Divided power structures and chain complexes

DIVIDED PO WER STR UCTURES AND CHAIN COMPLEXES BIRGIT RICHTER Abstra ct. W e interpret d ivided p o wer structures on the homotopy groups of simplicial com- mutativ e rings as ha ving a counterpart i n divided p ow er structures on chain complexes coming from a n on-standard symmetric monoidal structu re. 1. Introduction Ev ery comm utativ e simplicial algebra has a divided p o we r stru cture on its homotop y groups. The Dold-Kan corresp ondence compares simp licial mo dules to n on-negativ ely graded c h ain complexes. It is an equiv alence of categories, but its multiplica tiv e prop erties do not in teract w ell with comm u tativit y: comm utativ e simp licial al gebras are sen t to homotop y comm utativ e differen tial graded alg ebras, bu t in general not to differen tial graded a lgebras that are com- m utativ e on the nose. The aim of this note is to gain a b etter un derstanding when s u itable m ultiplicativ e structures on a c hain complex actually d o giv e r ise to divided p ow er structures. T o this end, we use the equiv alence of categories b etw een sim p licial mo d ules and non-n egativ ely graded c h ain complexes and transfer the tensor pro duct of simplicial mo dules to a s y m metric monoidal catego ry structure on the category of c hain complexes. W e start with a brief o v erview on divided p o w er algebras in section 2. W e prov e a general transfer result for symmetric monoidal catego ry structures in section 6. That s u c h a transf er of monoidal str u ctures is p ossible is a f olklore r esult and constructions like ours are used in other con texts, see for instance [Sc h01, p.263] and [Q ∞ ]. W e consider the case of c hain complexes in section 7 where w e u se this monoidal structure to gain our main results: in C orollary 7.2 we giv e a criterion w hen a c hain complex has a divided p o w er str ucture on its homology groups and in Theorem 7.5 we describ e wh en w e can actually gain a divided p o w er stru cture on the differen tial graded comm utativ e algebra that in teracts nicely with the different ial. As an example we giv e an alternativ e d escription of the well-kno wn ([C54, BK94]) divid ed p o we r structure on Ho c hsc hild homology: instead of working with the bar construction in th e differen tial graded setting, we co nsider the simplicial bar construction of a comm utativ e algebra. This is naturally a simp licial commutat iv e algebra and hence th e redu ced differen tial graded bar construction inherits a divided p o w er c hain algebra structure from its simplicial relativ e. 2. Divided power algebras Let R b e a commuta tiv e rin g with unit an d let A ∗ b e an N 0 -graded comm utativ e algebra with A 0 = R . W e denote the p ositiv e part of A ∗ , L i> 0 A i , by A ∗ > 0 . Definition 2.1. A system of divide d p owers in A ∗ consists of a collecti on of functions γ n for n > 0 th at are defined on A i for i > 0 su c h that the follo wing conditions are satisfied. (a) γ 0 ( a ) = 1 and γ 1 ( a ) = a for all a ∈ A ∗ > 0 . (b) Th e degree of γ i ( a ) is i times the d egree of a . (c) γ i ( a ) = 0 if the degree of a is o dd and i > 1. (d) γ i ( λa ) = λ i γ i ( a ) for all a ∈ A ∗ > 0 and λ ∈ R . 2000 Mathematics Subje ct Classific ation. Primary 18G30, 18G35; S econdary 18D50 . Key wor ds and phr ases. Divided p ow er structures, Dold-Kan corresp ondence, simplicial comm utative rings. I th ank Benoit F resse, Haynes Miller and T eim uraz Pirashvili for help with some of the references. Octob er 23, 2018. 1 (e) F or all a ∈ A ∗ > 0 γ i ( a ) γ j ( a ) =  i + j i  γ i + j ( a ) . (f ) F or all a, b ∈ A ∗ > 0 γ i ( a + b ) = X k + ℓ = i γ k ( a ) γ ℓ ( b ) . (g) F or all a, b ∈ A ∗ > 0 γ i ( ab ) = i ! γ i ( a ) γ i ( b ) = a i γ i ( b ) = γ i ( a ) b i . (h) F or all a ∈ A ∗ > 0 γ i ( γ j ( a )) = ( ij )! i !( j !) i γ ij . If w e wan t to sp ecify a fixed sy s tem of divided p o w ers ( γ i ) i > 0 on A ∗ , w e use the notation ( A ∗ , γ ). F or basics ab out systems of d ivid ed p o w ers s ee [C54, Exp os ´ e 7, 8], [GL69, sectio n 7], [E95, App end ix 2], [Be74, Chapitre I] an d [Ro68 ]. Some pr op erties. Condition (e) implies that for all i > 1 the i -fold p o we r of an elemen t a ∈ A ∗ > 0 is related to its i -th divid ed p o w er via (1) a i = i ! γ i ( a ) . Therefore, if the underlying R -mo dules A i are torsion-free, then there is at m ost one system of divided p ow ers on A ∗ , and if R is a fi eld of c haracteristic zero, then the assignmen t γ i ( a ) = a i /i ! defines a u nique system of divided p o w ers on ev ery A ∗ . If sq u ares of o d d degree elemen ts are zero and if w e are in a torsion-free context, then condition (c) is of course tak en care of by condition (e). Some authors ( e.g. [C54]) demand that the und erlying graded comm utativ e algebra is strict , i.e. , that a 2 = 0 wh enev er a has o dd degree. Not ev ery N 0 -graded comm utativ e algebra p ossesses a system of divid ed p o w ers. Consider for in stance the p olynomial r ings Z [ x ] o v er Z an d F 2 [ x ] o v er F 2 , where x is a generator in degree t w o. In F 2 [ x ], x 2 is not zero, but if there w ere a system of divided p o w ers, the equation x 2 = 2 γ 2 ( x ) would force x 2 to v anish. Ov er the inte gers, the existence of γ 2 ( x ) w ould imply that x 2 w ere divisible b y t w o and that’s not the case. Note, that the follo wing useful pro duct form ula (2) ( ij )! i !( j !) i = i Y r =2  r j − 1 j − 1  holds. W e s aw that o v er the rationals, divided p ow ers can b e expr essed in terms of the u nderlying m ultiplication of the N 0 -graded comm utativ e algebra. If the ground ring R is a fi eld of c har- acteristic p for s ome prime num b er p > 2, then for any system of divided p o w ers on A ∗ the relation a p = p ! γ p ( a ) forces the p -th p o w ers of elemen ts in A ∗ to b e trivial. There are m ore relations implied b y divided p o wer structures, for ins tance an y iteration of the form γ i ( γ p ( a )) is equal to γ ip ( ip )! i !( p !) i , but using relation (2) it is easy to see that the co efficien t of γ ip is congruen t to one and h ence γ i ( γ p ( a )) = γ ip ( a ) . F or a more thorough treatmen t of divided p ow ers in pr im e charact eristic see [C54, Exp os´ e 7, §§ 7,8], [A76], and [G90]. 2 Divide d p ower structur e with r esp e ct to an ide al. The o ccurence of divided p o w er stru ctures is not limited to the graded setting. In an N 0 -graded commutat iv e R -algebra A ∗ the p ositiv e part is an ideal. In the con text of un grad ed comm utativ e rings, divided p ow er structures can b e defined relativ e to an ideal. The follo win g definition is taken fr om [Be74, Ch apitre I Definition 1.1.]. Definition 2.2. Let A b e a commutativ e ring and I and ideal in A . A divide d p ower structur e on I consists of a family of maps γ i : I → A, i > 0 whic h satisfy the follo wing conditions (a) F or all a ∈ I , γ 0 ( a ) = 1 and γ 1 ( a ) = a . The image of the γ i for i > 2 is con tained in I . (b) F or all elemen ts a ∈ A and b ∈ I , γ i ( ab ) = a i γ i ( b ). (c) Con d itions (e), (f ) and (h) of Definition 2.1 app ly in an adapted sens e. An imp ortant example of divided p o w er structures on u ngraded rings is th e case of discrete v aluation r in gs of mixed charact eristic. If p is the characte ristic of the resid u e field, π is a uniformizer and p = uπ e with u a unit, then f or the existence of a divided p o we r structure on the discrete v aluation r ing is is necessary and sufficient that the ramification in dex e is less than or equal to p − 1 (see [Be74, Chapitre I, Prop osition 1.2.2]). Morphisms and fr e e obje cts. Morphisms are straigh tforw ard to define: Definition 2.3. Let ( A ∗ , γ ) and ( B ∗ , γ ′ ) b e t w o N 0 -graded comm utativ e algebras with systems of divided p o wers. A morph ism of N 0 -graded comm utativ e algebras f : A ∗ → B ∗ is a morphism of divi de d p ower structur es , if f ( γ i ( a )) = γ ′ i ( f ( a )) , for all i > 2 . An analogo us d efinition w orks in the u ngraded case. W e w ill describ e the free divided p o w er algebra generated by a N 0 -graded mo d ule M ∗ whose comp onent s M i are free R -mo dules. Definition 2.4. Consider the free R -mo du le generated b y an elemen t x of degree m . • If m is o d d, then the fr e e divide d p ower algebr a on x o v er R is the exterior alge bra ov er R generated b y x , Λ R ( x ). I n this case the γ i are trivial for i > 2. • If m is ev en, the fr e e divide d p ower algebr a on x over R is R [ X 1 , X 2 , . . . ] /I . Here the X n are p olynomial generators in degree nm and I is the ideal generated b y X i X j −  i + j i  X i + j . As the tensor pro d uct o v er R is the copro d u ct in the category of N 0 -graded comm utativ e R - algebras, we get a notion of a free d ivided p o w er algebra on a finitely generated mo d ule M ∗ whose M i are free as R -mo dules by taking care of Condition (g) of Defin ition 2.1. If M ∗ is not finitely generated we tak e the colimit of the free divid ed p o we r algebras on fin itely man y generators. Compare [GL69, Prop osition 1.7.6]. If M ∗ is an N 0 -graded mo d ule that is freely generated by elemen ts x 1 , . . . , x n , then it is common to denote the free divided p o wer algebra o v er R on these generators b y Γ R ( x 1 , . . . , x n ). Occurences of free divided p ow er algebras are ample. F or in stance, the cohomology ring of the lo op space on a sp here, Ω( S n ), for n > 2 is a f ree divided p o w er algebra. Using the Serre sp ectral sequence for the path-loop fibration Ω S n → P S n → S n one gets H ∗ (Ω( S n ); Z ) ∼ =  Γ Z ( a ) | a | = n − 1 , n od d Γ Z ( a, b ) ∼ = Λ Z ( a ) ⊗ Γ Z ( b ) | a | = n − 1 , | b | = 2 n − 2 , n ev en 3 Often, f ree divided p o wer algebras arise as duals of symm etric algebras. F or a N 0 -graded R -mo du le M ∗ , its R -du al, M ∗ , is the N 0 -graded R -mo d ule with M i = Hom R ( M i , R ). The symmetric algebra generated b y a N 0 -graded R -mo dule M ∗ is S ( M ∗ ) = T R ( M ∗ ) /C where C is the ideal generated b y the graded commutativi t y relation ab − ( − 1) | a || b | ba. Here, | a | den otes the degree of a and T R ( M ∗ ) = L i > 0 M ⊗ i ∗ is the tensor algebra generated b y M ∗ . The d iagonal map M ∗ → M ∗ ⊕ M ∗ , a 7→ ( a, a ) induces a cocommutat iv e coal gebra str ucture ∆ : S ( M ∗ ) − → S ( M ∗ ⊕ M ∗ ) ∼ = S ( M ∗ ) ⊗ S ( M ∗ ) . On the graded dual of S ( M ∗ ), S ( M ∗ ) ∗ , this com ultiplication yields a graded commutat iv e m ultiplication m = ∆ ∗ : S ( M ∗ ) ∗ ⊗ S ( M ∗ ) ∗ − → S ( M ∗ ) ∗ . If M ∗ is a N 0 -graded R -mo dule whose comp onents are fr ee o ver R , th en S ( M ∗ ) ∗ has a system of divided p o w ers (see for instance [E95, A2.6]). Let Σ n denote the symmetric group on n -letters. If N ∗ is an is an N 0 -graded mo d u le with Σ n -action, then w e denote by N Σ n ∗ the in v ariants in N ∗ with resp ect to the Σ n -action. If M is a free R -mo dule, then one can describ e the free d ivided p o we r algebra on M as (3) M n > 0 ( M ⊗ n ∗ ) Σ n . This is a classical result and is for instance pr o v ed in [C54, Exp os ´ e 8, Prop osition 4]. See Roby [Ro68, Remarqu e p . 103] for an example wh er e th e t w o notions differ if one considers a mo du le that is not free. Divided p o w er structures can in fact b e d escrib ed via (3): a graded mo dule with free comp onent s, M ∗ , with M 0 = 0 has a d ivided p o w er structure if there is a m ap (4) M n > 1 ( M ⊗ n ∗ ) Σ n → M ∗ that satisfies the axioms of a monad action (see [F00]). T he monad s tructure that is ap p lied in the desciption via (4) uses the in v ertibilit y of the n orm map on reduced symmetric sequences of the form M ⊗ n ∗ [F00, 1.1.16 and 1.1.18] . The in v ertibilit y of the n orm map in this case wa s disco v ered earlier by Sto v er [St93, 9.10]. 2.0.1. Divide d p ower structur es in the simplicial c ontext. On the h omotop y group s of sim p licial comm utativ e rings there are divided p o w er op erations and it is this instance of d ivid ed p o w er structures that w e will in v estigat e in this pap er. In the con text of the action of the Steenro d algebra on cohomology groups of spaces, the top oper ation in th e p -th p o w er map. On the homotop y groups of simplicial comm utativ e F 2 - algebras, there are analogous op eratio ns δ i of degree i > 2 su ch that the h ighest op eration is the divided square. These op erations w ere in v estigated by C artan [C54, Exp os ´ e no 8] and w ere in tensely studied b y many p eople ([Bo67, sectio n 8], [D80], [G90, c hapter 2], [T99]). In [Bo67, 8.8 onw ards] a family of op erations for od d p rimes is d iscussed as well. Notation. With ∆ we den ote th e category wh ose ob jects are the sets [ n ] = { 0 , . . . , n } with their natur al ordering and morphisms from in ∆ are monotone maps. A simplicial ob ject in a catego ry C is a fu n ctor fr om the opp osite category of ∆, ∆ op , to C . W e denote the category of simplicial ob jects in C by s C . In th e category of simp licial sets, the rep resen table functors ∆( n ) : ∆ op → S ets are the ones that send [ m ] ∈ ∆ to ∆([ m ] , [ n ]). If δ i : [ n ] → [ n + 1] denotes the map that is the inclusion that misses i and is strictly monotone everywhere else and if σ i : [ n ] → [ n − 1] is the 4 surjection that sends i and i + 1 to i and is strictly monotone elsewhere, then we denote their opp osite maps b y d i = ( δ i ) op and s i = ( σ i ) op . If S is a set and R is a rin g, then we d enote the free R -mo dule generated b y S by R [ S ]. T he tensor pro duct of t wo R -mo du les N and M , N ⊗ R M , will b e abbr eviated b y N ⊗ M . 3. The Dold-Kan corr esponde nce The Dold-Kan corresp ondence [Do5 8, Theorem 1.9] compares the category of sim p licial ob- jects in an ab elian category A to th e non-n egativ ely graded c hain complexes o v er A via a sp ecific equiv alence of categories. In the follo wing w e fix an arb itrary comm utativ e ring with unit R and w e will fo cus on the corresp ondence b et ween simp licial R -mo dules , smo d R , and n on-negativ ely graded c h ain complexes of R -mo du les, Ch R > 0 . The equiv alence is giv en b y the normaliza tion functor , N : smo d R → Ch R > 0 , and we d enote its in v erse by Γ N : smo d R / / Ch R > 0 : Γ o o In p articular the f unctor N is a left adj oint to Γ. The v alue of N on a simplicial R -mo du le X • in c hain degree n is N n ( X • ) = n \ i =1 k er( d i : X n − → X n − 1 ) where the d i are the s implicial structure maps. The differential d : N n ( X • ) → N n − 1 ( X • ) is giv en b y the remaining face map d 0 . Recall that for a c hain complex C ∗ , Γ n ( C ∗ ) = n M p =0 M  : [ n ] ։ [ p ] C  p where  is an order p reserving surjection and C  p = C p . The n orm alized chain complex has an alternativ e d escription via a quotien t constru ction where one reduces mo du lo d egenerate elemen ts [W94, lemma 8.3.7]. There is a canonical ident ification ϕ C ∗ : N Γ C ∗ ∼ = C ∗ : if  is not the iden tit y map, then the simplicial structure of Γ( C ∗ ) id en tifies elemen ts in C  p as b eing degenerate. F or a simplicial R -mo d u le A • the isomorphism ψ A • : Γ N ( A • ) ∼ = A • is induced by the map that sends N ( A • )  p ⊂ A p via  to A n . The tensor p ro duct of c hain complexes ( C ∗ , d ) and ( C ′ ∗ , d ′ ) is defined as usual via ( C ∗ ⊗ C ′ ∗ ) n = M p + q = n C p ⊗ C ′ q with differen tial D ( c ⊗ c ′ ) = ( dc ) ⊗ c ′ + ( − 1) p c ⊗ d ′ c ′ for c ∈ C p , c ′ ∈ C ′ q . Let ( R, 0) denote the c hain complex th at has R as degree zero part and that is trivial in all other d egrees. Th ere is a t wist isomorphism τ C ∗ ,C ′ ∗ : C ∗ ⊗ C ′ ∗ → C ′ ∗ ⊗ C ∗ that is induced b y τ C ∗ ,C ′ ∗ ( c ⊗ c ′ ) = ( − 1) pq c ′ ⊗ c for c and c ′ as ab o v e. The structure ( Ch R > 0 , ⊗ , ( R, 0) , τ ) tur ns Ch R > 0 in to a symm etric monoidal catego ry . F or tw o arbitrary simplicial R -mo dules A • and B • , let A • ˆ ⊗ B • denote the d egree-wise tensor pro du ct of A • and B • , i.e. , ( A • ˆ ⊗ B • ) n = A n ⊗ k B n . Here, the simp licial structure m aps are applied in eac h comp onen t; in particular, th e differentia l on N ∗ ( A • ˆ ⊗ B • ) in degree n is d 0 ⊗ d 0 . The constan t simplicial ob ject R whic h consists of R in every degree is the u n it with resp ect to ˆ ⊗ and the twist ˆ τ A • ,B • : A • ˆ ⊗ B • − → B • ˆ ⊗ A • , ˆ τ A • ,B • ( a ⊗ b ) = b ⊗ a giv es ( smo d R , ˆ ⊗ , R , ˆ τ ) the structure of a sy m metric monoidal category . Note that N ( R ) ∼ = ( R, 0). There are n atural maps, the shuffle maps , sh : N ( A • ) ⊗ N ( B • ) − → N ( A • ˆ ⊗ B • ) 5 (see [ML95, VI I I.8]) that turn the normalization into a lax symmetric monoidal fun ctor, i.e. , the sh uffle maps are associativ e in a suitable sense and the diagram N ( A • ) ⊗ N ( B • ) sh / / τ   N ( A • ˆ ⊗ B • ) N ( ˆ τ )   N ( B • ) ⊗ N ( A • ) sh / / N ( B • ˆ ⊗ A • ) comm utes for all A • , B • ∈ smo d R . Ho w ev er, th e in v erse of N , Γ, is not lax symmetric monoidal. In order to compare Γ( C ∗ ) ˆ ⊗ Γ( C ′ ∗ ) and Γ( C ∗ ⊗ C ′ ∗ ) one uses the Alexander-Whitney m ap a w : N ( A • ˆ ⊗ B • ) − → N ( A • ) ⊗ N ( B • ) and this natural map is n ot symmetric, i.e. , th e diagram N ( A • ˆ ⊗ B • ) aw / / N ( ˜ τ )   N ( A • ) ⊗ N ( B • ) τ   N ( B • ˆ ⊗ A • ) aw / / N ( B • ) ⊗ N ( A • ) do es not comm ute. Sc h w ede and Shipley pro v ed, that the Dold-Kan corresp ondence passes to a Quillen equiv a- lence b etw een the category of asso ciativ e simplicial rings and the category of differentia l graded asso ciativ e algebras that are concent rated in non-n egativ e degrees. They consider the normal- ization fun ctor and construct an adjoint on the lev el of m onoids which then giv es r ise to a monoidal Quillen equiv alence [Sc hSh 03]. If one starts with a d ifferential graded commutati v e algebra, then Γ send s this algebra to a simplicial E ∞ -algebra [R03, T heorem 4.1]. In general, th e Dold-Kan corresp ond ence giv es rise to a Quillen ad j unction b et ween simplicial homotopy O -alg ebras and d ifferen tial graded homotop y O -algebras for op erads O in R -mo dules [R06, Theorem 5.5.5 ]. 4. The divided power structure on ho m otopy groups o f commut a t ive simpl icial algebras In the follo wing we view Σ ni as the group of bijections of the set { 0 , . . . , n i − 1 } . F or a p ermutatio n σ w e use ε ( σ ) for its sign um. W e consider the set of shuffle p erm utations, Sh( n, . . . , n | {z } i ) ⊂ Σ ni . This set consists of p erm utations σ ∈ Σ ni suc h that σ (0) < . . . < σ ( n − 1) , . . . , σ (( i − 1) n ) < . . . < σ ( ni − 1) . Let j denote the blo ck of n umbers ( j − 1) n < . . . < j n − 1 for 1 6 j 6 i and let [ ni − 1] \ j denote the complemen t of j with its inh er ited ord er in g fr om the one of [ ni − 1]. W e use the abbreviation s σ ([ ni ] \ j ) for the comp osition of the d egeneracy maps s σ ( k ) where k ∈ [ ni − 1] \ j and th e order of the comp osition uses sm all ind ices fi rst. F or example, let σ ∈ Sh(2 , 2 , 2) b e the p ermutatio n σ = (0 , 2)(1 , 4)(3 , 5) 0 N N N N N N N N N N 1 T T T T T T T T T T T T T T T 2 p p p p p p p p p p 3 O O O O O O O O O O 4 j j j j j j j j j j j j j j j 5 p p p p p p p p p p 0 1 2 3 4 5 . In this case, s σ ([5] \ 2 ) = s 4 ◦ s 3 ◦ s 2 ◦ s 1 . Let A • b e a comm utativ e simplicial R -algebra, i.e. , a commutati v e monoid in ( smo d R , ˆ ⊗ , R , ˆ τ ). The homotop y groups of A • , π ∗ ( A • ), are th e h omology groups of the normalization of A • , H ∗ ( N ( A • )). Starting with a cycle a ∈ N n ( A • ) w e can m ap a it to its i -fold tensor p o wer N n ( A • ) ∋ a 7→ a ⊗ i ∈ N n ( A • ) ⊗ i . 6 The i -fold iterated s huffle map sen ds a ⊗ i to X σ ∈ Sh ( n,...,n ) ε ( σ ) s σ ([ ni − 1] \ 1 ) ( a ) ⊗ . . . ⊗ s σ ([ ni − 1] \ n ) ( a ) so that the outcome is an elemen t in A ni ⊗ . . . ⊗ A ni = ( A • ˆ ⊗ . . . ˆ ⊗ A • ) ni . As none of th e d egeneracy maps arises n times, w e consider the image as an elemen t of N ni ( A • ˆ ⊗ . . . ˆ ⊗ A • ). If w e comp ose the i -fold diagonal map with the i -fold iterated shuffle map follo wed by the comm utativ e multiplicat ion in A • , we can view the comp osite as a map P i : N n ( A • ) − → N ni ( A • ) . A tedious calculation sho ws that P i is actually a chain map. On the level of homology , th is comp osite sends a h omology class to its i -fold p o w er. The group Σ i acts on the set of shuffles Sh( n, . . . , n | {z } i ) by p erm uting the i blo c ks of size n . If ξ ∈ Σ i w e denote the corresp on d ing blo c k p ermutatio n by ξ b . F or a σ ∈ S h( n, . . . , n | {z } i ) and ξ ∈ Σ i , σ ◦ ξ b is again an elemen t of S h( n, . . . , n | {z } i ). As A • is commutati ve , we ha v e that the m ultiplication applied to a summ and s σ ([ ni − 1] \ 1 ) ( a ) ⊗ . . . ⊗ s σ ([ ni − 1] \ n ) ( a ) gives the same output as the multiplicatio n applied to the su mmand corresp onding to σ ◦ ξ b . If the characte ristic of the ground ring is not t w o, then i -fold p ow ers for i > 2 are trivial unless n is ev en. The signum of the p erm utation σ ◦ ξ b is the signum of ξ b m ultiplied by ε ( σ ). F or eac h crossin g in ξ the blo c k p ermutatio n ξ b has n 2 crossings, so ξ b is in the alternating group in this case. If the characte ristic of R is 2, then signs do not matter. Definition 4.1. F or a simplicial comm utativ e R -algebra A • , the i -th divide d p ower of [ a ] ∈ π n ( A • ) = H n ( N ( A • )) is defined as the class of µ ◦ X σ ∈ Sh ( n,...,n ) / Σ i ε ( σ ) s σ ([ ni ] \ 1 ) ( a ) ⊗ . . . ⊗ s σ ([ ni ] \ n ) ( a ) where w e c ho ose a system of rep resen ting elemen ts σ ∈ Sh( n, . . . , n ) / Σ i . W e denote the i -th divided p ow er of a ∈ π n ( A • ) by γ i ( a ). As w e kno w that all elements σ ′ in the same coset as σ giv e r ise to the same v alue un der th e map P i , we obtain that a i = i ! γ i ( a ) . With the con v en tions γ 0 ( a ) = 1 and γ 1 ( a ) = a w e obtain the follo wing (see for instance [F00, § 2.2] for a p ro of ). Prop osition 4.2. The system of divide d p owers in the homotopy gr oups of a c ommutative simplicial R -algebr a A • , ( π ∗ ( A • ) , γ ) satisfies the pr op erties fr om Definition 2.1. 5. A large s ymmetric monoidal pr oduct on chain com plexes W e w ill use the follo wing pro d uct later in order to in v estigate divided p o we r s tructures on c hain complexes. Definition 5.1. W e defin e the lar ge tensor pr o duct of t w o chai n complexes C ∗ and C ′ ∗ to b e C ∗ ˜ ⊗ C ′ ∗ := N (Γ( C ∗ ) ˆ ⊗ Γ( C ′ ∗ )) . Note that the large tensor pro du ct deserv es its name: the d egenerate elements in Γ( C ∗ ) ˆ ⊗ Γ( C ′ ∗ ) are only the on es that are images of the maps s i ˆ ⊗ s i , and in general N (Γ( C ∗ ) ˆ ⊗ Γ( C ′ ∗ )) is muc h larger than the ordinary tensor p ro duct C ∗ ⊗ C ′ ∗ ∼ = N (Γ( C ∗ )) ⊗ N (Γ( C ′ ∗ )). 7 As a concrete example, consider the norm alized c hain complex on the standard simplex Z [∆(1)]. Let us denote a monotone map f : [ n ] → [1] b y an ( n + 1)-tup el corresp onding to its image, so that for instance f : [3] → [1] , f (0) = 0 , f (1) = f (2) = f (3) = 1 is represen ted by (0 , 1 , 1 , 1). As Z [∆(1)] has non-d egenerate simp lices only in degrees zero and one corresp ond ing to the mon otone maps (0) and (1) ∈ ∆([1] , [0]) and (0 , 1) in ∆ ([1] , [1]), its normalization C ∗ = N ( Z [∆(1)]) is the chain complex Z ⊕ Z ← Z ← 0 ← . . . and the b oun dary map sends the generator (0 , 1) to (0) − (1). Therefore, C ∗ ⊗ C ∗ is a c hain complex, that is concen tr ated in degrees zero, one and t w o w ith c h ain groups of rank f our, four and one r esp ectiv ely . Note th at N (Γ( N ( Z [∆(1)])) ˆ ⊗ Γ( N ( Z [∆(1)]))) ∼ = N ( Z [∆(1)] ˆ ⊗ Z [∆(1)]) . Th us for instance in degree one, C ∗ ˜ ⊗ C ∗ is of rank sev en . 6. Equiv alences of ca te gories and transf er o f monoidal str uctur es If F : C → D and G : D → C is a pair of fu nctors that constitute an equiv alence of categories and if ( D , ˆ ⊗ , 1 , ˆ τ ) is symmetric monoidal, then we can transfer the symmetric monoidal s tructure on D to one on C in the follo wing w a y . • As for c hain complexes, one defines a pro d uct ˜ ⊗ via C 1 ˜ ⊗ C 2 = G ( F C 1 ˆ ⊗ F C 2 ) for ob jects C 1 , C 2 of C . • As an equiv alence ˜ τ C 1 ,C 2 : C 1 ˜ ⊗ C 2 = G ( F C 1 ˆ ⊗ F C 2 ) → G ( F C 2 ˆ ⊗ F C 1 ) = C 2 ˜ ⊗ C 1 w e take G ( ˆ τ F C 1 ,F C 2 ). • The unit for the symmetric monoidal structure is G (1) . F or later reference, w e sp ell out some of th e structural isomorp hisms. Recall that an y equiv alence of catego ries giv es rise to an adjoint equiv alence [ML95, I V.4]; in particular th e u nit and counit of the adjunction are isomorp hisms. W e w an t to denote the natural isomorphism from GF C to C for C an ob ject of C by ϕ C and th e one fr om F GD to D by ψ D for all ob jects D in D . T hen the iden tities (5) F ( ϕ C ) = ψ F C and G ( ψ D ) = ϕ GD hold for all C and D . F or the left unit we ha v e to iden tify C with G (1) ˜ ⊗ C and to this end we use the morphism ˜ ℓ : C ϕ − 1 / / G ( F ( C )) G ( ˆ ℓ ) / / G (1 ˆ ⊗ F ( C )) G ( ψ − 1 ˆ ⊗ id) / / G ( F ( G (1)) ˆ ⊗ F ( C )) = G (1) ˜ ⊗ C where ˆ ℓ is the left un it isomorp hism for ˆ ⊗ . T he righ t unit is d efined s im ilarly . The asso ciativit y isomorphism ˜ α is giv en in terms of the one for ˆ ⊗ , ˆ α as ˜ α := G (id ˆ ⊗ ψ ) − 1 ◦ G ( ˆ α ) ◦ G ( ψ ˆ ⊗ id) : G ( F G ( F ( C ) ˆ ⊗ F ( C )) ˆ ⊗ F ( C )) G ( ψ ˆ ⊗ id)   ˜ α / / G ( F ( C ) ˆ ⊗ F G ( F ( C ) ˆ ⊗ F ( C ))) G (id ˆ ⊗ ψ )   G (( F ( C ) ˆ ⊗ F ( C )) ˆ ⊗ F ( C )) G ( ˆ α ) / / G ( F ( C ) ˆ ⊗ ( F ( C ) ˆ ⊗ F ( C ))) 8 Then it is a tedious, b ut straight forwa rd task to sho w the follo wing resu lt. A pro of in the non-symmetric setting can b e found in [Q ∞ , T heorem 3]. Prop osition 6.1. The c ate gory ( C , ˜ ⊗ , G (1) , ˜ τ ) is a symmetric monoida l c ate gory. If C already has a symmetric monoidal structure, then w e can compare the old one to the new one as follo ws. Prop osition 6.2. If ( C , ⊗ , G (1) , τ ) i s a symmetric monoid al structur e and if G : ( D , ˆ ⊗ , 1 , ˆ τ ) → ( C , ⊗ , G (1) , τ ) is lax symmetric monoidal, then the i dentity functor id : ( C , ˜ ⊗ , G (1) , ˜ τ ) − → ( C , ⊗ , G (1) , τ ) is lax symmetric monoidal. Pr o of. W e ha v e to constru ct maps λ C 1 ,C 2 : C 1 ⊗ C 2 → C 1 ˜ ⊗ C 2 that are natural in C 1 and C 2 and that render the diagrams C 1 ⊗ C 2 τ   λ C 1 ,C 2 / / C 1 ˜ ⊗ C 2 ˜ τ C 1 ,C 2   C 2 ⊗ C 1 λ C 2 ,C 1 / / C 2 ˜ ⊗ C 1 comm utativ e for all C 1 , C 2 ∈ C . Let Υ b e th e transformation that turn s G in to a lax symmtric monoidal functor. W e defin e λ C 1 ,C 2 : C 1 ⊗ C 2 ϕ − 1 C 1 ⊗ ϕ − 1 C 2 / / GF ( C 1 ) ⊗ GF ( C 2 ) Υ F ( C 1 ) ,F ( C 2 ) / / G ( F ( C 1 ) ˆ ⊗ F ( C 2 )) = C 1 ˜ ⊗ C 2 .  Corollary 6.3. E very c ommutat ive monoid in ( C , ˜ ⊗ , G (1) , ˜ τ ) is a c ommutative monoid i n ( C , ⊗ , G (1) , τ ) . The fu nctor F compares comm utativ e monoids in the catego ries C and D as follo ws . Theorem 6.4. An obje ct F ( C ) is a c ommutative monoid in ( D , ˆ ⊗ , 1 , ˆ τ ) i f and only if C is a c ommutative monoid i n ( C , ˜ ⊗ , G (1) , ˜ τ ) . M or e over, the assignment C 7→ F ( C ) is a f unctor fr om the c ate gory of c ommutative mono ids in ( C , ˜ ⊗ , G (1) , ˜ τ ) to the c ate gory of c ommutative monoids in ( D , ˆ ⊗ , 1 , ˆ τ ) . Pr o of. If w e assume that C is a commutativ e monoid in ( C , ˜ ⊗ , G (1) , ˜ τ ), then C has an asso ciativ e m ultiplication ˜ µ : C ˜ ⊗ C = G ( F ( C ) ˆ ⊗ F ( C )) − → C that satisfies ˜ µ ◦ ˜ τ = ˜ µ and there is a unit map j : G (1) → C . W e consider the comp osition ϕ − 1 ◦ ˜ µ : G ( F ( C ) ˆ ⊗ F ( C )) → GF ( C ). As G is an equiv alence of categories, it is a full functor, i.e. , the morph ism ϕ − 1 ◦ ˜ µ is of the f orm G ( ˆ µ ) for s ome morphism ˆ µ : F ( C ) ˆ ⊗ F ( C ) → F ( C ) in D . W e w ill sh o w that ˆ µ turns F ( C ) in to a comm utativ e m on oid. W e defi ne the un it map as i = F ( j ) ◦ ψ − 1 : 1 → F ( C ). As ˜ τ = G ( ˆ τ ), the comm utativit y of ˆ µ follo ws from th e one of ˜ µ and the fact that the fun ctor G is f aithful. In ord er to chec k the unit prop ert y of i w e ha v e to show that the follo wing diagram comm utes: (6) F ( C ) ˆ ℓ / / 1 ˆ ⊗ F ( C ) ψ − 1 ˆ ⊗ id / / F ( G (1)) ˆ ⊗ F ( C ) F ( j ) ˆ ⊗ id   F ( C ) F ( C ) ˆ ⊗ F ( C ) . ˆ µ o o As j is a unit for the m ultiplication ˜ µ w e kno w that ˜ µ ◦ ( j ˜ ⊗ id) ◦ G ( ψ − 1 ˆ ⊗ id) ◦ G ( ˆ ℓ ) ◦ ϕ − 1 C = id C . 9 Applying the faithful functor F to this identit y and u sing the defi nition of ˆ µ , we get that F G ( ˆ µ ) ◦ F ( j ˜ ⊗ id) ◦ F G ( ψ − 1 ˆ ⊗ id) ◦ F G ( ˆ ℓ ) = id F GF ( C ) . By the v er y defin ition, F ( j ˜ ⊗ id) is F ( G ( F ( j ) ˆ ⊗ id)) and th us v ia the faithfu lness of F G w e can conclude that diagram (6) commutes. The analogous statemen t for the right unit can b e sho wn similarly and hence i is a un it. F or the associativit y of the multiplicat ion ˆ µ we ha v e to sho w that the inner p en tagon in the follo w ing diagram comm utes. G ( F G ( F ( C ) ˆ ⊗ F ( C )) ˆ ⊗ F ( C )) ˜ µ ˜ ⊗ id ' ' G ( ψ ˆ ⊗ id)   ˜ α / / G ( F ( C ) ˆ ⊗ F G ( F ( C ) ˆ ⊗ F ( C ))) G (id ˆ ⊗ ψ )   id ˜ ⊗ ˜ µ w w G (( F ( C ) ˆ ⊗ F ( C )) ˆ ⊗ F ( C )) G ( ˆ µ ˆ ⊗ id)   G ( ˆ α ) / / G ( F ( C ) ˆ ⊗ ( F ( C ) ˆ ⊗ F ( C ))) G (id ˆ ⊗ ˆ µ )   G ( F ( C ) ˆ ⊗ F ( C )) G ( ˆ µ ) ( ( P P P P P P P P P P P P ˜ µ - - G ( F ( C ) ˆ ⊗ F ( C )) G ( ˆ µ ) v v n n n n n n n n n n n n ˜ µ q q GF ( C ) ϕ   C The outer d iagram comm utes b ecause ˜ µ is asso ciativ e and the upp er square commutes b ecause ˜ α is giv en in terms of ˆ α in this wa y . The only thing that remains to b e prov en is that th e outer wings comm ute. W e prov e th e claim for the left wing. Using the definitions of the maps in v olv ed, we hav e to sh o w th at G ( F G ( F ( C ) ˆ ⊗ F ( C )) ˆ ⊗ F ( C )) ˜ µ ˜ ⊗ id= G ( F ( ˜ µ ) ˆ ⊗ id) * * U U U U U U U U U U U U U U U U U G ( ψ F ( C ) ˆ ⊗ F ( C ) ˆ ⊗ id) / / G (( F ( C ) ˆ ⊗ F ( C )) ˆ ⊗ F ( C )) G ( ˆ µ ˆ ⊗ id) t t j j j j j j j j j j j j j j j j G ( F ( C ) ˆ ⊗ F ( C )) comm utes. W e know that F G ( ˆ µ ) = F ( ϕ − 1 C ) ◦ F ( ˜ µ ) and therefore ψ F ( C ) ◦ F G ( ˆ µ ) = F ( ˜ µ ) . The naturalit y of ψ implies th at ψ F ( C ) ◦ F G ( ˆ µ ) = ˆ µ ◦ ( ψ F ( C ) ˆ ⊗ F ( C ) ) and the claim follo ws. If F ( C ) is a comm utativ e m onoid with resp ect to ˆ ⊗ with m ultiplication ˆ µ and u nit i : 1 → F ( C ), then w e claim that ˜ µ := ϕ C ◦ G ( µ ) and j := ϕ C ◦ G ( i ) giv e C the s tr ucture of a comm utativ e monoid with resp ect to ˜ ⊗ . The fact that ˜ µ is comm utativ e follo ws directly b ecause ˜ τ = G ( ˆ τ ). Th e pro of of the unit axiom and of associativit y use the same diagrams as ab o v e with the arguments rev ersed. It remains to show that a morphism f : C 1 → C 2 of comm utativ e monoids with resp ect to ˜ ⊗ giv es rise to a morphism F ( f ) : F ( C 1 ) → F ( C 2 ) of commutativ e m onoids with resp ect to ˆ ⊗ . It 10 suffices to sho w th at the outer diagram in G ( F ( C 1 ) ˆ ⊗ F ( C 1 )) G ( ˆ µ ) $ $ G ( F ( f ) ˆ ⊗ F ( f )) / / ˜ µ   G ( F ( C 2 ) ˆ ⊗ F ( C 2 )) ˜ µ   G ( ˆ µ ) z z C 1 f / / C 2 GF ( C 1 ) ϕ C 1 O O GF ( f ) / / GF ( C 2 ) ϕ C 2 O O comm utes. The upp er square commute s by assump tion, the low er square comm u tes b ecause ϕ is natural and the win gs commute b y the very defin tion of ˆ µ in terms of ˜ µ .  7. Divided power structures and com m ut a tive monoids If w e apply the ab o v e resu lts to the Dold-Kan corresp ondence with Γ = F , N = G , C = Ch R > 0 and D = smo d R and the large tensor pro d uct of c hain complexes, th en we obtain the follo win g statemen ts that w e collect in one th eorem. Theorem 7.1. (a) The c ate gory of c hain c omplexes with the lar ge tensor pr o duct is a symmetric monoidal c ate gory with N ( R ) b eing the unit of the monoidal structur e and ˜ τ = N ( ˆ τ Γ( C ∗ ) , Γ( C ′ ∗ ) ) : N (Γ( C ∗ ) ˆ ⊗ Γ( C ′ ∗ )) − → N (Γ( C ′ ∗ ) ˆ ⊗ Γ( C ∗ )) as the twist. (b) The identity functor id : ( Ch R > 0 , ˜ ⊗ , N ( R ) , ˜ τ ) − → ( Ch R > 0 , ⊗ , ( R, 0) , τ ) is lax symmetric mono idal. (c) Ev ery c ommutative monoid in ( Ch R > 0 , ˜ ⊗ , N ( R ) , ˜ τ ) is a differ e ntial gr ade d c ommutative R -algebr a. (d) A simplicial R -mo dule Γ( C ∗ ) is a simplicial c ommutative R -algebr a if and only if the chain c ompl ex C ∗ is a c ommutative monoid in ( Ch R > 0 , ˜ ⊗ , N ( R ) , ˜ τ ) . The assignment C ∗ 7→ Γ( C ∗ ) is a functor fr om the c ate gory of c ommutative monoids in ( Ch R > 0 , ˜ ⊗ , N ( R ) , ˜ τ ) to the c ate gory of simplicial c ommutative R -algebr as. Corollary 7.2. E v ery c ommutative monoid C ∗ in ( Ch R > 0 , ˜ ⊗ , N ( R ) , ˜ τ ) has a divide d p ower struc- tur e on its homolo gy. The con v erse of statemen t (c) of T h eorem 7.1 is not true: not ev er y differential graded comm utativ e algebra p ossesses a divided p ow er s tructure on its homology , so these algebras cannot b e commutati v e monoids in ( Ch R > 0 , ˜ ⊗ , N ( R ) , ˜ τ ). F or instance the p olynomial ring Z [ x ] o v er the in tegers with x of degree t w o and trivial differen tial provides an example. Definition 7.3. W e denote by Com ˜ ⊗ the category whose ob jects are commutati v e monoids in ( Ch R > 0 , ˜ ⊗ , N ( R ) , ˜ τ ) and w hose m orphisms are multi plicativ e c hain m ap s , i.e . , if C ∗ , D ∗ are ob jects of Com ˜ ⊗ with unit maps j C ∗ : N ( R ) → C ∗ and j D ∗ : N ( R ) → D ∗ , then a morp hism is a c hain map f : C ∗ → D ∗ with f ◦ j C ∗ = j D ∗ and suc h that the d iagram C ∗ ˜ ⊗ C ∗ ˜ µ C ∗   f ˜ ⊗ f / / D ∗ ˜ ⊗ D ∗ ˜ µ D ∗   C ∗ f / / D ∗ comm utes. 11 If we start with a C ∗ ∈ Com ˜ ⊗ , then in particular, C ∗ is a differentia l graded algebra, i.e. , the differen tial d on the underlyin g chain complex C ∗ is compatible with the pro duct stru cture: it satisfies the Leibniz rule d ( ab ) = d ( a ) b + ( − 1) | a | ad ( b ) , for all a, b ∈ C ∗ . If there are divided p o w er structures on the underlying graded comm u tative algebra C ∗ , then w e w an t these to b e compatible with the differen tial. Definition 7.4. 1) A comm utativ e differentia l algebra C ∗ with divided p ow er op eratio ns is called a divide d p ower chain algebr a , if th e d ifferen tial d of C ∗ satisfies (a) d ( γ i ( c )) = d ( c ) · γ i − 1 ( c ) for all c ∈ C ∗ (b) If c is a b oundary , then γ i ( c ) is a b oundary for all i > 1. 2) A m orphism of comm utativ e different ial algebras f : C ∗ → D ∗ is a morphism of divide d p ower chain algebr as , if f satisfies f ( γ i ( c )) = γ i ( f ( c )) for all c ∈ C ∗ , i > 0. The first condition in 1) ens u res that divided p o w ers resp ect cycles an d together w ith the second condition this guaran tees that the homology of C ∗ inherits a d ivided p ow er structure from C ∗ . A reform ulation of the criterium f or a divided p o w er c hain algebra is used in [AH86, definition 1.3]: they demand th at every elemen t of p ositiv e degree is in the image of a morphism of differen tial graded comm utativ e algebras with d ivid ed p o w er structure f : D ∗ → C ∗ suc h that D ∗ satisfies condition (a) of Definition 7.4 and has trivial h omology in p ositiv e degrees. F or a comm utativ e monoid with r esp ect to ˜ ⊗ , C ∗ , the i -th p o wer of an element c ∈ N Γ( C ∗ ) is giv en via the follo wing composition c ∈ _   N Γ( C ∗ ) n c 7→ c ⊗ i / / N Γ( C ∗ ) ⊗ i n ⊂ ( N Γ( C ∗ ) ⊗ i ) ni sh   N (Γ( C ∗ ) ˆ ⊗ . . . ˆ ⊗ Γ( C ∗ )) ni = C ∗ ˜ ⊗ . . . ˜ ⊗ C ∗ N ( ˆ µ ) t t h h h h h h h h h h h h h h h h h h h ˜ µ   c i ∈ N Γ( C ∗ ) ni ϕ / / C ∗ . W e defin e a d ivided p ow er structur e on N Γ( C ∗ ) by using a v arian t of the shuffle map as in Definition 4.1 sending c ⊗ i to X σ ∈ Sh ( n,...,n ) / Σ i ε ( σ ) s σ ([ ni ] \ 1 ) ( c ) ⊗ . . . ⊗ s σ ([ ni ] \ n ) ( c ) and applying N ( µ ). Theorem 7.5. The c omp osite N Γ is a func tor fr om the c ate gory Com ˜ ⊗ to the c ate gory of divide d p ower chain algebr as. Pr o of. Let C ∗ b e a comm utativ e monoid in ( Ch R > 0 , ˜ ⊗ , N ( R ) , ˜ τ ). L et us first p r o v e that d ( γ i ( c )) = γ i − 1 ( c ) · d ( c ) for all c ∈ C ∗ in p ositiv e degrees. I f we app ly the b oundary d = d 0 to (7) γ i ( c ) = N ( µ )( X σ ∈ Sh ( n,...,n ) / Σ i ε ( σ ) s σ ([ ni − 1] \ 1 ) ( c ) ⊗ . . . ⊗ s σ ([ ni − 1] \ n ) ( c )) then w e can use that d 0 is a m orphism in the simplicial category to obtain d 0 ◦ N ( µ ) = N ( µ ) ◦ d 0 ⊗ . . . ⊗ d 0 . Only one of the sets σ ([ ni − 1] \ j ) do es n ot con tain zero. Therefore th e simplicial iden tities d 0 ◦ s i = s i − 1 ◦ d 0 for i > 0 and d 0 ◦ s 0 = id ensure, that in the sum (7) th er e are i − 1 tensor 12 factors con taining just degeneracies applied to c and only one term conta ining degeneracies applied to d 0 ( c ). A sh uffle p ermutation in Sh( n, . . . , n | {z } i − 1 ) tensorized with the id en tit y map follo w ed by a sh uffle in Sh( n ( i − 1) , n ) giv es a shuffle in Sh( n, . . . , n | {z } i ) and ev ery sh uffle in Sh( n, . . . , n | {z } i ) is decomp osable in the ab o v e w a y . Th ere are n Y j =2  nj n  i ! elemen ts in Sh( n, . . . , n | {z } i ) / Σ i and n − 1 Y j =2  nj n  ( i − 1)! ·  ni − 1 n − 1  elemen ts in the pro duct of Sh( n, . . . , n | {z } i − 1 ) / Σ i − 1 and Sh( n ( i − 1) , n ). As these n um b ers are equal, w e ob tain that the t w o sets are in bijection and d 0 ( γ i ( c )) can b e expr essed as th e pro d uct of γ i − 1 ( c ) and d 0 ( c ). The b oundary criterium for the d ivid ed p o wer stru cture can b e seen as follo ws: if c is of the form d 0 ( b ) for some b ∈ N Γ( C ∗ ) n +1 , th en γ i ( c ) is N ( µ )( X σ ∈ Sh ( n,...,n ) / Σ i ε ( σ ) s σ ([ ni − 1] \ 1 ) ( d 0 ( b )) ⊗ . . . ⊗ s σ ([ ni − 1] \ n ) ( d 0 ( b ))) . The simplicial iden tit y s j − 1 d 0 = d 0 s j for all j > 0 allo ws us to mo v e the d 0 -terms in fron t b y increasing the ind ices of the degeneracy maps. T herefore γ i ( d 0 ( b )) is equal to an expression of the f orm N ( µ ) N ( d 0 ⊗ . . . ⊗ d 0 )( x ) for some suitable x . As N ( µ ) N ( d 0 ⊗ . . . ⊗ d 0 ) is equal to d 0 ◦ N ( µ ) we obtain the desir ed result. F or f : C ∗ → D ∗ a morph ism of commutativ e monoids in ( Ch R > 0 , ˜ ⊗ , N ( R ) , ˜ τ ) we ha v e to sh o w that N Γ( f ) is a morphism of divid ed p ow er c hain algebras, i.e. , that it is a m ultiplicativ e c hain map that preserv es units and divid ed p o w ers. As f is a c hain m ap , Γ( f ) is a m ap of simp licial R -mo du les and N Γ( f ) is a c hain map. If j C ∗ and j D ∗ are the units for C ∗ and D ∗ , we ha v e f ◦ j C ∗ = j D ∗ and this implies N Γ( f ) ◦ N Γ( j C ∗ ) ◦ ϕ − 1 N ( R ) = N Γ ( j D ∗ ) ◦ ϕ − 1 N ( R ) and th us the unit cond ition holds. In order to establish that N Γ( f ) is multiplica tiv e w e ha v e to sho w that the bac k face in the diagram N Γ( C ∗ ) ⊗ N Γ( C ∗ ) µ / / sh + + V V V V V V V V V V V V V V V V V V N Γ( f ) ⊗ N Γ( f )   N Γ( C ∗ ) N Γ( f )   N (Γ( C ∗ ) ˆ ⊗ Γ( C ∗ )) ϕ − 1 ◦ ˜ µ = N ( ˆ µ ) 4 4 j j j j j j j j j j j j j j j j N (Γ( f ) ˆ ⊗ Γ( f ))   N Γ( D ∗ ) ⊗ N Γ( D ∗ ) µ / / sh + + V V V V V V V V V V V V V V V V V V N Γ( D ∗ ) N (Γ( D ∗ ) ˆ ⊗ Γ( D ∗ )) ϕ − 1 ◦ ˜ µ = N ( ˆ µ ) 4 4 j j j j j j j j j j j j j j j j 13 comm utes. The top and b ottom tr iangle comm ute by d efiniton, the left f ron t squ are commutes b ecause the shuffle map is natural and the right front square comm utes b ecause w e kn o w from Theorem 6.4 that Γ( f ) is multiplic ativ e. The fact that f preserves divided p ow ers can b e seen directly: using n aturalit y of ϕ and the m ultiplicativit y of f w ith resp ect to ˜ ⊗ , the only thin g we h a v e to v erify is the compatibilit y of f with the v arian t of the shuffle map. Bu t this map is a sum of tensors of degeneracy maps and as Γ( f ) r esp ects the simplicial s tructure, the claim follo ws.  Remark 7.6. W e can tran s fer the mo del structur e on simplicial commutati v e R -algebras as in [Qui67, I I , Theorem 4] to th e category Com ˜ ⊗ of comm utativ e m on oids in ( Ch R > 0 , ˜ ⊗ , N ( R ) , ˜ τ ) b y declaring that a map f : C ∗ → D ∗ is a we ak equiv alence, fibration resp. cofibration in C om ˜ ⊗ if and only if Γ( f ) is a wea k equiv alence, fibration resp. cofib ration in the mo d el stru cture on comm utativ e simplicial R -algebras. 8. Bar constructions and Hochsch ild com p lex One wel l-kno wn example of a d ivid ed p ow er c hain algebra is th e normalization of a bar construction of a comm utativ e R -algebra (see for instance [C54, Exp os ´ e n o 7] and [BK94, § 3]). Let A b e a comm utativ e R -algebra. The b ar construction of A is the simplicial comm utativ e R -algebra, B • ( A ), with B n ( A ) = A ⊗ ( n +2) . The simplicial structure maps are giv en by insertin g the multiplic ativ e unit 1 ∈ R for degenera- cies and by multiplic ation for face maps. As A is commutativ e, w e can m ultiply comp onen t wise B n ( A ) ⊗ B n ( A ) → B n ( A ) , ( a 0 ⊗ . . . ⊗ a n +1 ) ⊗ ( a ′ 0 ⊗ . . . ⊗ a ′ n +1 ) 7→ a 0 a ′ 0 ⊗ . . . ⊗ a n +1 a ′ n +1 . F rom Theorems 6.4 and 7.5 it follo ws that the norm alizati on B ∗ ( A ) := N ( B • ( A )) is a divided p o we r c hain algebra. The Ho chsc h ild complex of the comm utativ e R -algebra A ist defin ed as C ∗ ( A ) = A ⊗ A ⊗ A B ∗ ( A ) where the A -bim o dule s tructure on B n ( A ) is induced b y ( a ⊗ ˜ a ) ( a 0 ⊗ . . . ⊗ a n +1 ) := aa 0 ⊗ . . . ⊗ a n +1 ˜ a. If A is fl at o v er R , the homology of this complex is T or A ⊗ A ∗ ( A, A ). As B ∗ ( A ) is acyclic and surjects on to C ∗ ( A ), the Ho c hsc hild complex inherits a structure of a divided p o wer c hain algebra from B ∗ ( A ). Cartan sho w ed [C54, Exp os´ e 7], th at th e bar construction of strict differenti al graded comm utativ e algebras h as a d ivided p o w er structur e. 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