The cyclosyntomic regulator of a number field

THE CYCLOSYNTOMIC REGULA TOR OF A NUMBER FIELD tess bouis ∗ and q - uentin gazd a † W e construct a q -deformation of the p -adic regulator of a n umber ﬁeld, called the cyclosyntomic r e gulator , building on the Habiro ring of Garoufalidis–Scholze–Wheeler– Zagier. The k ey new ingredien t in our construction is a reﬁnement of Sulyma’s norm maps in prismatic cohomology , whic h interpolate b et ween classical p o w ers and F rob e- nius maps at v arious prime n um b ers p . F urthermore, w e compute the v alues of the cyclosyn tomic regulator at units of the form 1 − ζ , where ζ is a ro ot of unit y . Contents 1 Intro duction 1 2 No rm maps on q -Witt vectors 5 2.1 Review of Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Review of q -Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Alternativ e approach to q -ghost maps . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Norm maps on q -Witt v ectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Cyclotomic rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 The cyclotomic loga rithm 13 3.1 F rob enius maps on the Habiro ring of a num b er ﬁeld . . . . . . . . . . . . . . . . . 13 3.2 The cyclotomic logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4 Cyclosyntomic regulator 20 4.1 The cyclosyntomic cohomology of a n umber ﬁeld . . . . . . . . . . . . . . . . . . . 20 4.2 The ﬁrst Chern class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.3 The ﬁrst q -p olylogarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1 Intro duction Motiv ated by questions on the emerging theory of p -adic L -functions, Leop oldt [ Leo62 ] deﬁned the p -adic regulator map of a num ber ﬁeld K reg p : O × K  − → Y σ ∈ Hom(K , C p ) C × p log p − − − → Y σ ∈ Hom(K , C p ) C p as a non-archimedean analogue of the classical regulator map. This p -adic regulator, via the p -adic class num ber form ula prov ed by Colmez [ Col88 ], is in particular related to the residue at s = 1 of the p -adic Dedekind zeta function of K . The Leop oldt conjecture [ Leo62 , Leo75 ], stating that the image of the p -adic regulator map is a lattice of rank r 1 + r 2 − 1 , is how ev er still op en b eyond the case where K is an ab elian extension of Q or of an imaginary quadratic num ber ﬁeld (see [ NSW08 , Chapter X.3] for a review). The modern theory of regulators, as initiated by Beilinson [ Be ˘ ı84 ], recasts the regulator map of a num b er ﬁeld as an instance of higher Chern class maps to Deligne cohomology . In the p -adic setting, it was ﬁrst noted by Kato [ Kat87 , Remark 3 . 5 ] that syn tomic cohomology should b e a go o d analogue of Deligne cohomology , and such a cohomological approach to p -adic regulators was ∗ Institute for A dv anced Study , Princeton, tbouis@ias.edu † Sorbonne Université and Université Paris Cité, CNRS, IMJ-PRG, 75005 Paris, quentin@gazda.fr 1 TESS BOUIS AND QUENTIN GAZDA successfully used by Gros and Kurihara [ Gro90 , Gro94 ] to op en the study of the p -adic Beilinson conjectures. Recall that syntomic cohomology was ﬁrst introduced by F on taine–Messing [ FM87 ] in terms of p -adic de Rham cohomology , and that a well-behav ed in tegral reﬁnement of their deﬁnition was in tro duced b y Bhatt–Morrow–Sc holze [ BMS19 ] and Bhatt–Scholze [ BS22 ] in terms of prismatic cohomology . F ollowing the latter approach, the p -adic regulator map of a num b er ﬁeld K is then induced by the syntomic ﬁrst Chern class c syn 1 : G m (R) − → H 1 syn (R , Z p (1)) of R := O K , where the weigh t one syntomic complex of R is deﬁned, in terms of the Breuil–Kisin t wisted ∆ R { 1 } and Nygaard ﬁltered N ⩾ 1 ∆ R { 1 } prismatic cohomology of R , b y RΓ syn (R , Z p (1)) := ﬁb  N ⩾ 1 ∆ R { 1 } can − F rob ∆ p { 1 } − − − − − − − − − → ∆ R { 1 }  . The goal of this article is to introduce and study a decompleted reﬁnemen t of this story ( i.e. , where the cohomology groups are not p -complete), whic h in particular allows us to in terp olate the p -adic regulator maps b etw een diﬀeren t prime num b ers p . Deﬁnition 1.1 (Cyclosyntomic cohomology; see Deﬁnition 4.7 ) . Let K be a num ber ﬁeld, ∆ K b e the discriminan t of K , R b e the étale Z -algebra O K [∆ − 1 K ] , and d ⩾ 2 b e an in teger. The cyclosyntomic c omplex of R is the complex RΓ CycSyn (R , Z (1) ( d ) ) :=  N ⩾ 1 C R { 1 } can − F rob cyc d { 1 } − − − − − − − − − − → C ( d ) R { 1 }  in the deriv ed category D ( Z ) , where N ⩾ 1 C R { 1 } and C ( d ) R { 1 } are Z [ q ] -mo dules constructed as “decompleted versions” of the prismatic ob jects N ⩾ 1 ∆ R { 1 } and ∆ R { 1 } . T o deﬁne the ob jects app earing in Deﬁnition 1.1 , we use in a crucial wa y the recen t deﬁnition of Habiro ring H R of Garoufalidis–Scholze–Wheeler–Zagier [ GSWZ24 ], which can b e seen as a ring of “analytic functions in one v ariable q at roots of unity”. F or instance, H Z := lim m ⩾ 1 Z [ q ] ∧ ( q m − 1) where the limit of the completions is taken ov er the set of integers m ⩾ 1 partially ordered b y divisibility . W e refer to Section 3.1 for the deﬁnition of these Habiro rings, and to [ Sc h25 , W ag25 , GW25 ] for recen t progress tow ards a more general theory of Habiro cohomology . More precisely , w e introduce the notion of cyclotomic ring C R of a commutativ e ring R , as a reduced version of the Habiro ring H R (Examples 2.23 (2)), in terms of the theory of big q -Witt v ectors developed b y W agner [ W ag24 ]. F or instance, C Z := lim m ⩾ 1 Z [ q ] / ( q m − 1) . These cyclotomic rings C R are comm utative Z [ q ] -algebras, and w e deﬁne its cousins N ⩾ 1 C R { 1 } and C ( d ) R { 1 } as C R -mo dules which are compatible with the deﬁnitions of the Nygaard ﬁltered and Breuil–Kisin t wisted prismatic complexes [ BMS19 , BS22 , BL22 , AKN23 ]. Theorem A (First Chern class; see Sections 4.1 and 4.2 ) . L et K b e a numb er ﬁeld, and R b e the étale Z -algebr a O K [∆ − 1 K ] . F or every prime numb er p , ther e is a natur al p -adic realisation map RΓ CycSyn (R , Z (1) ( p ) ) − → RΓ q syn (R , Z p (1)) to the syntomic c ohomolo gy of R r elative to the q -prism ( Z p [ [ q − 1] ] , [ p ] q ) . Mor e over, for every inte ger d ⩾ 2 , ther e exists a natur al cyclosyntomic ﬁrst Chern class c CycSyn 1 : G m (R)[ − 1] − → RΓ CycSyn (R , Z (1) ( d ) ) which r e c overs, for d = p and after p ost-c omp osing with the pr evious p -adic r e alisation map, the syntomic ﬁrst Chern class of R . By the work of Bhatt–Lurie [ BL22 ], the theory of in tegral syntomic cohomology is equipp ed with a natural theory of Chern classes, which is built out of their notion of prismatic logarithm log ∆ . A derived reﬁnement dlog ∆ of their construction was more recently introduced by Mao [ Mao24 ], and we similarly deﬁne the notion of cyclotomic lo garithm dlog cyc : G m (R)[ − 1] − → N ⩾ 1 C R { 1 } 2 CYCLOSYNTOMIC REGULA TOR to construct the cyclosyntomic ﬁrst Chern class of Theorem A . The key to deﬁne this cyclotomic logarithm is a reﬁnement of Sulyma’s norm maps in prismatic cohomology [ Sul23 , Mao24 ], which w e call the cyclotomic norms (Prop osition 2.20 ). Note that, although one may deﬁne the ob jects N ⩾ 1 C R { 1 } and C R { 1 } purely in terms of the q -Witt vectors of [ W ag24 ], we do need to use the Habiro rings of [ GSWZ24 , W ag25 ] to deﬁne the previous cyclotomic logarithm (Construction 3.15 ). Relatedly , one can prov e that this cyclotomic logarithm cannot b e lifted to a non trivial notion of Habiro logarithm (Remark 3.19 ), which is our main reason for working at the level of cyclotomic rings rather than at the lev el of Habiro rings. Using these cohomological constructions, we now return to the n umber theoretic questions that motiv ated the study of these regulators. In the p -adic setting, an imp ortan t application of syntomic cohomology is Kato’s cohomological interpretation of explicit recipro cit y laws [ Kat91 ]. In this pap er, Kato gives in particular the fundamental expression c syn 1 : u 7− → 1 p log  u p F rob p ( u )  ∈ H 1 syn (R , Z p (1)) for the syntomic ﬁrst Chern class of a unit u ∈ R × ([ Kat91 , Corollary 2.9], see also [ Gro90 , Prop osition 4.1]). The following result is the analogous computation in our context. Theorem B (See Corollary 4.18 ) . L et K b e a numb er ﬁeld, R b e the étale Z -algebr a O K [∆ − 1 K ] , and d ⩾ 2 b e an inte ger. F or every unit u ∈ R × , the cyclosyntomic ﬁrst Chern class at u is given by c CycSyn 1 : u 7− → 1 d log  ˜ Π d ( ˜ u ) F rob Hab d ( ˜ u )  ∈ H 1 CycSyn (R , Z (1) ( d ) ) wher e ˜ Π d is a q -deformation of the d th p ower map (Pr op osition 3.17 ), F rob Hab d is the d th Habir o F r ob enius (Construction 3.7 ), and ˜ u := (Π Hab m ( u )) m ⩾ 1 is any se quenc e of Habir o lifts Π Hab m ( u ) in H R ,m / ( q m − 1) 2 of the cyclotomic norms Π m ( u ) ∈ H R ,m / ( q m − 1) of u (Pr op osition 2.20 ). Un winding the deﬁnitions, Theorem B b oils down to constructing suitable homotopies of the desired form mo dulo q m − 1 which are compatible b et ween diﬀerent integers m ⩾ 1 . W e do so in Construction 4.12 , after introducing the relev ant F rob enius map on the Habiro ring of R in Section 3.1 . Although sligh tly intractable for completely general computations, the previous formula for the (cyclo)syn tomic ﬁrst Chern class can b e made more explicit on the subgroup of cyclotomic units, i.e. , on the subgroup generated b y elemen ts of the form 1 − ζ ∈ K × for ζ a ro ot of unity . W e refer to [ W as97 , Chapter 8] for a survey on cyclotomic units, and simply p oint out that cyclotomic units form a subgroup of ﬁnite index in the group of units of a cyclotomic ﬁeld. In the context of regulators, the role of these cyclotomic units w as highlighted b y Deligne’s construction of a v ariation of mixed Hodge–T ate structure L o g H on P 1 ( C ) \ { 0 , 1 , ∞} , called the p olylo garithm variation , which splits at ro ots of unity [ Del89 ]. More concretely , this sp eciﬁc phenomenon at ro ots of unity implies that the sp ecial v alues of the ﬁrst p olylogarithm function Li 1 ( z ) := X k ⩾ 1 z k k at ro ots of unity can b e seen as natural classes in weigh t one Deligne cohomology [ BD94 , HW98 ]. A similar p -adic story of p olylogarithms was initiated by Coleman [ Col82 ] (see also [ Del89 , 3.2]), who in particular deﬁned the ﬁrst p -adic p olylogarithm function Li ( p ) 1 ( z ) := X k ⩾ 1 p ∤ k z k k whose sp ecial v alues at ro ots of unity were later prov ed to app ear naturally as classes in w eight one syntomic cohomology [ Gro90 ]. Giv en the prominent role of ro ots of unity in the context of Habiro cohomology , the following result can b e seen as an attempt to b etter understand the relation b etw een p olylogarithms and cyclotomic units. 3 TESS BOUIS AND QUENTIN GAZDA Theorem C (Firs t q -p olylogarithm as cyclosyn tomic Chern class; see Theorem 4.25 ) . L et K b e a numb er ﬁeld, R b e the étale Z -algebr a O K [∆ − 1 K ] , and d ⩾ 2 b e an inte ger. F or every r o ot of unity ζ ∈ R \ {± 1 } , the cyclosyntomic ﬁrst Chern class at 1 − ζ ∈ R × is given by c CycSyn 1 : 1 − ζ 7− → − Li ( d ) 1 ([ ζ ]) q ∈ H 1 CycSyn (R , Z (1) ( d ) ) wher e Li ( d ) 1 ( z ) q := X k ⩾ 1 d ∤ k z k [ k ] q is the ﬁrst q -p olylo garithm se en as a c onver gent q -p ower series in a suitable sense (Se ction 4.3 ) and [ ζ ] is a natur al se quenc e of Habir o lifts of the cyclotomic norms of ζ (Notation 4.20 ). In particular, via the p -adic realisation map of Theorem A , the cyclosyntomic cohomology classes of Theorem C also induce natural q -p olylogarithm classes in the syntomic cohomology of R relative to the q -prism. The pro of of Theorem C essentially relies on the fact that cyclotomic units u admit a natural explicit lift ˜ u to the Habiro ring, which mak es it p ossible to compute the Habiro F rob enius in Theorem B . W e end this introduction by mentioning t wo further p otential directions of research. First, it is p ossible to assemble the cyclosyn tomic complexes of Deﬁnition 1.1 for v arious d ⩾ 2 into a non trivial complex of ab elian groups. The resulting cyclosyntomic complex computes extensions of T ate ob jects in a category of C R -mo dules equipp ed with compatible d th F rob enius maps for all in tegers d ⩾ 2 . This phenomenon will b e discussed by the second author in an epilogue to this article [ Gaz26 ]. Moreo ver, we exp ect that the ﬁrst cyclosyntomic Chern class constructed here should b e part of a more general theory of higher Chern classes, which would relate the higher K -groups of a n umber ﬁe ld, cyclosyntomic complexes of higher weigh ts, and higher q -p olylogarithms. While such a general theory seems currently out of reach, we exp ect that ongoing progress in the theory of Habiro cohomology may provide insight for further progress in this direction. Notation. W e denote by ∆ K the discriminant of a num ber ﬁeld K . The q -analog 1 + q + · · · + q m − 1 ∈ Z [ q ] of an integer m ⩾ 1 is denoted b y [ m ] q . F or every integer e ⩾ 1 , we denote by Φ e ( q ) ∈ Z [ q ] the e th cyclotomic p olynomial. Given an integer m ⩾ 1 and a prime num ber p , we denote b y v p ( m ) the p -adic v aluation of m . Giv en integers a, b ⩾ 1 , we denote by ( a, b ) (resp. [ a, b ] ) the greatest common divisor (resp. the least common multiple) of a and b . Giv en a commutativ e ring R , we denote b y D (R) the derived ∞ -category of R -mo dules. W e refer to [ A T69 , Sections I.1 and I.4] for the deﬁnition and basic prop erties of Λ -rings. W e call a Λ -ring R p erfect if its Adams op erations ψ n : R → R are isomorphisms. Finally , ﬁx an integer N ⩾ 1 for the rest of this article, which represents a ﬁnite set of prime n umbers that will b e discarded in our constructions. Limits of the form lim m ⩾ 1 will typically b e tak en ov er the set of integers m ⩾ 1 that are coprime to N ( i.e. , ( m, N) = 1 ), partially ordered by divisibilit y . The choice of the integer N will b e imp ortan t only in Section 4 , where we will need the fact that the ro ot of unity ζ ∈ R is of order dividing N ; b efore this p oin t, the reader is welcome to assume that N = 1 . A ckno wledgements. W e would like to thank Michel Gros, P eter Scholze, F erdinand W agner, and Campb ell Wheeler for many inspiring and insightful discussions surrounding this pro ject. This pro ject has received funding from the SFB 1085 Higher In v arian ts, the Institute for A dv anced Study , and the Simons F oundation. This work also marks the launch of the ANR-25-CE40-3307-01 Al K traZ: Algebr aic K -the ory, tr ac e maps, and the Zagier c onje ctur e , of whic h b oth authors are mem b ers. W e are pleased to express our gratitude to the ANR and to the anonymous experts for their v aluable supp ort of the pro ject. 4 CYCLOSYNTOMIC REGULA TOR 2 No rm maps on q -Witt vecto rs In this section, we dev elop the necessary to ols on big q -Witt vectors that we will use in the next sections. More precisely , we construct norm maps on the big q -Witt v ectors of a commutativ e ring (Section 2.4 ), following the analogous theory of Angeltveit for classical big Witt v ectors [ Ang15 ], and introduce the notions of cyclotomic rings and of cyclotomic F rob enius (Section 2.5 ). 2.1 Review of Witt vecto rs In this subsection, w e review the classical theory of big Witt v ectors, and refer to the more extensiv e review [ Hes15 , Section 1] for more details. Let R b e a commutativ e ring. Given an integer m ⩾ 1 , the m -trunc ate d big Witt ve ctors W m (R) of R is the commutativ e ring characterised 1 b y the facts that it is abstractly isomorphic to the set Q e | m R , and that the map gh : W m (R) − → Y e | m R , x := ( x e ) e | m 7− → (gh e ( x )) e | m :=   X d | e dx e/d d   e | m where the commutativ e ring structure on the target is deﬁned component wise, is a ring homo- morphism. The map gh e : W m (R) → R is called the e th ghost c o or dinate 2 (whereas the map W m (R) → R , ( x e ) e | m 7→ x e is called the e th Witt c o or dinate ). Big Witt vectors are related to lifts of F rob enius via the following imp ortant lemma ([ Hes15 , Lemma 1.1]). Lemma 2.1 (Dwork’s lemma) . L et R b e a c ommutative ring, and m ⩾ 1 b e an inte ger. Assume that for every prime numb er p dividing m , ther e exists a ring homomorphism ϕ p : R → R /p v p ( m ) R such that ϕ p ( x ) ≡ x p mo dulo p R . 3 Then for every element ( x e ) e | m ∈ Q e | m R , the fol lowing ar e e quivalent: (1) ( x e ) e | m ∈ Q e | m R is in the image of the ghost map gh ; (2) for every prime numb er p dividing m and every inte ger e ⩾ 1 satisfying p | e | m , we have x e ≡ ϕ p ( x e/p ) mo dulo p v p ( e ) R . Deﬁnition 2.2 (Big Witt v ectors) . Let R b e a commutativ e ring. Given integers m, m ′ ⩾ 1 satisfying m | m ′ , there exists a natural r estriction map of commutativ e rings R m ′ /m : W m ′ (R) − → W m (R) , ( x e ) e | m ′ 7− → ( x e ) e | m , and the big Witt ve ctors W (R) of R is the commutativ e ring W (R) := lim m ⩾ 1 W m (R) where the transition maps are given by these restriction maps. Similarly , if one restricts to in tegers m ⩾ 1 which are the p ow ers of a given prime num ber p , then this limit deﬁnes the ring of p -typic al Witt ve ctors of R . Construction 2.3 (F rob enii and V erschiebungen) . Let R b e a comm utative ring. Given integers m, m ′ ⩾ 1 satisfying m | m ′ , and d := m ′ m , there also exist natural F r ob enius and V erschiebung maps F d : W m ′ (R) − → W m (R) V d : W m (R) − → W m ′ (R) of comm utative rings and of additiv e ab elian groups, resp ectiv ely . These maps are given, in ghost co ordinates, by F d : ( x e ) e | m ′ 7→ ( x de ) e | m and V d : ( x e ) e | m 7→ ( dx e d 1 d | e ) e | m ′ . They satisfy the relations F d ◦ V d = d , and F d ◦ V d ′ = V d ′ ◦ F d for every integer d ′ ⩾ 1 coprime to d . 1 The fact that this indeed characterises W m (R) is a consequence of Lemma 2.1 . 2 This terminology is justiﬁed by the fact that the ghost map gh : W m (R) → Q e | m R is injective for every ﬂat Z -algebra R (or, slightly more generally , for every commutative ring R which is p -torsionfree for every prime num ber p dividing m ). 3 The statement [ Hes15 , Lemma 1.1] is stated with the slightly stronger assumption that there exists a ring ho- momorphism φ p : R → R lifting the F rob enius endomorphism of R . How ev er, the pro of of [ Hes15 , Lemma 1.1] adapts readily to our context, which has the adv an tage to include all smo oth Z -algebras. 5 TESS BOUIS AND QUENTIN GAZDA Remark 2.4. The big Witt v ectors W m (R) of a commutativ e ring R corresp onds to the zeroth term of the big de Rham–Witt complex of R ([ Hes15 ]). Remark 2.5. The limit lim m ⩾ 1 W m (R) along the F rob enius maps of Construction 2.3 do es not usually ha v e a name. How ever, when R := O C p is the ring of in tegers of the p -adic complex num bers, and the limit is taken ov er the p ow ers of the prime num ber p , this construction corresp onds to F on taine’s p erio d ring A inf in p -adic Ho dge theory ([ BMS18 , Lemma 3.2]). Notation 2.6 (T eic hmüller lift) . Let R b e a comm utative ring, and m ⩾ 1 b e an integer. The T eichmül ler lift on R is the natural morphism of multiplicativ e monoids Π m : R − → W m (R) giv en by x 7→ ( x m/e ) e | m in ghost co ordinates. 4 In particular, the T eichm üller lift is a section of the ﬁrst ghost co ordinate gh 1 : W m (R) → R . 2.2 Review of q -Witt vectors In this subsection, we review W agner’s theory of q -Witt vectors, as introduced in [ W ag24 , Sec- tion 2]. Deﬁnition 2.7 (Big q -Witt vectors) . Let R b e a commutativ e ring. F or ev ery in teger m ⩾ 1 , the m -trunc ate d big q -Witt ve ctors q - W m (R) of R is the commutativ e Z [ q ] -algebra q - W m (R) := W m (R)[ q ] / I m where I m ⊆ W m (R)[ q ] is the ideal generated by (i) ( q e − 1) im (V m/e ) , for every integer e ⩾ 1 satisfying e | m , and (ii) im ([ e/d ] q d V m/e − V m/d ◦ F e/d ) , for all integers d, e ⩾ 1 satisfying d | e | m . Examples 2.8. Let m ⩾ 1 b e an integer. The big q -Witt vectors of R can b e made explicit in the follo wing cases. (1) ( R = Z ) q - W m ( Z ) ∼ = Z [ q ] / ( q m − 1) . More generally , for every p erfect Λ -ring R ( e.g. , R = Z p for an y prime n umber p , R any commutativ e Q -algebra, R any p erfect prism, or R = Z [T Q ] ), q - W m (R) ∼ = R[ q ] / ( q m − 1) ([ W ag24 , Corollary 2.37]). (2) ( R = O K [∆ − 1 K ] ) q - W m (R) ∼ = H R ,m / ( q m − 1) for every num b er ﬁeld K , where R = O K [∆ − 1 K ] and H R ,m is the m -truncated Habiro ring of R ([ W ag25 , Theorem 2.9]). (3) ( R = Z [T] ) q - W m ( Z [T]) ∼ = P e | m [ e ] q m/e Z [T m/e , q ] / ( q m − 1) ⊆ Z [T , q ] / ( q m − 1) ([ W ag24 , Prop osition 2.36]). Equiv alently , q - W m (R) is naturally iden tiﬁed with the k ernel of the op erator Z [T , q ] / ( q m − 1) − → Z [T , q ] / ( q m − 1) , f (T , q ) 7− → ( q − 1) ∇ q ( f (T , q )) where ∇ q ( f (T , q )) is the q -deriv ative of f (T , q ) (see [ W ag24 , Theorem 4.27] for a similar computation after ( q − 1) -completion). Construction 2.9 (F rob enii and V erschiebungen) . Let R b e a comm utative ring. Given integers m, m ′ ⩾ 1 satisfying m | m ′ , and d := m ′ m , the big q -Witt vectors of R are equipp ed with natural F r ob enius and V erschiebung maps F d : q - W m ′ (R) − → q - W m (R) V d : q - W m (R) − → q - W m ′ (R) of Z [ q ] -algebras and of Z [ q ] -mo dules, resp ectiv ely ([ W ag24 , pro of of Lemma 2.9]). These are c haracterised by their Z [ q ] -linearity and compatibility with the F rob enius and V erschiebung maps on big Witt vectors (Construction 2.3 ). They satisfy the relations F d ◦ V d = d and V d ◦ F d = [ d ] q m . 4 The T eichm üller lift of x is usually denoted by [ x ] . W e choose the notation Π m ( x ) to av oid a p otential confusion with the notation for q -integers [ k ] q ∈ Z [ q ] . 6 CYCLOSYNTOMIC REGULA TOR Remark 2.10. The big q -Witt v ectors q - W m (R) of a commutativ e ring R , together with their F rob enius and V erschiebung maps, satisfy a universal prop ert y: namely , the system of Z [ q ] -algebras ( q - W m (R)) m ⩾ 1 is the initial q -FV-system ov er R ([ W ag24 , Deﬁnition 2.8 and Lemma 2.9]). Also note that for every in teger m ⩾ 1 , we hav e q m − 1 ∈ I m (Deﬁnition 2.7 (i) for e = m ), so that q - W m (R) is a commutativ e Z [ q ] / ( q m − 1) -algebra. W arning 2.11. Despite the name, big q -Witt vectors are not a q -deformation of big Witt v ectors, i.e. , the natural morphism of commutativ e rings W m (R) → q - W m (R) / ( q − 1) is in general not an isomorphism ([ W ag24 , Remark 2.11]). Remark 2.12. The restriction maps R m ′ /m : W m ′ (R) → W m (R) of Deﬁnition 2.2 do not extend to morphisms of commutativ e Z [ q ] -algebras R m ′ /m : q - W m ′ (R) → q - W m (R) for general comm utative rings R ([ W ag24 , 2.14]). In particular, it is not p ossible to deﬁne the ring of big q -Witt vectors q - W (R) := lim m ⩾ 1 q - W m (R) as in Deﬁnition 2.2 . Instead, in ligh t of Remark 2.5 and of the relation established in [ W ag25 , Section 2.2] b etw een q -Witt v ectors and Habiro rings, we will rather consider the limit lim m ⩾ 1 q - W m (R) along the F rob enius maps of Construction 2.9 in this pap er, which we will typically refer to as cyclotomic rings (Section 2.5 ). 2.3 Alternative app roach to q -ghost maps In [ W ag24 , 2.13], W agner constructs q -ghost maps for the big q -Witt vectors as some explicit quotien t b y the images of some V erschiebung maps. In this subsection, w e giv e an alternativ e description of these q -ghost maps (Deﬁnition 2.14 ), using the universal property of big q -Witt v ectors (Remark 2.10 ). Construction 2.13 (Cyclotomic q -FV-system) . Let R b e a commutativ e ring. Here we deﬁne a structure of q -FV-system o v er R on the system of commutativ e Z [ q ] -algebras ( Q e | m R[ q ] / Φ e ( q )) m ⩾ 1 , in the sense of [ W ag24 , Deﬁnition 2.8]. T o do this, for every in teger m ⩾ 1 , let W m (R)[ q ] / ( q m − 1) − → Y e | m R[ q ] / Φ e ( q ) b e the morphism of commutativ e Z [ q ] -algebras characterised by the fact that it is given by x 7− → (gh m/e ( x )) e | m on the subring W m (R) ⊆ W m (R)[ q ] / ( q m − 1) . 5 Giv en integers m, m ′ ⩾ 1 satisfying m | m ′ , and d := m ′ m , let F d : Y e | m ′ R[ q ] / Φ e ( q ) − → Y e | m R[ q ] / Φ e ( q ) V d : Y e | m R[ q ] / Φ e ( q ) − → Y e | m ′ R[ q ] / Φ e ( q ) b e the morphisms given by F d : ( c e ( q )) e | m ′ 7→ ( c e ( q )) e | m and V d : ( c e ( q )) e | m 7→ ( dc e ( q ) 1 e | m ) e | m ′ , resp ectiv ely . These morphisms satisfy the relations F d ◦ V d = d and V d ◦ F d = [ d ] q m , and are compatible with the F rob enius and V erschiebung maps on big Witt vectors, th us endowing the system ( Q e | m R[ q ] / Φ e ( q )) m ⩾ 1 with the structure of q -FV-system ov er R . Deﬁnition 2.14 ( q -ghost maps) . Let R b e a comm utativ e ring. F or every integer m ⩾ 1 , the q -ghost map on big q -Witt vectors is the morphism of commutativ e Z [ q ] -algebras q - gh : q - W m (R) − → Y e | m R[ q ] / Φ e ( q ) , c 7− → ( c e ( q )) e | m := ( q - gh e ( c )) induced b y the universal prop erty of ( q - W m (R)) m ⩾ 1 as the initial q -FV-system o ver R (Remark 2.10 and Construction 2.13 ). The map q - gh e : q - W m (R) → R[ q ] / Φ e ( q ) is called the e th q -ghost c o or di- nate . 6 5 Note that the order of the ghost coordinates is reversed compared to the con v ention used for ghost maps in Section 2.1 . This is related to the fact that there are no restriction maps in the theory of big q -Witt vectors (Remark 2.12 ). 6 This terminology is justiﬁed by the result [ W ag24 , Lemma 2.23], stating that the q -ghost map q - gh : q - W m (R) → Q e | m R[ q ] / Φ e ( q ) is injective for every ﬂat Z -algebra R (or, slightly more generally , for every commutativ e ring R which is p -torsionfree for every prime num ber p dividing m ). 7 TESS BOUIS AND QUENTIN GAZDA Remark 2.15. The q -ghost maps of Deﬁnition 2.14 agree with the q -ghost maps deﬁned in [ W ag24 , 2.13]. Indeed, one can prov e that the q -ghost maps of [ W ag24 , 2.13] deﬁne a map of q -FV-systems o ver R from ( q - W m (R)) m ⩾ 1 to Q e | m R[ q ] / Φ e ( q ) (equipp ed with the q -FV-system structure from Construction 2.13 ), and such a map must b e unique by univ ersality (Remark 2.10 ). The following result is an analogue of Lemma 2.1 for big q -Witt vectors. Lemma 2.16 ( q -Dwork’s lemma) . L et R b e a c ommutative ring, and m ⩾ 1 b e an inte ger. Assume that for every prime numb er p dividing m , ther e exists a ring homomorphism ϕ p : R → R /p v p ( m ) R such that ϕ p ( x ) ≡ x p mo dulo p R . Then for every element ( c e ( q )) e | m ∈ Q e | m R[ q ] / Φ e ( q ) , the fol lowing ar e e quivalent: (1) ( c e ( q )) e | m ∈ Q e | m R[ q ] / Φ e ( q ) is in the image of the q -ghost map q - gh ; (2) ther e exists a lift ( ˜ c e ( q )) e | m ∈ Q e | m R[ q ] of ( c e ( q )) e | m ∈ Q e | m R[ q ] / Φ e ( q ) such that, for every prime numb er p and every inte ger e ⩾ 1 satisfying pe | m , we have ˜ c e ( q ) ≡ ϕ p (˜ c pe ( q )) in (R /p v p ( m/e ) )[ q ] , wher e ϕ p : R[ q ] → (R /p v p ( m/e ) )[ q ] is the unique Z [ q ] -line ar ring homomor- phism extending ϕ p : R → R /p v p ( m/e ) R . Pr o of. By construction, there is a commutativ e diagram of Z [ q ] -algebras W m (R)[ q ] Q e | m R[ q ] Q e | m R[ q ] / Φ e ( q ) q - W m (R) Q e | m R[ q ] / Φ e ( q ) gh q -gh where the top horizontal map is induced by the ghost map of Section 2.1 and the right vertical map is given by ( c e ( q )) e | m 7→ ( c m/e ( q )) e | m . The desired result is then a consequence of Dwork’s lemma (Lemma 2.1 ). Remark 2.17. Let R b e a ﬂat Z -algebra. Lemma 2.16 suggests that big q -Witt vectors q - W m (R) are a sort of “F rob enius t wisted” v ersion of R[ q ] / ( q m − 1) . 7 More precisely , one can prov e that the image of the natural pro jection map R[ q ] / ( q m − 1) → Q e | m R[ q ] / Φ e ( q ) is giv en by the elements ( c e ( q )) e | m ∈ Q e | m R[ q ] / Φ e ( q ) for whic h there exists a lift (˜ c e ( q )) e | m ∈ Q e | m R[ q ] suc h that, for ev ery prime n umber p and every integer e ⩾ 1 satisfying pe | m , w e hav e ˜ c e ( q ) ≡ ˜ c pe ( q ) in (R /p v p ( m/e ) )[ q ] . Notation 2.18 ( q -T eichm üller lift) . Let R b e a commutativ e ring, and m ⩾ 1 b e an in teger. The q -T eichmül ler lift on R is the morphism of multiplicativ e monoids Π m : R − → q - W m (R) deﬁned as the composition of the T eichm üller lift Π m : R → W m (R) of Notation 2.6 with the canonical map W m (R) → q - W m (R) . In particular, the q -T eichm üller lift is a section of the ﬁrst q -ghost co ordinate q - gh 1 : q - W m (R) → R[ q ] / Φ 1 ( q ) ∼ = R , and is in general given by x 7→ ( x m/e ) e | m in q -ghost co ordinates. 2.4 No rm maps on q -Witt vectors In [ Ang15 ], Angeltv eit constructs multiplicativ e norm maps on big Witt vectors, as a natural generalisation of the T eichm üller lift. In this subsection, w e pro ve that big q -Witt vectors similarly admit multiplicativ e norm maps Π m ′ /m : q - W m (R) → q - W m ′ (R) (Prop osition 2.20 ). Lemma 2.19. L et R b e a c ommutative ring, and m, m ′ ⩾ 1 b e inte gers satisfying m | m ′ . Assume that for every prime numb er p dividing m , ther e exists a ring homomorphism ϕ p : R → R /p v p ( m ′ ) R such that ϕ p ≡ x p mo dulo p R . Then for every element c ∈ q - W m (R) with q -ghost c o or dinates ( c e ( q )) e | m , the element Π m ′ /m (( c e ( q )) e | m ) :=  c ( e,m ) ( q e ( e,m ) ) m ′ [ e,m ]  e | m ′ ∈ Y e | m ′ R[ q ] / Φ e ( q ) is in the image of the q -ghost map q - gh : q - W m ′ (R) → Q e | m ′ R[ q ] / Φ e ( q ) . 7 This is directly related to the deﬁnition of the Habiro rings of a num b er ﬁeld in [ GSWZ24 , W ag25 ], where the Habiro ring H R ,m of an étale Z -algebra R is a sort of “F rob enius twisted” version of R[ q ] ∧ ( q m − 1) . 8 CYCLOSYNTOMIC REGULA TOR Pr o of. First note that the morphism of multiplicativ e monoids Π m ′ /m : Y e | m R[ q ] / Φ e ( q ) → Y e | m ′ R[ q ] / Φ e ( q ) is well-deﬁned, b ecause Φ e ( q ) divides Φ ( e,m ) ( q e ( e,m ) ) in the commutativ e ring Z [ q ] . Moreov er, for an y integers m, m ′ , m ′′ ⩾ 1 satisfying m | m ′ | m ′′ , we hav e the equality Π m ′′ /m ′ ◦ Π m ′ /m = Π m ′′ /m . Indeed, this is a consequence of the series of equalities c (( e,m ′ ) ,m )  q ( e,m ′ ) (( e,m ′ ) ,m ) · e ( e,m ′ )  m ′ [( e,m ′ ) ,m ] · m ′′ [ e,m ′ ] = c ( e,m ) ( q e ( e,m ) ) m ′ [( e,m ′ ) ,m ] · m ′′ [ e,m ′ ] = c ( e,m ) ( q e ( e,m ) ) m ′′ [ e,m ] where the equalit y m ′ [( e,m ′ ) ,m ] · [ e,m ′ ] = 1 [ e,m ] follo ws from the fact that for an y integers a, b, c ⩾ 0 satisfying c ⩽ b , w e hav e the equality b − max(min( a, b ) , c ) − max( a, b ) = − max( a, c ) . In particular, it suﬃces to pro v e the desired claim when m ′ /m is a prime num b er, whic h we assume from now on. Let ( c e ( q )) e | m ∈ Q e | m R[ q ] / Φ e ( q ) b e an element in the image of the q -ghost map. W e use the q -Dwork lemma (Lemma 2.16 ) to pro ve that Π m ′ /m (( c e ( q )) e | m ) ∈ Q e | m ′ R[ q ] / Φ e ( q ) is in the image of the q -ghost map. Let ( ˜ c e ( q )) e | m ∈ Q e | m R[ q ] b e a lift of ( c e ( q )) e | m ∈ Q e | m R[ q ] / Φ e ( q ) such that, for every prime num ber p and every integer e ⩾ 1 satisfying pe | m , w e hav e ˜ c e ( q ) ≡ ϕ p (˜ c pe ( q )) in (R /p v p ( m/e ) )[ q ] . It then suﬃces to prov e that, for every prime num ber p and every integer e ⩾ 1 satisfying pe | m ′ , we hav e ϕ p  ˜ c ( pe,m ) ( q pe ( pe,m ) ) m ′ [ pe,m ]  ≡ ˜ c ( e,m ) ( q e ( e,m ) ) m ′ [ e,m ] in (R /p v p ( m ′ /e ) )[ q ] . W e distinguish several cases. Assume ﬁrst that v p ( e ) > v p ( m ) . Because m ′ /m is a prime num b er, this implies that m ′ /m = p , so the condition pe | m ′ can not b e satisﬁed in this case, and the desired statement is v acuously true. Assume now that v p ( e ) = v p ( m ) . If v p ( e ) = v p ( m ′ ) , then v p ( m ′ /e ) = 0 and the desired statement is again v acuously true, so we assume that v p ( e ) = v p ( m ) < v p ( m ′ ) . Because m ′ /m is a prime n umber, this implies that m ′ = pm and that v p ( m ′ /e ) = 1 . The desired statement then follows from the series of equalities ϕ p  ˜ c ( pe,m ) ( q pe ( pe,m ) ) m ′ [ pe,m ]  = ϕ p  ˜ c ( e,m ) ( q pe ( e,m ) )  m ′ p · [ e,m ] ≡ ˜ c ( e,m ) ( q e ( e,m ) ) p · m ′ p · [ e,m ] = ˜ c ( e,m ) ( q e ( e,m ) ) m ′ [ e,m ] where the congruence holds mo dulo p , using that ϕ p : R → R /p v p ( m ′ ) R is a lift of F rob enius. Assume ﬁnally that v p ( e ) < v p ( m ) . In this case, the desired statement follo ws similarly from the series of equalities ϕ p  ˜ c ( pe,m ) ( q pe ( pe,m ) ) m ′ [ pe,m ]  = ϕ p  ˜ c p · ( e,m ) ( q pe p · ( e,m ) )  m ′ [ e,m ] ≡ ˜ c ( e,m ) ( q e ( e,m ) ) m ′ [ e,m ] = ˜ c ( e,m ) ( q e ( e,m ) ) m ′ [ e,m ] where the congruence holds mo dulo p v p ( m ′ /e ) , as a consequence of the assumption on (˜ c e ( q )) e | m . More precisely , either m ′ /m is diﬀerent from p , in whic h case v p ( m ′ /e ) = v p ( m/ ( e, m )) and this is indeed a consequence of the assumption, or m ′ /m = p , in whic h case we use the fact that a ≡ b mo dulo p v p ( m/e ) implies a p ≡ b p mo dulo p v p ( m ′ /e ) . Prop osition 2.20 (Cyclotomic norms) . L et R b e a c ommutative ring. Then for any inte gers m, m ′ ⩾ 1 satisfying m | m ′ , ther e exists a natur al morphism of multiplic ative monoids Π m ′ /m : q - W m (R) − → q - W m ′ (R) given by ( c e ( q )) e | m 7→ ( c ′ e ( q )) e | m ′ :=  c ( e,m ) ( q e ( e,m ) ) m ′ [ e,m ]  e | m ′ in q -ghost c o or dinates. This morphism is uniquely determine d by the fact that it is functorial in R and given by the pr evious formula on q -ghost c o or dinates. Pr o of. First assume that R = Z [T i , i ∈ I] is a polynomial Z -algebra (p otentially in inﬁnitely man y v ariables). F or every prime num ber p , let ϕ p : R → R b e the lift of F rob enius given b y T i 7→ T p i . The Z -algebra R is ﬂat, so the q -ghost maps of Deﬁnition 2.14 are injective ([ W ag24 , Lemma 2.23]). 9 TESS BOUIS AND QUENTIN GAZDA The form ula in q -ghost co ordinates is moreov er m ultiplicative by construction, so it suﬃces to prov e that an elemen t ( c e ( q )) e | m in the image of the q -ghost map is sent to an element ( c ′ e ( q )) e | m ′ in the image of the q -ghost map. Using the lifts of F rob enius ϕ p , this is a consequence of Lemma 2.19 . Assume now that R is a general commutativ e ring. Let P ↠ R b e a surjectiv e morphism of comm utative rings, where P is a p olynomial Z -algebra. There is a natural diagram of commutativ e Z [ q ] -algebras q - W m (P) q - W m ′ (P) q - W m (R) q - W m ′ (R) Π m ′ /m Π m ′ /m where the vertical maps are surjective ([ W ag24 , Corollary 2.29]). In particular, given an elemen t c ∈ q - W m (R) , an y lift ˜ c ∈ q - W m (P) induces an elemen t Π m ′ /m ( c ) ∈ q - W m ′ (R) , whic h is compatible with the desired formula on q -ghost co ordinates by the previous paragraph and the functoriality of q -ghost maps. Note that this construction a priori dep ends on the surjection P ↠ R and on the lift ˜ c ∈ q - W m (R) . T o pro ve that it is indep endent of these c hoices, let P 1 ↠ R , ˜ c 1 ∈ q - W m (P 1 ) and P 2 ↠ R , ˜ c 2 ∈ q - W m (P 2 ) b e tw o such choices. Let P := P 1 × P 2 . By [ W ag24 , Corollary 2.29], there is a natural isomorphism of commutativ e Z [ q ] -algebras q - W m (P) ∼ = q - W m (P 1 ) × q - W m (P 2 ) . In particular, there exists an elemen t ˜ c ∈ q - W m (R) mapping to ˜ c 1 ∈ q - W m (P 1 ) and ˜ c 2 ∈ q - W m (P 2 ) via the natural pro jection maps, and the element Π m ′ /m ( c ) ∈ q - W m ′ (R) is well-deﬁned. Remark 2.21. Let R b e a commutativ e ring. F or any integers m, m ′ , m ′′ ⩾ 1 satisfying m | m ′ | m ′′ , the cyclotomic norms of Prop osition 2.20 satisfy the identit y Π m ′′ /m ′ ◦ Π m ′ /m ◦ Π m ′′ /m . This is indeed a consequence of the ﬁrst paragraph of the proof of Lemma 2.19 . In particular, for any in tegers m, m ′ ⩾ 1 satisfying m | m ′ , we hav e the equality Π m ′ /m ◦ Π m = Π m ′ , where the notation here is compatible with that of the T eichm üller lift (Notation 2.18 ). 2.5 Cyclotomic rings In this subsection, w e introduce the notion of cyclotomic ring C R of a comm utativ e ring R (Deﬁnition 2.22 ). Deﬁnition 2.22 (Cyclotomic ring) . Let R b e a commutativ e ring. The cyclotomic ring of R is the commutativ e Z [ q ] -algebra C R := lim m ⩾ 1 q - W m (R) where the limit is taken along the F rob enius maps of Construction 2.9 . Examples 2.23. F ollowing Examples 2.8 , the cyclotomic ring of R can b e made explicit in the follo wing cases. (1) ( R = Z ) C Z ∼ = lim m ⩾ 1 Z [ q ] / ( q m − 1) where the limit is taken along the natural pro jection maps, and similarly for any p erfect Λ -ring R . (2) ( R = O K [∆ − 1 K ] ) C R ∼ = lim m ⩾ 1 H R ,m / ( q m − 1) for every num ber ﬁeld K , where R = O K [∆ − 1 K ] and the limit is taken along the natural pro jection maps ([ W ag25 , Remark 2.9]). (3) ( R = Z [T] ) C Z [T] ∼ = C Z (Examples 2.8 (3)). Remark 2.24. F or general commutativ e rings R , it may happ en that the derived inv erse limit Rlim m ⩾ 1 q - W m (R) , where the limit is taken along the F rob enius maps of Construction 2.9 , is not concen trated in degree zero. Ho wev er, if R is étale ov er Z , or ov er a p erfect Λ -ring, then one can use the q -Dwork lemma (Lemma 2.16 ) to prov e that the F rob enius maps of Construction 2.9 are surjectiv e, hence the deriv ed and classical limits coincide. Lemma 2.25. F or every prime numb er p and every inte ger r ⩾ 0 , the element q p r − 1 of Z [ q ] b elongs to the ide al ( p, q − 1) r ⊆ Z [ q ] . In p articular, for every p -torsionfr e e c ommutative ring R , the ( p, q − 1) -c ompletion of the c ommutative Z [ q ] -algebr a lim r ⩾ 0 R[ q ] / ( q p r − 1) is natur al ly identiﬁe d with R ∧ p [ [ q − 1] ] . 10 CYCLOSYNTOMIC REGULA TOR Pr o of. The second claim is a direct consequence of the ﬁrst claim. T o pro v e the ﬁrst claim, it suﬃces to prov e that [ p r ] q ∈ ( p, q − 1) r , b ecause q p r − 1 = ( q − 1) · [ p r ] q . This statement for [ p r ] q is a consequence of the formula [ p r ] q = Q r − 1 i =0 [ p ] q p i and of the congruence [ p ] q p i ≡ p mo dulo ( q − 1) for every integer i ⩾ 0 . Remark 2.26 ( p -adic realisation) . Let R b e a ﬂat Z -algebra, and p b e a prime num ber. If R ∧ p admits a lift of F rob enius ϕ p ( i.e. , a ring homomorphism ϕ p : R ∧ p → R ∧ p whose reduction mo dulo p is the F rob enius endomorphism of R /p ), then there is a natural morphism of comm utative Z [ q ] -algebras C R − → R ∧ p [ [ q − 1] ] . Using Lemma 2.16 and Remark 2.17 , this morphism is induced by the composite morphism of comm utative Z [ q ] -algebras Y e ⩾ 1 R[ q ] / Φ e ( q ) − → Y r ⩾ 0 R[ q ] / Φ p r ( q ) − → Y r ⩾ 0 R[ q ] / Φ p r ( q ) where the ﬁrst map is the natural pro jection ( c e ( q )) e ⩾ 1 7→ ( c p r ( q )) r ⩾ 0 and the second map is giv en by ( c p r ( q )) r ⩾ 0 7→ ( ϕ r p ( c p r ( q ))) . 8 More precisely , the image of the cyclotomic ring C R via this comp osite morphism is given by lim r ⩾ 0 R[ q ] / ( q p r − 1) , and the desired morphism is obtained by p ost-comp osing with the ( p, q − 1) -completion lim r ⩾ 0 R[ q ] / ( q p r − 1) → R ∧ p [ [ q − 1] ] (Lemma 2.25 ). Note that this map C R → R ∧ p [ [ q − 1] ] depends on the choice of F rob enius lift ϕ p ( e.g. , if R is smo oth o ver Z ). Also note that if R is étale ov er Z , then such a F rob enius lift ϕ p on R ∧ p is unique. W e now introduce the notion of cyclotomic F rob enius on these cyclotomic rings. T o do so, we will use the following v ariant of Deﬁnition 2.7 . Deﬁnition 2.27 (Big q -Witt v ectors at d ) . Let R b e a commutativ e ring, and d ⩾ 1 b e an in teger. F or every integer m ⩾ 1 , the m -trunc ate d big q -Witt ve ctors at d of R is the commutativ e Z [ q ] -algebra q - W ( d ) m (R) := W m (R)[ q ] / I ( d ) m where I ( d ) m ⊆ W m (R)[ q ] is the ideal generated by (i)  Q d | e ′ | e Φ e ′ ( q )  im (V m/e ) , for every integer e ⩾ 1 satisfying e | m , 9 and (ii) im ([ f /e ] q e V m/f − V m/e ◦ F f /e ) , for all integers e, f ⩾ 1 satisfying d | e | f | m . Remark 2.28. F or ev ery comm utative ring R and in teger m ⩾ 1 , we ha ve q - W (1) m (R) = q - W m (R) . Moreo ver, for every in teger d ⩾ 1 , one can chec k that I m ⊆ I ( d ) m . Remark 2.29. Let R b e a commutativ e ring, and d ⩾ 1 b e an integer. Arguin g as in [ W ag24 , Lemma 2.9], the system ( q - W ( d ) m (R)) m ⩾ 1 is the initial q -FV-system at d over R , i.e. , the initial system of comm utativ e Z [ q ] -algebras (W ( d ) m ) m ⩾ 1 indexed b y in tegers m ⩾ 1 , together with the follo wing structure: (i) for every integer m ⩾ 1 , a morphism of Z [ q ] -algebras W m (R)[ q ] / Q d | e | m Φ e ( q ) → W ( d ) m (in particular W ( d ) m = 0 unless d | m ); (ii) for any integers m, m ′ ⩾ 1 satisfying d | m | m ′ , morphisms F m ′ /m : W ( d ) m ′ → W ( d ) m of Z [ q ] -algebras and V m ′ /m : W ( d ) m → W ( d ) m ′ of Z [ q ] -mo dules, compatible with the F rob enius and V erschiebung maps on big q -Witt vectors (Construction 2.9 ), and such that F m ′ /m ◦ V m ′ /m = m ′ /m and V m ′ /m ◦ F m ′ /m = [ m ′ /m ] q m . Deﬁnition 2.30 (Canonical maps) . Let R b e a commutativ e ring, and d ⩾ 1 b e an integer. F or ev ery integer m ⩾ 1 , the c anonic al map can : q - W m (R) − → q - W ( d ) m (R) is the morphism of commutativ e Z [ q ] -algebras induced b y the inclusion of ideals I m ⊆ I ( d ) m of W m (R)[ q ] (Deﬁnitions 2.7 and 2.27 ). 8 F ollowing Lemma 2.16 , we denote by φ p : R[ q ] / Φ p r ( q ) → R[ q ] / Φ p r ( q ) the unique Z [ q ] -linear ring homomorphism extending φ p : R ∧ p → R ∧ p . 9 Note here that Q d | e ′ | e Φ e ′ ( q ) = 1 if d ∤ e . 11 TESS BOUIS AND QUENTIN GAZDA Remark 2.31. Let R b e a commutativ e ring, and d ⩾ 1 b e an integer. An y q -FV-system at d o ver R is in particular a q -FV-system ov er R , and the canonical maps of Deﬁnition 2.30 are induced b y the universal prop ert y of ( q - W m (R)) m ⩾ 1 (Remarks 2.10 ). Remark 2.32 ( q -ghost maps at d ) . Let R b e a commutativ e ring, and d ⩾ 1 b e an in teger. F ollo wing Construction 2.13 and Deﬁnition 2.14 , there is a natural q -ghost map q - gh : q - W ( d ) m (R) − → Y d | e | m R[ q ] / Φ e ( q ) , c 7− → ( c e ( q )) d | e | m on big q -Witt vectors at d for every integer m ⩾ 1 , which is compatible with the canonical map of Deﬁnition 2.30 . Prop osition 2.33 (Cyclotomic F rob enii) . L et R b e a c ommutative ring, and d ⩾ 1 b e an inte ger. Then for every inte ger m ⩾ 1 , ther e exists a natur al morphism of c ommutative rings F rob cyc d : q - W m (R) − → q - W ( d ) dm (R) given by ( c e ( q )) e | m 7→ ( c ′ e ( q )) d | e | dm := ( c e/d ( q d )) d | e | dm in q -ghost c o or dinates. This morphism is uniquely determine d by the fact that it is functorial in R and by the pr evious formula on q -ghost c o or dinates. Pr o of. First note that the morphism of comm utative rings Q e | m R[ q ] / Φ e ( q ) → Q d | e | dm R[ q ] / Φ e ( q ) giv en by ( c e ( q )) e | m 7→ ( c e/d ( q d )) d | e | dm is well-deﬁned, b ecause Φ e ( q ) divides Φ e/d ( q d ) in Z [ q ] . If R is a polynomial Z -algebra, then R admits lifts of F rob enius ϕ p : R → R for all prime n umbers p . In this case, we use the q -Dwork lemma (Lemma 2.16 ) 10 to prov e the desired result. T o do so, let ( c e ( q )) e | m ∈ Q e | m R[ q ] / Φ e ( q ) b e an elemen t in the image of the q -ghost map, and (˜ c e ( q )) e | m ∈ Q e | m R[ q ] b e a lift of this element such that, for every prime num b er p and ev ery in teger e ⩾ 1 satisfying pe | m , we ha ve ˜ c e ( q ) ≡ ϕ p (˜ c pe ( q )) in (R /p v p ( m/e ) )[ q ] . It suﬃces to prov e that, for every prime num b er p and every integer e ⩾ 1 satisfying d | e and pe | dm , we hav e ˜ c e/d ( q d ) ≡ ϕ p (˜ c pe/d ( q d )) in (R /p v p ( dm/e ) )[ q ] . This congruence holds as a conse- quence of the assumption on (˜ c e ( q )) e | m , where we use that p v p ( m/ ( e/d )) = p v p ( dm/e ) . If R is a general commutativ e ring, then arguing as in the second part of the pro of of Prop osi- tion 2.20 implies the desired result. The following v arian t of Deﬁnition 2.27 will b e used in Section 4 . V ariant 2.34 (Cyclotomic ring at d ) . Let R b e a commutativ e ring, and d ⩾ 1 b e an integer. F or ev ery integer m ⩾ 1 , the m -trunc ate d cyclotomic ring of R at d is the commutativ e Z [ q ] -algebra C ( d ) R ,m := q - W ( d ) m (R)  Φ e ( q ) − 1   d ∤ e > 1  where the localisation is indexed by integers e > 1 satisfying d ∤ e and e | m . In particular, the morphisms of Deﬁnition 2.30 and Prop osition 2.33 on big q -Witt vectors induce natural morphisms can : C R ,m − → C ( d ) R ,m F rob cyc d : C R ,m − → C ( d ) R ,dm of commutativ e rings, which w e call the c anonic al map and the d th cyclotomic F r ob enius . Similarly , the cyclotomic ring of R at d is the comm utativ e Z [ q ] -algebra C ( d ) R deﬁned as the limit of the truncated cyclotomic rings C ( d ) R ,m o ver integers m ⩾ 1 satisfying ( m, N) = 1 , along the F rob enius maps F m ′ /m of Remark 2.29 . Note in particular that C ( d ) R = 0 if ( d, N) > 1 . The map F rob cyc d in V ariant 2.34 is called cyclotomic F r ob enius (despite the map F d in Construc- tions 2.3 and 2.9 already b eing called F rob enius) b ecause of the following remarks, which states that F rob cyc d b eha ves as a “universal q -lift of F rob enius”, at least when R can b e equipp ed with a Λ -ring structure. 10 Note that the statement and the proof of the q -Dwork lemma also hold for big q -Witt vectors at d . 12 CYCLOSYNTOMIC REGULA TOR Remark 2.35. Let R b e a Λ -ring with Adams op erations ( ψ n ) n ⩾ 1 , and d ⩾ 1 b e an integer. F or ev ery integer m ⩾ 1 , there are natural commutativ e diagrams of commutativ e rings q - W m (R) R[ q ] / ( q m − 1) q - W ( d ) m (R) R[ q ] / Q d | e | m Φ e ( q ) c m can can c ( d ) m q - W m (R) R[ q ] / ( q m − 1) q - W ( d ) dm (R) R[ q ] / Q d | e | dm Φ e ( q ) c m F rob cyc d ψ d ⊗ ( q 7→ q d ) c ( d ) dm where the top horizontal maps are deﬁned in [ W ag24 , Lemma 2.34], the b ottom horizon tal maps are deﬁned as in [ W ag24 , Lemma 2.34], and the left vertical maps are deﬁned in Deﬁnition 2.30 and Prop osition 2.33 . In particular, there are similar comm utativ e diagrams for the cyclotomic rings of Deﬁnition 2.22 and V ariant 2.34 , and these commutativ e diagrams are compatible with the isomorphism of Examples 2.8 (1) when the Λ -ring R is p erfect. 3 The cyclotomic loga rithm Let K be a n umber ﬁeld, with ring of integers O K and discriminant ∆ K . In this section, w e deﬁne the (derive d) cyclotomic lo garithm dlog cyc : G m (R)[ − 1] − → N ⩾ 1 C R { 1 } of R := O K [∆ − 1 K ] (Deﬁnition 3.21 ). Our construction is a global analogue of Bhatt–Lurie’s prismatic logarithm [ BL22 , Section 2]. T o bypass the use of quasisyntomic descen t used in their construction, w e follow Mao’s alternative approach to the prismatic logarithm [ Mao24 , Section 2]. 3.1 F rob enius maps on the Habiro ring of a numb er ﬁeld In this subsection, w e recall the deﬁnition of the Habiro rings from the recent work of Garoufalidis– Sc holze–Wheeler–Zagier [ GSWZ24 ], and construct an analogue of the cyclotomic F rob enius maps of the previous section in this context (Construction 3.7 ). Deﬁnition 3.1 (Habiro ring, [ GSWZ24 , W ag25 ]) . Let R be an étale Z -algebra and, for ev ery prime n umber p , let ϕ R ∧ p b e the unique lift of F rob enius of the formally étale Z p -algebra R ∧ p . The Habir o ring of R is the commutativ e ring H R :=    ( f e ( q )) e ⩾ 1 ∈ Y e ⩾ 1 R[ q ] ∧ Φ e ( q )     f e ( q ) = ϕ p ( f pe ( q )) in R[ q ] ∧ (Φ e ( q ) , Φ pe ( q ))    where the condition is taken ov er all prime num b ers p and integers e ⩾ 1 , and where the endo- morphism ϕ p : R[ q ] ∧ (Φ e ( q ) , Φ pe ( q )) → R[ q ] ∧ (Φ e ( q ) , Φ pe ( q )) is deﬁned as the ( p, Φ e ( q )) -completion 11 of ϕ R ∧ p ⊗ id : R ∧ p ⊗ Z Z [ q ] → R ∧ p ⊗ Z Z [ q ] . W e will mostly use the following truncated v arian t of Deﬁnition 3.1 , which already app ears in [ W ag25 , 2.7]. V ariant 3.2 (T runcated Habiro ring) . Let R b e an étale Z -algebra, and m ⩾ 1 b e an integer. The m -trunc ate d Habir o ring of R is the commutativ e Z [ q ] -algebra H R ,m :=    ( f e ( q )) e | m ∈ Y e | m R[ q ] ∧ Φ e ( q )     f e ( q ) = ϕ p ( f pe ( q )) in R[ q ] ∧ (Φ e ( q ) , Φ pe ( q ))    where the condition is taken o ver all prime n umbers p and in tegers e ⩾ 1 such that pe | m . Remark 3.3. F or every étale Z -algebra R , w e hav e H R = lim m ⩾ 1 H R ,m . 11 Here we use the equality of ideals (Φ e ( q ) , Φ pe ( q )) = ( p, Φ e ( q )) in the commutativ e ring Z [ q ] ([ W ag24 , Lemma 2.1]). 13 TESS BOUIS AND QUENTIN GAZDA V ariant 3.4 (T runcated Habiro ring at d ) . Let R b e an étale Z -algebra, and d ⩾ 1 b e an in teger. F or every integer m ⩾ 1 , the m -trunc ate d Habir o ring of R at d is the commutativ e Z [ q ] -algebra H ( d ) R ,m :=    ( c e ) d | e | m ∈ Y d | e | m R[ q ] ∧ Φ e ( q )     c e = ϕ p ( c pe ) in R[ q ] ∧ (Φ e ( q ) , Φ pe ( q ))     Φ e ( q ) − 1   d ∤ e > 1  where the condition is taken ov er all prime n umbers p and integers e ⩾ 1 suc h that d | e , and the lo calisation is indexed by integers e > 1 satisfying d ∤ e and e | m . Remark 3.5. Inv erting the cyclotomic p olynomials Φ e ( q ) for d ∤ e > 1 in V ariant 3.4 seems to b e necessary to deﬁne the cyclosyntomic ﬁrst Chern class in Section 4.2 . Remark 3.6. Let R b e an étale Z -algebra. F or every integer m ⩾ 1 , it was prov ed b y W agner that there is a natural isomorphism of commutativ e Z [ q ] -algebras H R ,m / ( q m − 1) ∼ = q - W m (R) ([ W ag25 , Theorem 2 . 9 ]). Moreov er, for every in teger m ′ ⩾ 1 satisfying m | m ′ , this identiﬁcation induces a natural commutativ e diagram of Z [ q ] -algebras H R ,m ′ q - W m ′ (R) H R ,m q - W m (R) can F m ′ /m where the left v ertical map is the natural pro jection map and the righ t vertical map is the F rob enius map on big q -Witt v ectors ([ W ag25 , Remark 2.10]). Construction 3.7 (Habiro F rob enii) . Let R b e an étale Z -algebra, and d ⩾ 1 b e an integer. The d th F r ob enius on the Habir o ring H R ,m is the ring homomorphism F rob Hab d : H R ,m − → H ( d ) R ,dm giv en by ( f e ( q )) e | m 7→ ( f ′ e ( q )) d | e | dm := ( f e/d ( q d )) d | e | dm . More precisely , this map is induced by the comp osite map Y e | m R[ q ] ∧ Φ e ( q ) − → Y d | e | dm R[ q ] ∧ Φ e/d ( q d ) − → Y d | e | dm R[ q ] ∧ Φ e ( q ) where the ﬁrst map is given by ( f e ( q )) e | m 7→ ( f e/d ( q d )) d | e | dm and the second map is the Φ e ( q ) -com- pletion on each factor. 12 As a consequence of the series of equalities f ′ e ( q ) = f e/d ( q d ) = ϕ p ( f pe/d ( q d )) = ϕ p ( f ′ pe ( q )) in R[ q ] ∧ (Φ e ( q ) , Φ pe ( q )) , the image of H R ,m b y this comp osite map is indeed contained in H ( d ) R ,dm . Remark 3.8. Let R be an étale Z -algebra, m ⩾ 1 and r ⩾ 0 be integers, and p b e a prime num b er. F ollo wing Lemma 2.25 (and its pro of ) and Remark 2.26 , there is a natural morphism of commu- tativ e Z [ q ] -algebras C ( p ) pm → R[ q ] / ( p, q − 1) r . This morphism ﬁts naturally in the commutativ e diagram of commutativ e rings H R ,m C R ,m R[ q ] / ( p, q − 1) r H ( p ) R ,pm C ( p ) R ,pm R[ q ] / ( p, q − 1) r F rob Hab p F rob cyc p F rob ∆ p where the righ t vertical map is deﬁned as the unique morphism of comm utative rings induced b y ϕ p on R /p r and sending q to q p . Here we use that Φ e ( q ) is inv ertible in R[ q ] / ( p, q − 1) r for ev ery integer e ⩾ 1 satisfying p ∤ e . 12 Note that this second map is well-deﬁned as a consequence of the fact that Φ e ( q ) divides Φ e/d ( q d ) in Z [ q ] . 14 CYCLOSYNTOMIC REGULA TOR Lemma 3.9. L et R b e an étale Z -algebr a, and m ⩾ 1 b e an inte ger. Then for any se quenc e of nonne gative inte gers ( n e ) e | m , the morphism of c ommutative Z [ q ] -algebr as q - gh : H R ,m / Y e | m Φ e ( q ) n e − → Y e | m R[ q ] / Φ e ( q ) n e , f 7− → ( f e ( q )) e | m is inje ctive. Pr o of. Because the cyclotomic p olynomials Φ e ( q ) are irreducible in the unique factorisation domain Z [ q ] , the natural morphism of commutativ e rings Z [ q ] / Y e | m Φ e ( q ) n e − → Y e | m Z [ q ] / Φ e ( q ) n e is injective. Moreo ver, the quotient of the Habiro ring H R ,m / ( q m − 1) n is étale ov er the com- m utative ring Z [ q ] / ( q m − 1) n for every integer n ⩾ 0 ([ W ag25 , Theorem 2.9]). In particular, the comm utative ring H R ,m / Q e | m Φ e ( q ) n e is ﬂat o v er Z [ q ] / Q e | m Φ e ( q ) n e . T ensoring the previous injectiv e morphism with the ﬂat Z [ q ] / Q e | m Φ e ( q ) n e -mo dule H R ,m / Q e | m Φ e ( q ) n e then induces the desired injective morphism of commutativ e Z [ q ] -algebras H R ,m / Y e | m Φ e ( q ) n e − → Y e | m R[ q ] / Φ e ( q ) n e where we use the canonical iden tiﬁcation ( H R ,m ) ∧ Φ e ( q ) ∼ = R[ q ] ∧ Φ e ( q ) ([ W ag25 , 2.7]). Corollary 3.10. L et R b e an étale Z -algebr a. Then for any inte gers m, m ′ ⩾ 1 satisfying m | m ′ , the natur al morphism of c ommutative Z [ q ] -algebr as H R ,m ′ / ( q m − 1)( q m ′ − 1) − → H R ,m / ( q m − 1) 2 × Y e | m ′ e ∤ m R[ q ] / Φ e ( q ) (3.10.1) is inje ctive. Pr o of. The p ost-comp osition of the morphism ( 3.10.1 ) with the morphism of Z [ q ] -algebras H R ,m / ( q m − 1) 2 × Y e | m ′ e ∤ m R[ q ] / Φ e ( q ) − → Y e | m R[ q ] / Φ e ( q ) 2 × Y e | m ′ e ∤ m R[ q ] / Φ e ( q ) is injective by Lemma 3.9 , so the morphism ( 3.10.1 ) is also injective. 3.2 The cyclotomic loga rithm In this subsection, w e use the Habiro ring of Section 3.1 and (a reﬁnement of ) the norms on big q -Witt v ectors of Section 2.4 to construct the cyclotomic logarithm of an étale Z -algebra (Construction 3.15 and Deﬁnition 3.21 ). Lemma 3.11. L et m, m ′ ⩾ 1 b e inte gers satisfying m | m ′ . (1) The c anonic al map of Z [ q ] -mo dules ( q m ′ − 1) / ( q m ′ − 1) 2 → ( q m − 1) / ( q m − 1) 2 is divisible by m ′ m . (2) The induc e d map ( m ′ m ) − 1 : ( q m ′ − 1) / ( q m ′ − 1) 2 → ( q m − 1) / ( q m − 1) 2 is surje ctive. (3) In p articular, the pr evious map induc es an isomorphism ( m ′ m ) − 1 : ( q m ′ − 1) / ( q m − 1)( q m ′ − 1) ∼ = − → ( q m − 1) / ( q m − 1) 2 of Z [ q ] / ( q m − 1) -mo dules. 15 TESS BOUIS AND QUENTIN GAZDA Pr o of. This is a consequence of the congruence q m ′ − 1 q m − 1 =  m ′ m  q m ≡ m ′ m (mo d q m − 1) in the comm utative ring Z [ q ] , and of the fact that the mo dule ( q m − 1) / ( q m − 1) 2 is ﬂat o ver Z . Deﬁnition 3.12 (Nygaard twist) . Let R b e a commutativ e ring. F or ev ery integer m ⩾ 1 , the m -trunc ate d Nygaar d twist of R is the inv ertible q - W m (R) -mo dule N ⩾ 1 C R ,m { 1 } := q - W m (R) ⊗ Z [ q ] ( q m − 1) / ( q m − 1) 2 . If K is a num ber ﬁeld and R is the étale Z -algebra O K [∆ − 1 K ] , the Nygaar d twist of R is the Z [ q ] -mo dule N ⩾ 1 C R { 1 } := lim m ⩾ 1 ( m, N)=1 N ⩾ 1 C R ,m { 1 } where the limit is taken ov er in tegers m ⩾ 1 satisfying ( m, N) = 1 , 13 and along the maps induced b y the F rob enius maps F m ′ /m on q -Witt vectors (Construction 2.9 ) and the maps ( m ′ m ) − 1 on the second factor (Lemma 3.11 (2)). Remark 3.13. Let R b e an étale Z -algebra. F or every integer m ⩾ 1 , there is a natural isomor- phism of Z [ q ] -mo dules N ⩾ 1 C R ,m { 1 } ∼ = ( q m − 1) H R ,m / ( q m − 1) 2 H R ,m where the F rob enius maps big on q -Witt vectors corresp ond to the canonical restriction maps on Habiro rings (Remark 3.6 ). Construction 3.14 (Derived logarithm) . Given a commutativ e ring A and an ideal I ⊆ A , there is a natural short exact sequence of ab elian groups 0 − → I / I 2 exp − − → G m (A / I 2 ) − → G m (A / I) − → 0 where exp is given by x 7→ 1 + x . This induces a b oundary map dlog (A , I) : G m (A / I) − → I / I 2 [1] in the deriv ed category D ( Z ) (see also [ Mao24 , Construction 2.7] for a generalisation to animated rings). Construction 3.15 (T runcated cyclotomic logarithm) . Let R b e an étale Z -algebra, and m ⩾ 1 b e an integer. The m-trunc ate d cyclotomic lo garithm of R is the map dlog ( m ) cyc : G m (R)[ − 1] − → N ⩾ 1 C R ,m { 1 } in the derived category D ( Z ) deﬁned as the comp osite G m (R)[ − 1] Π m − − → G m ( q - W m (R))[ − 1] dlog − − − → N ⩾ 1 C R ,m { 1 } where the ﬁrst map is the cyclotomic norm of Prop osition 2.20 and the second map is the de- riv ed logarithm of Construction 3.14 for the pair (A , I) := ( H R ,m , ( q m − 1)) , where H R ,m and N ⩾ 1 C R ,m { 1 } are respectively deﬁned in V ariant 3.2 and in Deﬁnition 3.12 . Here we use the iden tiﬁcation H R ,m / ( q m − 1) ∼ = q - W m (R) (Remark 3.6 ). In the rest of this subsection, we construct the cyclotomic logarithm dlog cyc : G m (R)[ − 1] − → N ⩾ 1 C R { 1 } b y proving that the truncated cyclotomic logarithms of Construction 3.15 are compatible, in the deriv ed category D ( Z ) , b etw een diﬀeren t integers m ⩾ 1 . T o do so, we construct compatible homotopies h m,m ′ making the suitable transition diagrams comm ute (Prop osition 3.20 ). The homotopies h m,m ′ are in turn constructed out of a partial Habiro lift ˜ Π m ′ /m of the cyclotomic norms Π m ′ /m of Section 2.4 , which w e call the lifte d cyclotomic norms (Proposition 3.17 and Remark 3.19 ). 13 The restriction ( m, N) = 1 is not necessary in this section to deﬁne the cyclotomic logarithm. W e use this con ven- tion b ecause it will b e necessary to restrict to these integers m ⩾ 1 in order to deﬁne the ﬁrst q -p olylogarithm class of a cyclotomic unit in the next section (Deﬁnition 4.21 ). 16 CYCLOSYNTOMIC REGULA TOR Lemma 3.16. L et B 0 C 0 A 1 B 1 C 1 A 2 B 2 C 2 β 0 f B f C h C α 1 β 1 g A α 2 g B β 2 g C b e a c ommutative diagr am of ab elian gr oups whose midd le and b ottom lines ar e short exact se- quenc es. If the morphism g A is surje ctive, then the image of f B is c ontaine d in the image of g B . In p articular, if the morphism g B is inje ctive, then ther e exists a unique morphism of ab elian gr oups h B : B 0 → B 2 satisfying f B = g B ◦ h B . Pr o of. It suﬃces to prov e the inclusion f B (B 0 ) ⊆ g B (B 2 ) as subsets of B 1 . W e prov e ﬁrst that f B (B 0 ) ⊆ β − 1 1 ( g C (C 2 )) and then that β − 1 1 ( g C (C 2 )) ⊆ g B (B 2 ) . The ﬁrst inclusion follows from the existence of the morphism h C : giv en an element b 0 ∈ B 0 , we hav e β 1 ( f B ( b 0 )) = f C ( β 0 ( b 0 )) = g C ( h C ( β 0 ( b 0 ))) ∈ g C (C 2 ) . The second inclusion follows the snak e lemma and the surjectivity of the morphism g A . Prop osition 3.17 (Lifted cyclotomic norms) . L et R b e an étale Z -algebr a. Then for any inte gers m, m ′ ⩾ 1 satisfying m | m ′ , the morphism of multiplic ative monoids ˜ Π m ′ /m : Y e | m R[ q ] / Φ e ( q ) 2 − → Y e | m R[ q ] / Φ e ( q ) 2 × Y e | m ′ e ∤ m R[ q ] / Φ e ( q ) (3.17.1) given by ( f e ( q )) e | m 7→ ( f ′ e ( q )) e | m ′ :=  f ( e,m ) ( q e ( e,m ) ) m ′ [ e,m ]  e | m ′ is wel l-deﬁne d, and r estricts to a morphism of multiplic ative monoids ˜ Π m ′ /m : H R ,m / ( q m − 1) 2 − → H R ,m ′ / ( q m − 1)( q m ′ − 1) . Pr o of. The fact that the morphism ( 3.17.1 ) is well-deﬁned is a consequence of the fact that, for ev ery integer e ⩾ 1 satisfying e | m ′ , w e ha v e Φ e ( q ) 2 | Φ ( e,m ) ( q e ( e,m ) ) 2 in Z [ q ] . It is in addition m ultiplicative by construction. In the rest of the proof, we use the injectivity of Lemma 3.9 to interpret the comm utative Z [ q ] -algebras H R ,m / ( q m − 1) 2 and H R ,m ′ / ( q m − 1)( q m ′ − 1) as subrings of the source and target of the morphism ( 3.17.1 ). T o pro ve that the image of H R ,m / ( q m − 1) 2 b y the morphism ( 3.17.1 ) is con tained in H R ,m ′ / ( q m − 1)( q m ′ − 1) , ﬁrst note that if e | m , then f ′ e ( q ) = f e ( q ) m ′ m , so that ( f ′ e ( q )) e | m deﬁnes an element of H R ,m / ( q m − 1) 2 . Moreov er, w e hav e that H R ,m ′ / ( q m − 1) 2 ∼ = H R ,m / ( q m − 1) 2 b y [ W ag25 , 2.7], hence the isomorphism of commutativ e Z [ q ] -algebras H R ,m / (( q m − 1) 2 , ( q m ′ − 1)) ∼ = q - W m ′ (R) / ( q m − 1) 2 b y [ W ag25 , Theorem 2 . 9 ]. In particular, there is a commutativ e diagram of ab elian groups H R ,m ( q m − 1) 2 H R ,m q - W m (R) ( q m ′ − 1) H R ,m ′ ( q m − 1)( q m ′ − 1) H R ,m ′ H R ,m ( q m − 1) 2 H R ,m × Q e | m ′ e ∤ m R[ q ] Φ e ( q ) q - W m ′ (R) ( q m − 1) 2 × Q e | m ′ e ∤ m R[ q ] Φ e ( q ) ( q m ′ − 1) H R ,m ′ ( q m − 1)( q m ′ − 1) H R ,m ′ H R ,m ′ ( q m − 1)( q m ′ − 1) H R ,m ′ q - W m ′ (R) ˜ Π m ′ /m Π m ′ /m Π m ′ /m can can 17 TESS BOUIS AND QUENTIN GAZDA whose middle and bottom lines are short exact sequences, and where the comm utativity of the righ t part of the diagram is a consequence of Prop osition 2.20 . The natural morphism of ab elian groups H R ,m ′ / ( q m − 1)( q m ′ − 1) − → H R ,m / ( q m − 1) 2 × Y e | m ′ e ∤ m R[ q ] / Φ e ( q ) is moreo v er injective by Corollary 3.10 . So the desired result is a consequence of Lemma 3.16 applied to the previous diagram. Corollary 3.18. L et R b e an étale Z -algebr a. Then for any inte gers m, m ′ ⩾ 1 satisfying m | m ′ , the natur al diagr am of ab elian gr oups N ⩾ 1 C R ,m { 1 } G m  H R ,m ( q m − 1) 2 H R ,m  ( q m ′ − 1) H R ,m ′ ( q m − 1)( q m ′ − 1) H R ,m ′ G m  H R ,m ′ ( q m − 1)( q m ′ − 1) H R ,m ′  exp m ′ m ∼ = ˜ Π m ′ /m exp is c ommutative, wher e N ⩾ 1 C R ,m { 1 } is identiﬁe d with ( q m − 1) H R ,m / ( q m − 1) 2 H R ,m (R emark 3.13 ). Pr o of. By Lemma 3.9 , it suﬃces to prov e that ˜ Π m ′ /m ◦ exp and exp ◦ m ′ m agree as maps from Q e | m (Φ e ( q )) / (Φ e ( q )) 2 to Q e | m R[ q ] / Φ e ( q ) 2 × Q e | m ′ ,e ∤ m R[ q ] / Φ e ( q ) . Let (Φ e ( q ) c e ( q )) e | m b e an elemen t of Q e | m (Φ e ( q )) / (Φ e ( q ) 2 ) , and ( c ′ e ( q )) e | m ′ b e its image under the function ˜ Π m,m ′ ◦ exp. If e | m , then c ′ e ( q ) =  1 + Φ e ( q ) c e ( q )  m ′ m ≡ 1 + m ′ m Φ e ( q ) c e ( q ) = exp  m ′ m Φ e ( q ) c e ( q )  in R[ q ] / Φ e ( q ) 2 . And if e ∤ m , then the e th q -ghost comp onent of the comm utativ e Z [ q ] -mo dule ( q m ′ − 1) / ( q m − 1)( q m ′ − 1) is zero by Z [ q ] / ( q m − 1) -linearity , and c ′ e ( q ) =  1 + Φ ( e,m ) ( q e ( e,m ) ) c ( e,m ) ( q e ( e,m ) )  m ′ [ e,m ] ≡ 1 in R[ q ] / Φ e ( q ) , b ecause Φ e ( q ) | Φ ( e,m ) ( q e ( e,m ) ) . Remark 3.19. Contrary to what Prop osition 3.17 may suggest, the lifted cyclotomic norm ˜ Π m ′ /m cannot b e naturally lifted to a morphism of ab elian groups ˜ Π m ′ /m : G m  H R ,m / ( q m − 1) 2  − → G m  H R ,m ′ / ( q m ′ − 1) 2  . Indeed, in the case where ( m, m ′ ) := (1 , m ) , such a lift would induce a splitting of the extension [dlog ( m ) cyc ] ∈ Ext 1 Z ( N ⩾ 1 C R ,m { 1 } , G m ( q - W m (R))) of Construction 3.15 , and in particular that the corresp onding truncated cyclotomic logarithms are identically zero. How ever, we will pro ve that these cyclotomic logarithms are not iden tically zero in Section 4 . Prop osition 3.20. L et R b e an étale Z -algebr a. Then for every p air of p ositive inte gers ( m, m ′ ) satisfying m | m ′ , ther e exists an homotopy h m,m ′ making the diagr am G m (R)[ − 1] N ⩾ 1 C R ,m ′ { 1 } G m (R)[ − 1] N ⩾ 1 C R ,m { 1 } id dlog ( m ′ ) cyc F m ′ /m { 1 } dlog ( m ) cyc c ommute in the derive d c ate gory D ( Z ) , and such that h m ′ ,m ′′ ◦ h m,m ′ = h m,m ′′ for al l inte gers m, m ′ , m ′′ ⩾ 1 satisfying m | m ′ | m ′′ . Pr o of. Let m and m ′ b e p ositive integers satisfying m | m ′ . W e wan t to ﬁnd an homotopy making the diagram G m (R) G m ( q - W m ′ (R)) N ⩾ 1 C R ,m ′ { 1 } [1] G m (R) G m ( q - W m (R)) N ⩾ 1 C R ,m { 1 } [1] id Π m ′ dlog F m ′ /m { 1 } Π m dlog 18 CYCLOSYNTOMIC REGULA TOR comm ute in the deriv ed category D ( Z ) . By Remark 2.21 , the diagram of ab elian groups G m (R) G m ( q - W m ′ (R)) G m (R) G m ( q - W m (R)) id Π m ′ Π m Π m ′ /m is commutativ e, so it suﬃces to ﬁnd an homotopy making the diagram G m ( q - W m ′ (R)) N ⩾ 1 C R ,m ′ { 1 } [1] G m ( q - W m (R)) N ⩾ 1 C R ,m { 1 } [1] dlog F m ′ /m { 1 } dlog Π m ′ /m comm ute in the deriv ed category D ( Z ) . By Deﬁnition 3.12 and Remark 3.13 , the right vertical map of this diagram is naturally identiﬁed with (the shift of ) the comp osite N ⩾ 1 C R ,m ′ { 1 } ↠ ( q m ′ − 1) H R ,m ′ ( q m − 1)( q m ′ − 1) H R ,m ′ ∼ = − − → N ⩾ 1 C R ,m { 1 } where the ﬁrst map is induced by the canonical map ( q m ′ − 1) / ( q m ′ − 1) 2 ↠ ( q m ′ − 1) / ( q m − 1)( q m ′ − 1) of Z [ q ] -mo dules, and the second map is induced by the isomorphism of Lemma 3.11 (3). Moreo ver, the natural diagram G m ( q - W m ′ (R)) N ⩾ 1 C R ,m ′ { 1 } [1] G m ( q - W m ′ (R)) ( q m ′ − 1) H R ,m ′ ( q m − 1)( q m ′ − 1) H R ,m ′ [1] dlog id dlog in the derived category D ( Z ) , where the b ottom horizon tal map is the derived logarithm of Con- struction 3.14 for the pair (A , I) := ( H R ,m ′ / ( q m − 1)( q m ′ − 1) , ( q m ′ − 1)) , is commutativ e. It then suﬃces to pro duce an homotopy making the diagram G m ( q - W m ′ (R)) ( q m ′ − 1) H R ,m ′ ( q m − 1)( q m ′ − 1) H R ,m ′ [1] G m ( q - W m (R)) N ⩾ 1 C R ,m { 1 } [1] dlog Π m ′ /m dlog m ′ m ∼ = comm ute in the derived category D ( Z ) . Unwinding the deﬁnition of the derived logarithm (Con- struction 3.14 ), this amounts to ﬁnding a morphism of ab elian groups ˜ Π m ′ /m making the diagram of ab elian groups 0 ( q m ′ − 1) H R ,m ′ ( q m − 1)( q m ′ − 1) H R ,m ′ G m  H R ,m ′ ( q m − 1)( q m ′ − 1) H R ,m ′  G m ( q - W m ′ (R)) 0 0 N ⩾ 1 C R ,m { 1 } G m  H R ,m ′ ( q m ′ − 1) 2 H R ,m ′  G m ( q - W m (R)) 0 exp exp m ′ m ˜ Π m ′ /m Π m ′ /m comm ute. Suc h morphisms ˜ Π m ′ /m are constructed in Prop osition 3.17 . More precisely , these morphisms are well-deﬁned by Prop osition 3.17 , the right square is commutativ e by construction, and the left square is commutativ e by Corollary 3.18 . Finally , the fact that the induced homotopies h m,m ′ satisfy h m ′ ,m ′′ ◦ h m,m ′ = h m,m ′′ is a consequence of the form ula ˜ Π m ′′ /m ′ ◦ ˜ Π m ′ /m = ˜ Π m ′′ /m , whic h can b e chec ked as in the pro of of Lemma 2.19 . 19 TESS BOUIS AND QUENTIN GAZDA Deﬁnition 3.21 (Cyclotomic logarithm) . Let K b e a num b er ﬁeld, and R b e the étale Z -algebra O K [∆ − 1 K ] . The cyclotomic lo garithm of R is the map dlog cyc : G m (R)[ − 1] − → N ⩾ 1 C R { 1 } deﬁned as the inv erse limit ov er integers m ⩾ 1 satisfying ( m, N) = 1 , in the derived category D ( Z ) , of the m -truncated cyclotomic logarithms of Construction 3.15 . Note here that we use the com- patible homotopies of Prop osition 3.20 to make sense of this in verse limit. 14 Remark 3.22 (C omparison with [ Mao24 ]) . The general strategy to construct the cyclotomic log- arithm dlog cyc is inspired by Mao’s construction of the reﬁned prismatic logarithm dlog ∆ [ Mao24 , Section 2]. How ev er, note that Mao’s p -adic constructions are limited to the case of o dd prime num- b ers p . The explicit form ula for the lifted cyclotomic norms (Prop osition 3.17 ) allo ws us to b ypass this technical assumption, and in particular not to restrict our constructions to o dd integers m ⩾ 1 . 4 Cyclosyntomic regulato r Let K b e a num ber ﬁeld. In this section, we deﬁne, for each integer d ⩾ 2 , the cyclosyntomic c omplex RΓ CycSyn  R , Z (1) ( d )  ∈ D ( Z ) of R := O K [∆ − 1 K ] (Deﬁnition 4.7 ), its asso ciated ﬁrst Chern class c CycSyn 1 : RΓ ét (R , G m )[ − 1] − → RΓ CycSyn  R , Z (1) ( d )  as a map in the derived category D ( Z ) (Deﬁnition 4.17 ), and prov e that this cyclosyntomic ﬁrst Chern class is computed at cyclotomic units by a q -deformation of the ﬁrst p olylogarithm (Theo- rem 4.25 ). 4.1 The cyclosyntomic cohomology of a numb er ﬁeld In this subsection, we introduce the cyclosyntomic cohomology of R := O K [∆ − 1 K ] (Deﬁnition 4.7 ). It is deﬁned as a tw o term complex given in degree zero b y the Nygaard twist N ⩾ 1 C R { 1 } (Deﬁ- nition 3.12 ) and in degree one by the cyclotomic twist C ( d ) R { 1 } , which we deﬁne now. Deﬁnition 4.1 (Cyclotomic t wist at d ) . L et R b e a commutativ e ring, and d ⩾ 1 b e an integer. F or every integer m ⩾ 1 , the m -trunc ate d cyclotomic twist of R at d is the C ( d ) R ,m -mo dule C ( d ) R ,m { 1 } := C ( d ) R ,m ⊗ Z [ q ] ( q m − 1) / ( q m − 1) 2 where the cyclotomic ring C ( d ) R ,m is deﬁned in V ariant 2.34 . If K is a num ber ﬁeld and R is the étale Z -algebra O K [∆ − 1 K ] , the cyclotomic twist of R at d is the Z [ q ] -modu le C ( d ) R { 1 } := lim m ⩾ 1 ( m, ∆ K )=1 C ( d ) R ,m { 1 } where the limit is taken ov er integers m ⩾ 1 satisfying ( m, N) = 1 , and along the maps induced b y the F rob enius maps F m ′ /m on q -Witt vectors (Construction 2.9 ) and the maps ( m ′ m ) − 1 on the second factor (Lemma 3.11 (2)). Remark 4.2. Let R b e an étale Z -algebra, and d ⩾ 2 b e an integer. F or ev ery integer m ⩾ 1 , there are natural isomorphisms of Z [ q ] -mo dules C ( d ) R ,m { 1 } ∼ = ( q m − 1) H ( d ) R ,m / ( q m − 1) 2 H ( d ) R ,m ∼ = [ m ] q H ( d ) R ,m / [ m ] 2 q H ( d ) R ,m where H ( d ) R ,m is the m -truncated Habiro ring of R at d (V ariant 3.4 ), and where the F robenius maps on q -Witt vectors corresp ond to the canonical restriction maps on Habiro rings (Remark 3.6 ). Note that the ﬁrst isomorphism also holds for d = 1 and that, in general, one may replace ( q m − 1) and [ m ] q b y Q d | e | m Φ e ( q ) . 14 More precisely , to write down the limit of the maps dlog ( m ) cyc in the derived ∞ -category D ( Z ) , we use the 1 -homotopies h m,m ′ of Proposition 3.20 to encode the fact that the maps dlog ( m ) cyc are compatible b et ween diﬀerent integers m ⩾ 1 , and the equality h m ′ ,m ′′ ◦ h m,m ′ = h m,m ′′ , also prov ed in Prop osition 3.20 , to ensure that there are no higher coherences to chec k. 20 CYCLOSYNTOMIC REGULA TOR Remark 4.3. When d = p is a prime n um b er and m = p r is a p ow er of p , the iden tiﬁcation of Remark 4.2 in terms of [ m ] q is reminiscent of the notion of Breuil–Kisin twist in prismatic cohomology [ BL22 , Section 2.2], in the sense that we hav e the equality of ideals I ϕ ∗ A (I) · · · ( ϕ r − 1 A ) ∗ (I) =  [ p ] q [ p ] q p · · · [ p ] q p r − 1  = ([ p r ] q ) in the q -prism (A , I) := ( Z p [ [ q − 1] ] , [ p ] q ) . Notation 4.4. Giv en a num b er ﬁeld K , R := O K [∆ − 1 K ] , and integers d, m ⩾ 1 , we denote by can : N ⩾ 1 C R ,m { 1 } − → C ( d ) R ,m { 1 } the morphism of Z [ q ] -mo dules giv en by (Φ e ( q ) c e ( q )) e | m 7→ (Φ e ( q ) c e ( q )) d | e | m . These maps are compatible b etw een diﬀeren t integers m ⩾ 1 by construction, so they induce a morphism of Z [ q ] -mo dules can : N ⩾ 1 C R { 1 } − → C ( d ) R { 1 } . The following deﬁnition is a v arian t of Construction 3.7 . Construction 4.5 (T runcated twisted cyclotomic F rob enii) . Let R b e a commutativ e ring, and d, m ⩾ 1 b e integers. The m -trunc ate d d th twiste d cyclotomic F r ob enius is the morphism of ab elian groups F rob cyc d { 1 } : N ⩾ 1 C R ,m { 1 } − → C ( d ) R ,m { 1 } giv en by ( q m − 1)( c e ( q )) e | m 7→ ( q m − 1)( c e/d ( q d )) d | e | m . In terms of Habiro rings, if R is étale ov er Z , then this morphism is given b y the comp osite ( q m − 1) H R ,m ( q m − 1) 2 H R ,m F rob Hab d − − − − − → ( q dm − 1) H ( d ) R ,dm ( q dm − 1) 2 H ( d ) R ,dm 1 d − − → ( q m − 1) H ( d ) R ,m ( q m − 1) 2 H ( d ) R ,m where the ﬁrst morphism is induced by the Habiro F rob enius of Construction 3.7 . By construction, the truncated d th t wisted cyclotomic F rob enii are compatible b etw een diﬀeren t integers m ⩾ 1 . Deﬁnition 4.6 (T wisted cyclotomic F robenii) . Let K b e a num b er ﬁeld, R b e the étale Z -algebra O K [∆ − 1 K ] , and d ⩾ 1 b e an integer. The d th twiste d cyclotomic F r ob enius is the morphism of ab elian groups F rob cyc d { 1 } : N ⩾ 1 C R { 1 } − → C ( d ) R { 1 } deﬁned as the inv erse limit ov er integers m ⩾ 1 satisfying ( m, N) = 1 of the m -truncated twisted cyclotomic F rob enii of Construction 4.5 . Deﬁnition 4.7 (Cyclosyn tomic cohomology) . Let K b e a num ber ﬁeld, R b e the étale Z -algebra O K [∆ − 1 K ] , N b e a multiple of ∆ K , and d ⩾ 1 b e an integer. The cyclosyntomic c omplex of R at d is the ob ject RΓ CycSyn (R , Z (1) ( d ) ) :=  N ⩾ 1 C R { 1 } can − F rob cyc d { 1 } − − − − − − − − − − → C ( d ) R { 1 }  in the derived category D ( Z ) , where the ﬁrst term N ⩾ 1 C R { 1 } (Deﬁnition 3.12 ) sits in cohomo- logical degree zero, the second term C ( d ) R { 1 } (Deﬁnition 4.1 ) sits in cohomological degree one, and the morphism is deﬁned by Notation 4.4 and Deﬁnition 4.6 . Remark 4.8. Although v arian ts 15 of the previous deﬁnition would mak e sense for arbitrary com- m utative rings R , w e do not exp ect it to b e a reasonable deﬁnition beyond the case of étale Z -algebras R . This intuition is directly dra wn from the deﬁnition of syntomic cohomology in terms of absolute prismatic cohomology [ BS22 , BL22 ]. Remark 4.9 (Comparison to syn tomic cohomology) . Let K b e a n umber ﬁeld, R be the étale Z -algebra O K [∆ − 1 K ] , and p b e a prime num ber. Here we compare the cyclosyntomic complex of R of Deﬁnition 4.7 with the prismatic approach to syntomic cohomology , as developed in [ BMS19 , 15 F or instance, if one do es not restrict the limit ov er m ⩾ 1 to integers satisfying ( m, N) = 1 in Deﬁnitions 3.12 and 4.1 . 21 TESS BOUIS AND QUENTIN GAZDA BS22 , BL22 , AKN23 ]. F ollowing [ AKN23 , Section 7], the (w eight one) syn tomic complex of R relativ e to the q -prism ( Z p [ [ q − 1] ] , [ p ] q ) is the ob ject RΓ q syn (R , Z p (1)) := ﬁb  N ⩾ 1 ∆ R (1) / Z p [ [ q − 1] ] { 1 } can − F rob ∆ p { 1 } − − − − − − − − − → ∆ R (1) / Z p [ [ q − 1] ] { 1 }  of the deriv ed category D ( Z p ) , where R (1) := R ⊗ Z p Z p [ ζ p ] and F rob ∆ p { 1 } is the divided F rob enius on the ﬁrst Breuil–Kisin t wist of prismatic cohomology . Given that R = O K [∆ − 1 K ] is étale o ver Z , the prismatic cohomology of R (1) relativ e to Z p [ [ q − 1] ] is concentrated in degree zero, where it is given b y the initial prism R ∧ p [ [ q − 1] ] of the prismatic site (R (1) / Z p [ [ q − 1] ]) ∆ . Similarly , the Nygaard twist N ⩾ 1 ∆ R (1) / Z p [ [ q − 1] ] { 1 } is giv en by ( q − 1)R ∧ p [ [ q − 1] ] concentrated in degree zero ([ BS22 , Section 15], see also [ W ag25 , 3.20]). Unwinding the deﬁnitions, the p -adic realisation maps of Remarks 2.26 and 3.8 then induce a natural commutativ e diagram N ⩾ 1 C R { 1 } C ( p ) R { 1 } ( q − 1)R ∧ p [ [ q − 1] ] R ∧ p [ [ q − 1] ] can − F rob cyc d { 1 } can − F rob ∆ p { 1 } in the derived category D ( Z ) , where all the terms sit in cohomological degree zero. In particular, this commutativ e diagram induces a natural comparison map RΓ CycSyn (R , Z (1) ( p ) ) − → RΓ q syn (R , Z p (1)) in the derived category D ( Z ) . Remark 4.10 (T runcated cyclosyntomic cohomology) . Giv en a n umber ﬁeld K , R := O K [∆ − 1 K ] , a m ultiple N of ∆ K , and an in teger d ⩾ 1 , the cyclosyn tomic complex RΓ CycSyn (R , Z (1) ( d ) ) of Deﬁnition 4.7 is the limit o ver integers m ⩾ 1 satisfying ( m, N) = 1 of the analogous complexes RΓ CycSyn (R , Z (1) ( d ) m ) :=  N ⩾ 1 C R ,m { 1 } can − F rob cyc d { 1 } − − − − − − − − − − → C ( d ) R ,m { 1 }  in the derived category D ( Z ) , where the morphism is given by can − F rob cyc d { 1 } : (Φ e ( q ) c e ( q )) e | m 7→ (Φ e ( q ) c e ( q ) − Φ e ( q ) c d/e ( q d )) d | e | m . This is indeed a consequence of Remarks 4.2 , Notation 4.4 , and Construction 4.5 . 4.2 The ﬁrst Chern class In this subsection, we deﬁne the cyclosyntomic ﬁrst Chern class (Deﬁnition 4.17 ) as a reﬁnemen t of the cyclotomic logarithm dlog cyc : G m (R)[ − 1] → N ⩾ 1 C R { 1 } of Section 3.2 . More precisely , w e pro v e that the cyclotomic logarithm factors through the cyclosyn tomic complex deﬁned in Section 4.1 , by exhibiting suitable null-homotopies at the truncated level (Construction 4.12 and Prop osition 4.13 ). Notation 4.11. Giv en an étale Z -algebra R and an integer m ⩾ 1 , we denote by E m :=  ( y , z ) ∈ G m ( H R ,m / ( q m − 1) 2 ) × G m (R)   y = Π m ( z ) in H R ,m / ( q m − 1) ∼ = q - W m (R)  the ab elian group that corresp onds to the extension class 0 − → N ⩾ 1 C R ,m { 1 } exp × 1 − − − − → E m − → G m (R) − → 0 giv en b y the m -truncated cyclotomic logarithm dlog ( m ) cyc of Construction 3.15 . Here we use the isomorphism N ⩾ 1 C R ,m { 1 } ∼ = ( q m − 1) H R ,m / ( q m − 1) 2 H R ,m of Remark 3.13 to make sense of this short exact sequence. 22 CYCLOSYNTOMIC REGULA TOR The following construction will b e used to deﬁne the aforementioned null-homotop y (Prop osi- tion 4.13 ) and to give the formula presen ted in the introduction for the induced cyclosyntomic ﬁrst Chern class (Corollary 4.18 ). Construction 4.12. Let R b e an étale Z -algebra, and d ⩾ 2 and m ⩾ 1 b e integers. W e construct a morphism of ab elian groups s d : E m − → C ( d ) R ,m { 1 } where E m is deﬁned in Notation 4.11 . This morphism is given by the expression s d : ( y , z ) 7− → 1 d log  ˜ Π d ( y ) F rob Hab d ( y )  where the formula is understo o d via the following conv en tions: • ˜ Π d := can ◦ ˜ Π d is the comp osite map G m  H R ,m ( q m − 1) 2  ˜ Π d − − → G m  H R ,dm ( q m − 1)( q dm − 1)  can − − → G m  H ( d ) R ,dm ( q m − 1)( q dm − 1)  where the ﬁrst map ˜ Π d is the lifted cyclotomic norm of Prop osition 3.17 ; • F rob Hab d is the comp osite map G m  H R ,m ( q m − 1) 2  F rob Hab d − − − − − → G m  H ( d ) R ,dm ( q dm − 1) 2  − ↠ G m  H ( d ) R ,dm ( q m − 1)( q dm − 1)  where the ﬁrst map F rob Hab d is the Habiro F rob enius of Construction 3.7 ; • the logarithm is the morphism of ab elian groups log : G m  H ( d ) R ,dm ( q m − 1)( q dm − 1)  − → H ( d ) R ,dm ( q m − 1)( q dm − 1) , t 7− → t − 1; • “division by d ” is understo o d as the morphism of ab elian groups d − 1 = [ d ] − 1 q m : [ dm ] q H ( d ) R ,dm [ m ] q [ dm ] q H ( d ) R ,dm ∼ = − − → [ m ] q H ( d ) R ,dm [ m ] 2 q H ( d ) R ,dm ∼ = C ( d ) R ,m { 1 } giv en by division b y [ d ] q m = [ dm ] q / [ m ] q (Lemma 3.11 (3)). T o chec k that the morphism s d is well-deﬁned, it suﬃces to prov e, for ev ery ( y , z ) ∈ E m , that the elemen t ˜ Π d ( y ) F rob Hab d ( y ) − 1 ∈ H ( d ) R ,dm / [ m ] q [ dm ] q H ( d ) R ,dm b elongs to the ideal ([ dm ] q ) , or equiv alen tly that ˜ Π d ( y ) ≡ F rob Hab d ( y ) in H ( d ) R ,dm / [ dm ] q . W e no w use Lemma 3.9 , whose pro of adapts readily from the Habiro ring H R ,m (V arian t 3.2 ) to the v arian t H ( d ) R ,m (V arian t 3.4 ), in order to reduce this statemen t to a statement on q -ghost co ordinates. More precisely , it then suﬃces to prov e this iden tity on the e th q -ghost coordinate for in tegers e ⩾ 1 satisfying d | e | dm . 16 F or every such integer e , the desired identit y follows from the series of equalities in R[ q ] / Φ e ( q ) : ˜ Π d ( y ) e = Π d ( y ) e = Π d (Π m ( z )) e = Π dm ( z ) e = z dm e = z m e/d = F rob Hab d (Π m ( z )) e = F rob Hab d ( y ) e where we use that y ≡ Π m ( z ) (mo d Φ e ( q )) by deﬁnition of ( y , z ) ∈ E m (Notation 4.11 ). 16 Note that this is the critical step where we need Φ e ( q ) to b e inv ertible in H ( d ) R ,dm for d ∤ e , and the reason for taking this con ven tion in V ariant 2.34 , V arian t 3.4 , and Deﬁnition 4.1 . 23 TESS BOUIS AND QUENTIN GAZDA Prop osition 4.13. L et R b e an étale Z -algebr a, and d ⩾ 2 b e an inte ger. Then for every inte ger m ⩾ 1 , the map s d : E m − → C ( d ) R ,m { 1 } of Construction 4.12 induc es a natur al homotopy h ′ m making the c omp osite G m (R)[ − 1] dlog ( m ) cyc − − − − − → N ⩾ 1 C R ,m { 1 } can − F rob cyc d { 1 } − − − − − − − − − − → C ( d ) R ,m { 1 } homotopic to zer o in the derive d c ate gory D ( Z ) . Pr o of. This comp osite map corresp onds to an extension in Ext 1 Z ( G m (R) , C ( d ) R ,m { 1 } ) , given by the pushout of the extension [dlog ( m ) cyc ] ∈ Ext 1 Z ( G m (R) , N ⩾ 1 C R ,m { 1 } ) of Construction 3.15 along the map can − F rob cyc d { 1 } . Un winding the deﬁnition of the map dlog ( m ) cyc , a splitting of this extension is in turn equiv alen t to a morphism of ab elian groups s d : E m − → C ( d ) R ,m { 1 } suc h that the diagram of ab elian groups N ⩾ 1 C R ,m { 1 } E m C ( d ) R ,m { 1 } can − F rob cyc d { 1 } exp × 1 s d is commutativ e. W e claim that the morphism s d of Construction 4.12 satisﬁes this prop erty . T o pro ve this, let x b e an element of N ⩾ 1 C R ,m { 1 } . W e then hav e the series of equality 1 d log  ˜ Π d (exp( x )) F rob Hab d (exp( x ))  = 1 d  1 + d can ( x ) 1 + F rob Hab d ( x ) − 1  = 1 d  (1 + d can ( x ))(1 − F rob Hab d ( x )) − 1  = 1 d  d can ( x ) − F rob Hab d ( x )  = can ( x ) − F rob cyc d { 1 } ( x ) where the ﬁrst equality is a consequence of Corollary 3.18 , the second and third equalities are consequences of the fact that d can ( x ) and F rob Hab d ( x ) are divisible by ( q dm − 1) , and the last equalit y is a consequence of Construction 4.5 . Construction 4.14 (T runcated cyclosyntomic ﬁrst Chern class) . Let R b e an étale Z -algebra, and d ⩾ 2 and m ⩾ 1 b e integers. By Prop osition 4.13 , the cyclotomic logarithm dlog ( m ) cyc : G m (R)[ − 1] − → N ⩾ 1 C R ,m { 1 } factors uniquely , in the derived category D ( Z ) , through the homotopy ﬁbre of the map can − F rob cyc d { 1 } : N ⩾ 1 C R ,m { 1 } − → C ( d ) R ,m { 1 } . W e denote by c CycSyn 1 : G m (R)[ − 1] − → RΓ CycSyn (R , Z (1) ( d ) m ) the induced map in the deriv ed category D ( Z ) , where the target is deﬁned in Remark 4.10 . Remark 4.15. The homotopies h ′ m of Prop osition 4.13 are compatible b etw een diﬀeren t in tegers m ⩾ 1 . T o prov e this, it indeed suﬃces to prov e that the diagram of ab elian groups E m ′ E m C ( d ) R ,m ′ { 1 } C ( d ) R ,m { 1 } h m,m ′ s d s d F m ′ /m { 1 } is commutativ e for all in tegers m, m ′ ⩾ 1 satisfying m | m ′ , and this is a consequence of the com- patibilit y b etw een Construction 4.12 and the homotopies h m,m ′ of Prop osition 3.20 . 24 CYCLOSYNTOMIC REGULA TOR Remark 4.16 (Étale descen t) . By [ W ag24 , Prop osition 2.15], any descen t prop ert y satisﬁed by the iden tity functor on comm utative rings is also satisﬁed by the functor R 7→ W m (R) for every integer m ⩾ 1 . Unwinding the deﬁnitions, this in particular implies that the cyclosyn tomic complex of étale Z -algebras R (Deﬁnition 4.7 and Remark 4.10 ) satisﬁes étale descent on R , and the cyclosyntomic ﬁrst Chern class of Construction 4.14 then naturally factors as a map c CycSyn 1 : RΓ ét (R , G m )[ − 1] − → RΓ CycSyn (R , Z (1) ( d ) m ) in the derived category D ( Z ) , for any in tegers d ⩾ 2 and m ⩾ 1 . Note here that we use the fact that the higher coherent cohomology of aﬃne schemes v anishes. Deﬁnition 4.17 (Cyclosyn tomic ﬁrst Chern class) . Let K b e a n umber ﬁeld, R b e the étale Z -algebra O K [∆ − 1 K ] , N be a multiple of ∆ K , and d ⩾ 2 be an integer. The cyclosyntomic ﬁrst Chern class of R at d is the map c CycSyn 1 : G m (R)[ − 1] − → RΓ CycSyn (R , Z (1) ( d ) ) in the derived category D ( Z ) , deﬁned as the in verse limit ov er integers m ⩾ 1 satisfying ( m, N) = 1 of the m -truncated maps of Construction 4.14 . Note here that we use Remark 4.15 to make sense of this inv erse limit. The following result is a global analogue of the fundamental p -adic computation of Kato ([ Kat91 , Corollary 2.9], see also [ Gro90 , Prop osition 4.1 and App endix] or [ Som99 , page 289]). Corollary 4.18. L et K b e a numb er ﬁeld, R b e the étale Z -algebr a O K [∆ − 1 K ] , and d ⩾ 2 b e an inte ger. F or every unit u ∈ G m (R) , the cyclosyntomic ﬁrst Chern class at u is given by c CycSyn 1 : u 7− →  1 d log  ˜ Π d (Π Hab m ( u )) F rob Hab d (Π Hab m ( u ))  m ⩾ 1 ∈ H 1 CycSyn (R , Z (1) ( d ) ) wher e Π Hab m ( u ) ∈ H R ,m / ( q m − 1) 2 is any lift of the element Π m ( u ) ∈ q - W m (R) via the surje ctive map of c ommutative Z [ q ] -algebr as H R ,m / ( q m − 1) 2 ↠ q - W m (R) (R emark 3.6 ), ˜ Π d is the lifte d cyclotomic norm (Pr op osition 3.17 ), and F rob Hab d is the Habir o F r ob enius (Construction 3.7 ). Pr o of. By Deﬁnition 4.17 , it suﬃces to pro ve the result at the truncated lev el, where this is a consequence of Constructions 4.12 and 4.14 . 4.3 The ﬁrst q -p olylogarithm In this subsection, w e compute the cyclosyntomic ﬁrst Chern class of the previous subsection at cyclotomic units in terms of a q -deformation of the ﬁrst polylogarithm (Theorem 4.25 ). T o do so, we ﬁrst introduce the relev an t q -deformation of the ﬁrst p olylogarithm, whose deﬁnition is motiv ated by Deligne’s notion of p -adic p olylogarithm ([ Del89 , 3.2.3]). Construction 4.19. Let d ⩾ 2 b e an integer, and consider the p ow er series Li ( d ) 1 (T) q := X k ⩾ 1 d ∤ k T k [ k ] q ∈ Z [ q ]  1 [ k ] q     k ⩾ 1 d ∤ k  [ [T] ] . Giv en any integer m ⩾ 1 , as a consequence of the identit y [ k + m ] q = [ k ] q + q k [ m ] q , the reduction mo dulo [ m ] q of this p ow ers series is Li ( d ) 1 (T) q ≡ X a ⩾ 0 X 0 1 Φ e ( q ) is inv ertible (b ecause each Φ e ( q ) is, if d ∤ k ) and that 1 − [ ζ ] m is inv ertible (b ecause the order of ζ is inv ertible in R , and one only considers the e th q -ghost co ordinates for ( e, N) = 1 , so that 1 − ζ 1 e is inv ertible in R ). 26 CYCLOSYNTOMIC REGULA TOR Pr o of. Let e ⩾ 1 b e an integer satisfying e | m , and in particular ( e, N) = 1 . By [ W ag24 , Lemma 2.23], it suﬃces to pro ve that Π Hab m (1 − [ ζ ]) ≡ Π m (1 − ζ ) mo dulo Φ e ( q ) , which follows from the series of equalities Y 0 ⩽ j 0 d ∤ k T k [ k ] q where w e use that [ m ] q · [ k ] q m = [ m ] q · k mo dulo [ m ] 2 q . The desired result is then a consequence of Construction 4.19 . Theorem 4.25. L et K b e a numb er ﬁeld, R b e the étale Z -algebr a O K [∆ − 1 K ] , N b e a multiple of ∆ K , and d ⩾ 2 b e an inte ger. F or every r o ot of unity ζ ∈ R \ {± 1 } , 18 the cyclosyntomic ﬁrst Chern class (Deﬁnition 4.17 ) sends the unit 1 − ζ ∈ G m (R) to the ﬁrst q -p olylo garithm class Li ( d ) 1 ([ ζ ]) q ∈ H 1 CycSyn (R , Z (1) ( d ) ) (Deﬁnition 4.21 ): c CycSyn 1 : 1 − ζ 7− → − Li ( d ) 1 ([ ζ ]) q . 18 F or ζ = − 1 , the same result up to replacing R by R[ 1 2 ] , in order to ensure that 1 − ζ ∈ G m (R) . 27 TESS BOUIS AND QUENTIN GAZDA Pr o of. First note that 1 − ζ is a unit in R . Indeed, if g is the order of ζ , then 1 − ζ divides Φ g (1) in R , and Φ g (1) divides g in Z ( Φ g (1) is either equal to p if g = p r is a prime p ow er, or equal to 1 otherwise), hence 1 − ζ divides g in R . Moreo v er, b ecause g > 2 , w e know that the prime supp orts of g and of ∆ Q ( ζ ) agree. Using that ∆ Q ( ζ ) divides ∆ K in Z , this implies that g is inv ertible in R , and so is 1 − ζ . By Corollary 4.18 and Deﬁnition 4.21 , it suﬃces to prov e that for every integer m ⩾ 1 whic h is coprime to N (and in particular to the order of ζ ), one has the equality 1 d log  ˜ Π d (Π Hab m (1 − ζ )) F rob Hab d (Π Hab m (1 − ζ ))  = −  1 1 − [ ζ ] m X 0

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