R , Z ( C ) , R ( n ))[ − · · · R Hom( R Γ( U ´et , Z c ( n )) , R )[ − ⊕ R Γ c ( G R , U ( C ) , R ( n ))[ − R Hom( R Γ( X ´et , Z c ( n )) , R )[ − ⊕ R Γ c ( G R , X ( C ) , R ( n ))[ − R Hom( R Γ( Z ´et , Z c ( n )) , R )[ − ⊕ R Γ c ( G R , Z ( C ) , R ( n ))[ − · · · R Γ W,c ( U, R ( n )) R Γ W,c ( X, R ( n )) R Γ W,c ( Z, R ( n )) · · · Reg ∨ U,n [ − ⊕ id ∼ = Reg ∨ X,n [ − ⊕ id ∼ = Reg ∨ Z,n [ − ⊕ id ∼ =split ∼ = split ∼ = split ∼ = For n < and r ≥ , let X be an arithmetic scheme satisfying L c ( X ´et , n − r ) and B ( X, n − r ) .Then there is a natural quasi-isomorphism of complexes R Γ W,c ( A rX , Z ( n )) ∼ = R Γ W,c ( X, Z ( n − r ))[ − r ] , (6.11) which after passing to the determinants makes the following diagram commute: R (det Z R Γ W,c ( A rX , Z ( n ))) ⊗ Z R (det Z R Γ W,c ( X, Z ( n − r ))) ⊗ Z R ∼ = λ A rX,n λ X,n − r ∼ = ∼ = (6.12) Proof. We refer to figure 3 below (page 27) that shows how the flat morphism p : A rX → X induces the desiredquasi-isomorphism (6.11). Everything comes down to the homotopy property of motivic cohomology, namelythe fact that p induces a quasi-isomorphism p ∗ : R Γ( X ´et , Z c ( n − r ))[2 r ] ∼ = −→ R Γ( A rX, ´et , Z c ( n ))—for this see e.g. [Mor2014, Lemma 5.11]. After passing to real coefficients, we obtain the following diagram: R Γ c ( G R , A rX ( C ) , R ( n ))[ − ⊕ R Γ c ( G R , A rX ( C ) , R ( n ))[ − R Γ c ( G R , X ( C ) , R ( n − r ))[ − − r ] ⊕ R Γ c ( G R , X ( C ) , R ( n − r ))[ − − r ] R Hom( R Γ( A rX, ´et , Z c ( n )) , R )[ − ⊕ R Γ c ( G R , A rX ( C ) , R ( n ))[ − R Hom( R Γ( X ´et , Z c ( n − r ))[2 r ] , R )[ − ⊕ R Γ c ( G R , X ( C ) , R ( n − r ))[ − − r ] R Γ W,c ( A rX , R ( n )) R Γ W,c ( X, R ( n − r ))[ − r ] Reg ∨ A rX ,n [ − ⊕ id ∼ = ∼ = Reg ∨ X,n − r [ − − r ] ⊕ id ∼ =split ∼ = ∼ = split ∼ = ∼ = Here the first square commutes by the compatibility of the regulator with affine bundles (lemma 6.2), andthe second square commutes because the quasi-isomorphism (6.11) gives compatible splittings (again, seefigure 3 below). Taking the determinants, we obtain the desired commutative diagram (6.12). For an arithmetic scheme X and n < , assume L c ( X ´et , n ) , B ( X, n ) , and the meromorphiccontinuation of ζ ( X, s ) around s = n .1) If X = ` ≤ i ≤ r X i is a finite disjoint union of arithmetic schemes, then C ( X, n ) ⇐⇒ C ( X i , n ) for all i. 2) For a closed-open decomposition Z ֒ → X ← ֓ U , if two out of three conjectures C ( X, n ) , C ( Z, n ) , C ( U, n ) hold, then the third holds as well. ) For any r ≥ , one has C ( A rX , n ) ⇐⇒ C ( X, n − r ) . Proof. Follows from the previous lemmas 6.5, 6.6, 6.7, together with the respective identities for zeta functions(6.1), (6.2), (6.3). As a formal consequence of compatibility with closed-open decompositions, if we apply itto the canonical closed embedding X red ֒ → X , then we conclude that R Γ W,c ( X, Z ( n )) ∼ = R Γ W,c ( X red , Z ( n )).This is not surprising, because Weil-´etale complexes are constructed from a variant of cycle complexes /higher Chow groups, and these do not distinguish X from X red (unlike, for instance, algebraic K -groups).This is actually the desired behavior for us, since neither the zeta function does: ζ ( X, s ) = ζ ( X red , s ). If X/ F q is a variety over a finite field, then the proof of theorem 6.8 simplifies drastically:we can work with the formula (5.1), and the following properties of motivic cohomology:1) R Γ( ` i X i, ´et , Z c ( n )) ∼ = L i R Γ( X i, ´et , Z c ( n ));2) triangles R Γ( Z ´et , Z c ( n )) → R Γ( X ´et , Z c ( n )) → R Γ( U ´et , Z c ( n )) → R Γ( Z ´et , Z c ( n ))[1] associated toclosed-open decompositions;3) homotopy invariance R Γ( X ´et , Z c ( n − r ))[2 r ] ∼ = R Γ( A rX, ´et , Z c ( n )).There are no regulators involved in this case, so we do not need the technical lemmas 6.5, 6.6, 6.7.Considering the projective space P rX = P r Z × X , we have a formula for the zeta function ζ ( P rX , s ) = Y ≤ i ≤ r ζ ( X, s − i ) . (6.13) Let X be an arithmetic scheme, n < , and r ≥ . For i = 0 , . . . , r assume the conjectures L c ( X ´et , n − i ) , B ( X, n − i ) , and meromorphic continuation of ζ ( X, s ) around s = n − i .Then C ( X, n − i ) for i = 0 , . . . , r = ⇒ C ( P rX , n ) . Proof. Applied to the closed-open decomposition P r − X ֒ → P rX ← ֓ A rX , theorem 6.8 gives C ( X, n − r ) and C ( P r − X , n ) = ⇒ C ( A rX , n ) and C ( P r − X , n ) = ⇒ C ( P rX , n ) . The claim follows by induction on r . (Note that the same inductive argument proves the formula (6.13) from(6.3).) 24 Γ W,c ( U, Z ( n ))[ R Γ( U ´et , Z c ( n )) , Q [ − R Γ c ( U ´et , Z ( n )) R Γ fg ( U, Z ( n )) [ R Γ( U ´et , Z c ( n )) , Q [ − R Γ c ( G R , U ( C ) , Z ( n )) R Γ W,c ( X, Z ( n ))[ R Γ( X ´et , Z c ( n )) , Q [ − R Γ c ( X ´et , Z ( n )) R Γ fg ( X, Z ( n )) [ R Γ( X ´et , Z c ( n )) , Q [ − R Γ c ( G R , X ( C ) , Z ( n )) R Γ W,c ( Z, Z ( n ))[ R Γ( Z ´et , Z c ( n )) , Q [ − R Γ c ( Z ´et , Z ( n )) R Γ fg ( Z, Z ( n )) [ R Γ( Z ´et , Z c ( n )) , Q [ − R Γ c ( G R , Z ( C ) , Z ( n )) R Γ W,c ( U, Z ( n ))[1][ R Γ( U ´et , Z c ( n )) , Q [ − R Γ c ( U ´et , Z ( n ))[1] R Γ fg ( U, Z ( n ))[1] [ R Γ( U ´et , Z c ( n )) , Q ] R Γ c ( G R , U ( C ) , Z ( n ))[1] α U,n u ∞ i ∞ ∃ ! α X,n u ∞ i ∞ ∃ ! α Z,n u ∞ i ∞ ∃ ! α U,n [1] u ∞ i ∞ [1] Figure 1: Diagram induced by a closed-open decomposition Z ֒ → X ← ֓ U Γ W,c ( U, Q ( n ))[ R Γ( U ´et , Z c ( n )) , Q [ − R Γ fg ( U, Q ( n )) [ R Γ( U ´et , Z c ( n )) , Q [ − R Γ c ( G R , U ( C ) , Q ( n )) R Γ W,c ( X, Q ( n ))[ R Γ( X ´et , Z c ( n )) , Q [ − R Γ fg ( X, Q ( n )) [ R Γ( X ´et , Z c ( n )) , Q [ − R Γ c ( G R , X ( C ) , Q ( n )) R Γ W,c ( Z, Q ( n ))[ R Γ( Z ´et , Z c ( n )) , Q [ − R Γ fg ( Z, Q ( n )) [ R Γ( Z ´et , Z c ( n )) , Q [ − R Γ c ( G R , Z ( C ) , Q ( n )) R Γ W,c ( U, Q ( n ))[1][ R Γ( U ´et , Z c ( n )) , Q [ − R Γ fg ( U, Q ( n ))[1] [ R Γ( U ´et , Z c ( n )) , Q ] R Γ c ( G R , U ( C ) , Q ( n ))[1] ∃ ! ∼ =0 ∃ ! ∼ =0 ∃ ! ∼ =0 ∼ = Figure 2: Diagram induced by a closed-open decomposition Z ֒ → X ← ֓ U , tensored with Q Γ W,c ( A rX , Z ( n ))[ R Γ( A rX, ´et , Z c ( n )) , Q [ − R Γ c ( A rX, ´et , Z ( n )) R Γ fg ( A rX , Z ( n )) [+1] R Γ c ( G R , A rX ( C ) , Z ( n )) R Γ W,c ( X, Z ( n − r ))[ − r ][ R Γ( X ´et , Z c ( n − r ))[2 r ] , Q [ − R Γ c ( X ´et , Z ( n − r ))[ − r ] R Γ fg ( X, Z ( n − r ))[ − r ] [+1] R Γ c ( G R , X ( C ) , Z ( n − r ))[ − r ] ⊗ Z Q = R Γ W,c ( A rX , Q ( n ))[ R Γ( A rX, ´et , Z c ( n )) , Q [ − R Γ fg ( A rX , Q ( n )) [+1] R Γ c ( G R , A rX ( C ) , Q ( n )) R Γ W,c ( X, Q ( n − r ))[ − r ][ R Γ( X ´et , Z c ( n − r ))[2 r ] , Q [ − R Γ fg ( X, Q ( n − r ))[ − r ] [+1] R Γ c ( G R , X ( C ) , Q ( n − r ))[ − r ] ∼ = α A rX,n ( p ∗ ) ∨ ∼ = u ∞ p ∗ ∼ = i ∞ ∼ = ∼ = α X,n − r [ − r ] u ∞ i ∞ p ∗ ∼ = ∼ =( p ∗ ) ∨ ∼ = 0 ∼ = ∼ =0 p ∗ ∼ = Figure 3: Isomorphism R Γ W,c ( A rX , Z ( n )) ∼ = R Γ W,c ( X, Z ( n − r ))[ − r ] and its splitting after tensoring with Q Unconditional results Now we apply theorem 6.8 from the previous section in order to prove the main theorem stated in theintroduction: the validity of VO ( X, n ) and C ( X, n ) for all n < C ( Z ) whose elements satisfythe conjectures. This approach is motivated by [Mor2014, § Let C ( Z ) be the full subcategory of the category of arithmetic schemes generated by thefollowing objects: • the empty scheme ∅ , • Spec F q for a finite field, • Spec O F for an abelian number field F/ Q , • curves over finite fields C/ F q ,and the following operations. C X lies in C ( Z ) if and only if X red lies in C ( Z ). C 1) A finite disjoint union ` ≤ i ≤ r X i lies in C ( Z ) if and only if each X i lies in C ( Z ). C 2) Let Z ֒ → X ← ֓ U be a closed-open decomposition such that Z red , C , X red , C , U red , C are smooth andquasi-projective. Then if two out of three schemes Z, X, U lie in C ( Z ), then the third lies as well. C 3) If X lies in C ( Z ), then the affine space A rX also lies in C ( Z ) for each r ≥ X red , C is smooth and quasi-projective is needed to ensure that theregulator morphism exists (see remark 2.8). The conjectures VO ( X, n ) and C ( X, n ) hold for any X ∈ C ( Z ) and n < .Proof. Finite fields satisfy C ( X, n ) by example 5.6.If X = Spec O F for an abelian number field F/ Q , then the conjecture C ( X, n ) is equivalent to the conjec-ture of Flach and Morin [FM2018, Conjecture 5.11], which holds unconditionally in this particular case, viareduction to the Tamagawa number conjecture; see [FM2018, § VO ( X, n ) is also true in this case (see example 3.6).If X = C/ F q is a curve over a finite field, then C ( X, n ) holds thanks to theorem 5.4. The conjecture R ( F q , d ) that appears in the statement is true for d = 1. Similarly, L c fin ( X/ F q , n ) is well-known for curvesand essentially goes back to Soul´e; see for instance [Gei2017, Proposition 4.3] * .Then the fact that the conjectures L c ( X ´et , n ), B ( X, n ), VO ( X, n ), C ( X, n ) are closed under the opera-tions C C 3) is lemma 6.1, lemma 6.2, proposition 6.3, and theorem 6.8 respectively. Any -dimensional arithmetic scheme X lies in C ( Z ) .Proof. Since X is a noetherian scheme of dimension 0, it is a finite disjoint union of Spec A i for some artinianlocal rings A i . Thanks to C X = Spec A , and thanks to C X is reduced. But then A = k is a field. Since X is a scheme of finite type over Spec Z , we conclude that X = Spec F q ∈ C ( Z ). Let B be a -dimensional arithmetic scheme. Assume that each of the generic points η ∈ B satisfies one of the following properties: * Assuming X is smooth, the quoted result states that H i ( X ´et , Z c ( n )) = H i +2 ( X ´et , Z (1 − n )) is finite, except for i = − n, − n + 2, when it is possibly finitely generated or of cofinite type. However, Artin–Verdier duality [Bes2020, Theorem I]shows that these two “exceptional” groups are at most finite 2-torsion. If X is not smooth, the statement follows by resolutionof singularities. ) char κ ( η ) = p > ;b) char κ ( η ) = 0 , and κ ( η ) / Q is an abelian number field.Then B ∈ C ( Z ) .Proof. We will see that such a scheme can be obtained from Spec O F for an abelian number field F/ Q , or acurve over a finite field C/ F q , using the operations C C C 2) that appear in the definition of C ( Z ).Thanks to C B is reduced. Consider the normalization ν : B ′ → B . This is abirational morphism, and there exist open dense subsets U ′ ⊆ B ′ and U ⊆ B such that ν | U ′ : U ′ ∼ = −→ U . Now B \ U is 0-dimensional, and therefore B \ U ∈ C ( Z ) by the previous lemma. Thanks to C U ′ ∈ C ( Z ), and this would imply B ∈ C ( Z ).Now U ′ is a finite disjoint union of normal integral schemes, so by C 1) we may assume that U ′ is integral.Consider the generic point η ∈ U ′ and the residue field F = κ ( η ). There are two distinct cases to consider.a) If char F = p > 0, then U ′ is a curve over a finite field, so it lies in C ( Z ) by the definition.b) If char F = 0, then by our assumptions, F/ Q is an abelian number field.We note that if V ′ ⊆ U ′ is an affine open neighborhood of η , then U ′ \ V ′ ∈ C ( Z ) by the previouslemma. Therefore, we can assume without loss of generality that U ′ is affine.We have U ′ = Spec O , where O is a finitely generated integrally closed domain. All this means that O F ⊆ O = O F,S for some finite set S . Now U ′ = Spec O F \ S , and S ∈ C ( Z ), so again, everythingreduces to the case of U ′ = Spec O F , which lies in C ( Z ) by the definition. Schemes as above were considered by Jordan and Poonen in [JP2020], where the authorswrite down a special value formula at s = 1, generalizing the classical class number formula. Namely, theyconsider the case of B reduced and affine, albeit without requiring κ ( η ) / Q to be abelian. If B = Spec O for a nonmaximal order O ⊂ O F , where F/ Q is an abelian number field,then our formalism gives a cohomological interpretation of the special values of ζ O ( s ) at s = n < 0. Thisalready seems to be a new result. Let X → B be a B -scheme. We say that X is B -cellular if it admits a filtration by closedsubschemes X = Z N ⊇ Z N − ⊇ · · · ⊇ Z ⊇ Z − = ∅ (7.1)such that Z i \ Z i − ∼ = ` j A r ij B is a finite union of affine B -spaces.For instance, projective spaces P rB and in general Grassmannians Gr( k, ℓ ) B are cellular. Many interestingexamples of cellular schemes as above arise from actions of algebraic groups on varieties and Bia lynicki-Birulatheorem; for this see [Wen2010] and [Bro2005]. Let X be a B -cellular arithmetic scheme, where B ∈ C ( Z ) , and X red , C is smooth andquasi-projective. Then X ∈ C ( Z ) .Proof. Looking at the corresponding cellular decomposition (7.1), we better pass to open complements U i = X \ Z i , to obtain a filtration X = U − ⊇ U ⊇ · · · ⊇ U N − ⊇ U N = ∅ , with U i, C smooth and quasi-projective, being open subvarieties in X C . Now we have closed-open decomposi-tions ` j A r i,j B ֒ → U i ← ֓ U i +1 , and the claim follows by induction on the length of the cellular decomposition,using operations C C .9. Theorem. Let B be a -dimensional arithmetic scheme satisfying the assumptions of proposition 7.4.If X is a B -cellular arithmetic scheme with smooth and quasi-projective fiber X red , C , then the conjectures VO ( X, n ) and C ( X, n ) hold unconditionally for any n < .Proof. Follows from propositions 7.2, 7.4, 7.8. 30 Determinants of complexes In this appendix we include a brief overview of determinants of complexes. The original construction is dueto Knudsen and Mumford [KM1976], and other useful expositions may be found in [GKZ1994, Appendix A]and [Kat1993, § R be a commutative ring, which is an integral domain (we will be interested in R = Z , Q , R ). Denote by P is ( R ) * the category of graded invertible R -modules. It has as its objects pairs( L, r ), where L is an invertible R -module (= projective of rank 1) and r ∈ Z . The morphisms in this categoryare given by Hom P is ( R ) (( L, r ) , ( M, s )) = ( Isom R ( L, M ) , r = s, ∅ , r = s. This category is equipped with tensor products( L, r ) ⊗ R ( M, s ) = ( L ⊗ R M, r + s )with commutativity isomorphisms ψ : ( L, r ) ⊗ R ( M, s ) ∼ = −→ ( M, s ) ⊗ R ( L, r ) ,ℓ ⊗ m ( − r s m ⊗ ℓ for ℓ ∈ L , m ∈ M .The unit object with respect to this product is 11 = ( R, L, r ) ∈ P is ( R ) the inverse isgiven by ( L − , − r ) where L − = Hom R ( L, R ). The canonical evaluation morphism L ⊗ R Hom R ( L, R ) → R induces an isomorphism ( L, r ) ⊗ R ( L − , − r ) ∼ = 11 . A.1. Definition. We denote by C is ( R ) the category whose objects are finitely generated projective R -modules and whose morphisms are isomorphisms. For A ∈ C is ( R ) we define the corresponding determinantas an object in P is ( R ) given by det R ( A ) = (cid:16) rk R A ^ R A, rk R A (cid:17) . Here rk R A is the rank of A , so that the top exterior power V rk R AR A is an invertible R -module.This gives a functor det R : C is ( R ) → P is ( R ). The main result of [KM1976, Chapter I] is that thisconstruction may be generalized as follows. Let D ( R ) be the derived category of the category of R -modules.Recall that a complex A • is perfect if it is quasi-isomorphic to a bounded complex of finitely generatedprojective R -modules. We denote by P arf is ( R ) the subcategory of D ( R ) whose objects consist of perfectcomplexes, and whose morphisms are quasi-isomorphisms of complexes. A.2. Theorem (Knudsen–Mumford). The determinant may be extended to perfect complexes of R -modules as follows.I) For every ring R there exists a functor det R : P arf is ( R ) → P is ( R ) such that det R (0) = 11 . * P for “Picard”. I) For every short exact sequence of complexes in P arf is ( R )0 → A • α −→ B • β −→ C • → there exists an isomorphism i R ( α, β ) : det R A • ⊗ R det R C • ∼ = −→ det R B • . In particular, for the short exact sequence A • A • • (resp. • A • A • ) id id the isomorphism i R ( id, (resp. i R (0 , id ) ) is the canonical isomorphism det R A • ⊗ R ∼ = −→ det R A • Moreover, the following properties hold.i) Given an isomorphism of short exact sequences of complexes A • B • C • A ′• B ′• C ′• αu ∼ = βv ∼ = w ∼ = α ′ β ′ the diagram det R A • ⊗ R det R C • det R B • det R A ′• ⊗ R det R C ′• det R B ′• i ∗ ( α,β ) ∼ =det R ( u ) ⊗ det R ( w ) ∼ = det R ( v ) ∼ = i ∗ ( α ′ ,β ′ ) ∼ = commutes.ii) Given a commutative × diagram with rows and columns short exact sequences A • B • C • A ′• B ′• C ′• A ′′• B ′′• C ′′• 00 0 0 αu βu ′ u ′′ α ′ v β ′ v ′ v ′′ α ′′ β ′′ the diagram det R A • ⊗ R det R C • ⊗ R det R A ′′• ⊗ R det R C ′′• det R B • ⊗ R det R B ′′• (det R A • ⊗ R det R A ′′• ) ⊗ R (det R C • ⊗ R det R C ′′• )det R A ′• ⊗ R det R C ′• det R B ′• i R ( α,β ) ⊗ i R ( α ′′ ,β ′′ ) ∼ = id ⊗ ψ ⊗ id ∼ = i R ( u ′ ,v ′ ) ∼ = ∼ = i R ( u,v ) ⊗ i R ( u ′′ ,v ′′ ) i R ( α ′ ,β ′ ) ∼ = commutes. ii) det and i commute with base change. Namely, given a ring homomorphism f : R → S , there is anatural isomorphism η A = η A • ( f ) : det S ( A • ⊗ L R S ) ∼ = −→ (det R A • ) ⊗ R S, such that for every short exact sequence of complexes → A • α −→ B • β −→ C • → the diagram det S ( A • ⊗ L R S ) ⊗ S det S ( C • ⊗ L R S ) det S ( B • ⊗ L R S ) (cid:16) (det R A • ) ⊗ R S (cid:17) ⊗ S (cid:16) (det R C • ) ⊗ R S (cid:17) (det R B • ) ⊗ R S η A ⊗ η C ∼ = i S ( α ⊗ S,β ⊗ S ) ∼ = η B ∼ = i R ( α,β ) ⊗ S ∼ = commutes. Similarly, there is compatibility with compositions of base changes along R f −→ S g −→ T (weomit the corresponding commutative diagram). A.3. Remark. We refer to [KM1976] for the actual construction. In practice, the following considerationsare useful; see [ibid.] for the proofs.1) If A • is a bounded complex where each object A i is perfect (i.e. admits a finite length resolution byfinitely generated projective R -modules), thendet R A • ∼ = O i ∈ Z (det R A i ) ( − i . In particular, if each A i is already a finitely generated projective R -module, then det R A i in the aboveformula is given by A.1.2) If the cohomology modules H i ( A • ) are perfect, thendet R A • ∼ = O i ∈ Z (det R H i ( A • )) ( − i . The determinants also behave well not only with short exact sequences, but with distinguished triangles. A.4. Proposition ([KM1976, Proposition 7]). For a distinguished triangle of complexes in P arf is ( R ) A • u −→ B • v −→ C • w −→ A • [1] there is a canonical isomorphism i R ( u, v, w ) : det R A • ⊗ R det R C • ∼ = −→ det R B • , which is functorial in the following sense: given a (quasi-)isomorphism of distinguished triangles A • B • C • A • [1] A ′• B ′• C ′• A ′• [1] uf ∼ = vg ∼ = wh ∼ = f [1] ∼ = u v w the diagram det R A • ⊗ R det R C • det R B • det R A ′• ⊗ R det R C ′• det R B ′• i R ( u,v,w ) ∼ =det R ( f ) ⊗ det R ( h ) ∼ = det R ( g ) ∼ = i R ( u ′ ,v ′ ,w ′ ) ∼ = commutes. .5. Remark. In what follows, for ( L, r ) ∈ P is ( R ) we will forget about r and treat the determinant as aninvertible R -module.A particular very simple case of interest is when all cohomology groups H i ( A • ) are finite; then it is easyto understand what the determinant means. A.6. Lemma. 1) Let A be a finite abelian group. Then (det Z A ) ⊂ (det Z A ) ⊗ Z Q ∼ = det Q ( A ⊗ Z Q ) = det Q (0) ∼ = Q corresponds to the fractional ideal A Z ⊂ Q .2) In general, let A • be a perfect complex of abelian groups such that the cohomology groups H i ( A • ) areall finite. Then det Z A • corresponds to the fractional ideal m Z ⊂ Q , where m = Y i ∈ Z | H i ( A • ) | ( − i . Proof. Since det Z ( A ⊕ B ) ∼ = det Z A ⊗ Z det Z B , in part 1) it would be enough to consider the case of a cyclicgroup A = Z /m Z . Then we have a quasi-isomorphism of complexes Z /m Z [0] ∼ = h m Z deg. − ֒ → Z deg. 0 i . Therefore, det Z ( Z /m Z ) ∼ = Z ⊗ Z ( m Z ) − ∼ = ( m Z ) − , which corresponds to m Z inside Q . Part 2) follows immediately from 1) using the isomorphismdet Z A • ∼ = O i ∈ Z (det Z H i ( A • )) ( − i . A.7. Remark. The above argument works in a more general setting, assuming R is a regular noetherianring and A is a finitely generated torsion R -module (replacing Q with the total quotient field Q ( R )).34 eferences [Bes2020] Alexey Beshenov, Weil-´etale cohomology for arbitrary arithmetic schemes and n < . Part I:Construction of Weil-´etale complexes , 2020, arXiv.org preprint 2012.11034. https://arxiv.org/abs/2012.11034 [Blo1986a] Spencer Bloch, Algebraic cycles and higher K -theory , Adv. in Math. (1986), no. 3, 267–304.MR852815 https://doi.org/10.1016/0001-8708(86)90081-2 [Blo1986b] , Algebraic cycles and the Be˘ılinson conjectures , The Lefschetz centennial conference,Part I (Mexico City, 1984), Contemp. Math., vol. 58, Amer. Math. Soc., Providence, RI, 1986,pp. 65–79. MR860404 https://doi.org/10.1090/conm/058.1/860404 [Bro2005] Patrick Brosnan, On motivic decompositions arising from the method of Bia lynicki-Birula ,Invent. Math. (2005), 91–111. MR2178658 https://doi.org/10.1007/s00222-004-0419-7 [BS2015] Bhargav Bhatt and Peter Scholze, The pro-´etale topology for schemes , Ast´erisque (2015),no. 369, 99–201. MR3379634 https://arxiv.org/abs/1309.1198 [Del1980] Pierre Deligne, La conjecture de Weil. II , Inst. Hautes ´Etudes Sci. Publ. Math. (1980), no. 52,137–252. MR601520 [EV1988] H´el`ene Esnault and Eckart Viehweg, Deligne-Be˘ılinson cohomology , Be˘ılinson’s conjectures onspecial values of L -functions, Perspect. Math., vol. 4, Academic Press, Boston, MA, 1988,pp. 43–91. MR944991 [FM2012] Matthias Flach and Baptiste Morin, On the Weil-´etale topos of regular arithmetic schemes ,Doc. Math. (2012), 313–399. MR2946826 [FM2018] , Weil-´etale cohomology and zeta-values of proper regular arithmetic schemes , Doc.Math. (2018), 1425–1560. MR3874942 https://doi.org/10.25537/dm.2018v23.1425-1560 [FM2020] , Compatibility of special value conjectures with the functional equation of zeta functions ,2020, arXiv.org preprint 2005.04829. https://arxiv.org/abs/2005.04829 [FPR1994] Jean-Marc Fontaine and Bernadette Perrin-Riou, Autour des conjectures de Bloch et Kato:cohomologie galoisienne et valeurs de fonctions L , Motives (Seattle, WA, 1991), Proc. Sympos.Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 599–706. MR1265546[Gei2004a] Thomas Geisser, Motivic cohomology over Dedekind rings , Math. Z. (2004), no. 4, 773–794.MR2103541 https://doi.org/10.1007/s00209-004-0680-x [Gei2004b] , Weil-´etale cohomology over finite fields , Math. Ann. (2004), no. 4, 665–692.MR2102307 https://doi.org/10.1007/s00208-004-0564-8 Arithmetic cohomology over finite fields and special values of ζ -functions , Duke Math.J. (2006), no. 1, 27–57. MR2219269 https://doi.org/10.1215/S0012-7094-06-13312-4 [Gei2010a] , Arithmetic homology and an integral version of Kato’s conjecture , J. Reine Angew.Math. (2010), 1–22. MR2671773 https://doi.org/10.1515/CRELLE.2010.050 [Gei2010b] , Duality via cycle complexes , Ann. of Math. (2) (2010), no. 2, 1095–1126.MR2680487 https://doi.org/10.4007/annals.2010.172.1095 [Gei2017] , On the structure of ´etale motivic cohomology , J. Pure Appl. Algebra (2017), no. 7,1614–1628. MR3614969 https://doi.org/10.1016/j.jpaa.2016.12.019 [GKZ1994] I. M. Gel ′ fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and mul-tidimensional determinants , Mathematics: Theory & Applications, Birkh¨auser Boston, Inc.,Boston, MA, 1994. MR1264417 https://doi.org/10.1007/978-0-8176-4771-1 [Jan1988] Uwe Jannsen, Deligne homology, Hodge- D -conjecture, and motives , Be˘ılinson’s conjectures onspecial values of L -functions, Perspect. Math., vol. 4, Academic Press, Boston, MA, 1988,pp. 305–372. MR944998 [JP2020] Bruce W. Jordan and Bjorn Poonen, The analytic class number formula for 1-dimensionalaffine schemes , Bull. Lond. Math. Soc. (2020), no. 5, 793–806. MR4171403 https://doi.org/10.1112/blms.12357 [Kah2005] Bruno Kahn, Algebraic K -theory, algebraic cycles and arithmetic geometry , Handbook of K -theory. Vol. 1, 2, Springer, Berlin, 2005, pp. 351–428. MR2181827 https://doi.org/10.1007/3-540-27855-9_9 [Kat1993] Kazuya Kato, Lectures on the approach to Iwasawa theory for Hasse-Weil L -functions via B dR .I , Arithmetic algebraic geometry (Trento, 1991), Lecture Notes in Math., vol. 1553, Springer,Berlin, 1993, pp. 50–163. MR1338860 https://doi.org/10.1007/BFb0084729 [Kat1994] Nicholas M. Katz, Review of ℓ -adic cohomology , Motives (Seattle, WA, 1991), Proc. Sympos.Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 21–30. MR1265520[KL2007] Matt Kerr and James D. Lewis, The Abel-Jacobi map for higher Chow groups. II , Invent. Math. (2007), no. 2, 355–420. MR2342640 https://doi.org/10.1007/s00222-007-0066-x [KLL2018] Matt Kerr, James D. Lewis, and Patrick Lopatto, Simplicial Abel-Jacobi maps and reciprocitylaws , J. Algebraic Geom. (2018), no. 1, 121–172, With an appendix by Jos´e Ignacio Burgos-Gil. MR3722692 https://doi.org/10.1090/jag/692 [KLMS2006] Matt Kerr, James D. Lewis, and Stefan M¨uller-Stach, The Abel-Jacobi map for higher Chowgroups , Compos. Math. (2006), no. 2, 374–396. MR2218900 https://doi.org/10.1112/S0010437X05001867 The projectivity of the moduli space of stable curves.I. Preliminaries on “det” and “Div” , Math. Scand. (1976), no. 1, 19–55. MR437541 https://doi.org/10.7146/math.scand.a-11642 [Lev1994] Marc Levine, Bloch’s higher Chow groups revisited , K -theory (Strasbourg, 1992), no. 226,Soci´et´e math´ematique de France, 1994, pp. 235–320. MR1317122 [Lic1984] Stephen Lichtenbaum, Values of zeta-functions at nonnegative integers , Number theory, Noord-wijkerhout 1983 (Noordwijkerhout, 1983), Lecture Notes in Math., vol. 1068, Springer, Berlin,1984, pp. 127–138. MR756089 https://doi.org/10.1007/BFb0099447 [Lic2005] , The Weil-´etale topology on schemes over finite fields , Compos. Math. (2005),no. 3, 689–702. MR2135283 https://doi.org/10.1112/S0010437X04001150 [Lic2009a] , Euler characteristics and special values of zeta-functions , Motives and algebraic cy-cles, Fields Inst. Commun., vol. 56, Amer. Math. Soc., Providence, RI, 2009, pp. 249–255.MR2562461 https://doi.org/10.1090/fic/056 [Lic2009b] , The Weil-´etale topology for number rings , Ann. of Math. (2) (2009), no. 2, 657–683. MR2552104 https://doi.org/10.4007/annals.2009.170.657 [Lic2021] , Special values of zeta functions of schemes , 2021, arXiv.org preprint 1704.00062. https://arxiv.org/abs/1704.00062v2 [Mil1986] J. S. Milne, Values of zeta functions of varieties over finite fields , Amer. J. Math. (1986),no. 2, 297–360. MR833360 https://doi.org/10.2307/2374676 [Mor2014] Baptiste Morin, Zeta functions of regular arithmetic schemes at s = 0, Duke Math. J. (2014), no. 7, 1263–1336. MR3205726 https://doi.org/10.1215/00127094-2681387 [Nee1991] Amnon Neeman, Some new axioms for triangulated categories , J. Algebra (1991), no. 1,221–255. MR1106349 https://doi.org/10.1016/0021-8693(91)90292-G [Ram2016] Niranjan Ramachandran, Higher Euler characteristics: variations on a theme of Euler , Ho-mology Homotopy Appl. (2016), no. 1, 231–246. MR3491851 https://doi.org/10.4310/HHA.2016.v18.n1.a12 [Ser1965] Jean-Pierre Serre, Zeta and L functions , Arithmetical Algebraic Geometry (Proc. Conf. PurdueUniv., 1963), Harper & Row, New York, 1965, pp. 82–92. MR0194396[Ser1970] , Facteurs locaux des fonctions zˆeta des vari´et´es alg´ebriques (d´efinitions et conjectures) ,S´eminaire Delange-Pisot-Poitou. Th´eorie des nombres (1969–1970), no. 2. MR3618526 [Sie2019] Daniel A. Siebel, Special values of zeta-functions for proper regular arithmetic surfaces , 2019,PhD thesis, California Institute of Technology. https://doi.org/10.7907/YMHN-2T74 K -th´eorie et z´eros aux points entiers de fonctions zˆeta , Proceedings of theInternational Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), PWN, Warsaw, 1984,pp. 437–445. MR804699[Stacks] The Stacks Project Authors, Stacks project , 2021. https://stacks.math.columbia.edu/ [Wei2017] Thomas Weißschuh, A commutative regulator map into Deligne-Beilinson cohomology ,Manuscripta Math. (2017), no. 3-4, 281–315. MR3608294 https://doi.org/10.1007/s00229-016-0867-6 [Wen2010] Matthias Wendt, More examples of motivic cell structures , 2010, arXiv.org preprint 1012.0454. https://arxiv.org/abs/1012.0454 Alexey BeshenovCenter for Research in Mathematics (CIMAT), Guanajuato, MexicoURL: https://cadadr.org/https://cadadr.org/