Correction to: HZ-algebra spectra are differential graded algebras

CORRECTION TO “ H Z -ALGEBRA SPECTRA ARE DIFFERENTIAL GRADED ALGEBRAS” BROOKE SHIPLEY Abstract. This correction article is ac tually unnecessary . The proof of Theorem 1.2, concern- ing c ommut ativ e H Q -algebra spectra and commutativ e differen tial graded algebras, in the au- thor’s pap er [ Americ an Journal of Mathematics 129 (2007) 351-379 (arxiv:math/020921 5 v4)] is correct as originally stated. Neil Stric kland carefully prov ed that D is s ymmet ric monoidal [St1]; so Prop osition 4.7 and hence also Theorem 1.2 hold as stated. Strickland’s pro of will app ear i n joint work wi th Stefan Sc h we de [ScSt]; see related work in [arXiv:0810.1747] [St2] . Note here D is defined as a colimit of chain complexes; in con trast, non-symmetric monoidal functors analogous to D ar e defined as homotop y colimi ts of spaces in previous work of the author [S4 ]. W e leav e the old alternate approac h to Theorem 1.2 b elo w, with expository change s in the int ro duc tion, since it do es provide another slightly weak er, non-natural statemen t. In the author’s pap er [S1], the pro of o f Theo rem 1 .2 is c o rrect as stated; the functor D is symmetric monoidal. The author’s confusio n ab out this fact came fro m the comparison of this functor D , which is defined as a co limit o f chain co mplexes, with the functor D in [S4] whic h is defined as a homotopy co limit of spaces. See also the discussion of commutativ e I -monoids in section 2.2 of [Sc]. In the topo logical case D is not sy mmetr ic monoidal; in the algebra ic ca se though the functor D is symmetric. Since the pap er [S1] is mainly concer ned with ass o ciative a lgebras, the only place this issue arises is in the pro of of Theorem 1.2. As s tated in Remar k 2.11 in [S1], the main theor e ms (Theorem 1 .1, Cor ollary 2 .15 and Corollary 2.16 in [S1]) w ould als o hold with the “three step” functors H and Θ re placed by the “four step” functors H = U Lc C 0 f F 0 c and Θ = Ev 0 f i φ ∗ N Z c where c and f a re the appropriate co fibran t and fibra n t replacement functors. Since here the functors Ev 0 , i , φ ∗ N and Z ar e symmetric monoidal, we have the fo llo wing non-na tural version of Theorem 1.2 from [S1] with Θ replace d b y Θ. This statement first app eared as Theorem 1 .3 in [S3]. Theorem 1. F or C any c ommut ativ e H Q -algebr a, Θ C is we akly e quivalent to a c ommu tative differ en tial gr ade d Q -algebr a. Pr o of. As noted in the pro of of Theorem 1.2 from [S1], the r eason Θ is not symmetric monoidal is bec ause the cofibr an t and fibr a n t replacement functors inv olved in Θ a re not symmetric monoidal. This is why Θ C is only weakly equiv alent a nd not isomorphic to a commutativ e dg Q a lgebra. The method for dealing with the cofibrant replacement functor in Θ pro ceeds as in [S1]. As prov ed there, a natural zig- zag of weak equiv a lences exists b etw een Z c and the symmetric monoidal functor α ∗ e Q . Let Θ ′ = Ev 0 f i φ ∗ N α ∗ e Q . Then Θ C is naturally w eak ly equiv a len t to Θ ′ C . Next we need to consider the fibra n t replacement functor f which app ears in Θ ′ (and Θ). This is the fibrant replacement functor in the model category o f monoids in S p Σ ( C h Q ). As in [S3], we exchange f for the fibran t replacement functor f ′ in the mo del c a tegory of co mm utative Date : April 21, 2022; 2000 AMS M ath. Sub j. Class.: 55P43, 18G35, 55P62, 18D10. Researc h partially supp orted by NSF Gran ts No. 0417206 and No. 0706 877. 1 2 BR OO K E SHIPLEY monoids in S p Σ ( C h Q ) as established below in Pro position 3. F or any commutativ e monoid A in S p Σ ( C h Q ), w e thus have t w o weak equiv a lences A − → f A and A − → f ′ A . Since f ′ A is a lso fibrant as a monoid and A − → f A is a tr iv ial cofibra tio n of monoids, lifting provides a weak equiv alence f A − → f ′ A . If we let Θ ′′ C = Ev 0 f ′ iφ ∗ N α ∗ e Q , we then hav e a (non-na tural) weak equiv alence Θ ′ C − → Θ ′′ C . Since Θ C is w eak ly eq uiv alent to Θ ′ C a nd Θ ′′ C is a comm uta tive differential g raded Q - a lgebra, this c o mpletes the proo f.  Remark 2. Although Theorem 1 do es not give a na tural identification of Θ C with a c omm uta- tive DGA, for any small, fixed I -diagram D of c o mm utative H Q -alge bra sp ectra ther e will b e a map o f I -diagrams from Θ D to an I -diagram of c omm utative DGAs which is given by a v ar ian t of Θ ′′ D with f ′ replaced by the fibra n t replacement functor in the mo del catego ry o f I -diagr ams of commutativ e monoids in S p Σ ( C h Q ) given by [Hi, 11 .6.1]. Prop osition 3. Ther e is a mo del c ate gory st ructur e on t he c ate gory of c ommutative monoids in S p Σ ( C h Q ) in which a map is a we ak e quivalenc e or fibr ation if and only if t he underlying map in S p Σ ( C h Q ) is so. Let S Q denote the unit and let ⊗ S denote the monoidal pro duct in S p Σ ( C h Q ). T o establish this mo del catego ry we use the lifting prop ert y fro m [Sc Sh, 2.3(i)] applied to the fr e e c ommutative monoid functor P which is left adjoint to the forg etful functor from co mm utative monoids in S p Σ ( C h Q ) to the underlying ob ject in S p Σ ( C h Q ). Namely , P ( M ) = W n ≥ 0 M ( n ) / Σ n where M ( n ) = M ⊗ S · · · ⊗ S M is the n th tensor p o wer of M ov er S Q . Let I deno te the genera ting cofibrations and J denote the gener ating trivial cofibratio ns in S p Σ ( C h Q ); see [Ho, 7 ]. T o establish the lifting criterion in [ScSh, 2 .3 ], we fir st show that applying P to a ny map in J pr oduces a stable equiv alence. W e do this by showing that in the sour ce and tar get the orbit co nstructions can b e replaced by homoto py orbits without changing the homotopy type. Lemma 4 . L et X , Y b e in S p Σ ( C h Q ) and n ≥ 1 . (1) The map E Σ n ⊗ Σ n X ( n ) − → X ( n ) / Σ n is a level e quivalenc e. (2) The map ( E Σ n ⊗ Σ n X ( n ) ) ⊗ S Y − → ( X ( n ) / Σ n ) ⊗ S Y is also a level e qu ivalenc e. Pr o of. The first sta temen t follows directly fro m the fact that given an y Σ n -equiv ariant c o mplex A in C h Q , then E Σ n ⊗ Σ n A − → A/ Σ n is a quasi- isomorphism. The second statement follo ws a s well by extending the Σ n -action trivially to Y and shifting the parentheses.  Next we show that pushouts of maps in P ( J ) are sta ble equiv a lences and level cofibrations. Since directed colimits of suc h maps are a gain stable equiv alences, Prop osition 3 then follows from Lemma s 4 a nd 5 b y [ScSh, 2.3]. Lemma 5. L et f : T − → U b e a c ofibr ation in S p Σ ( C h Q ) and V b e a P T -mo dule. Then t he map q : V − → V ⊗ P T P U is a level c ofibr ation. If f is a trivial c ofibr ation, then q is a stable e quivalenc e. CORRECTION F OR “DGAS AND H Z -ALGEBRAS” 3 Pr o of. This follows from the fitration arguments of [Ma, 7.5, 8.6] using Le mma 4 instead of [Ma, 8.2, 8.10]. No te, her e one do es not need to res trict to the p ositive cofibrant ob jects since no such restriction is needed in Lemma 4.  References [Hi] P . S. Hirschhorn, Mo del c ate gories and their L o ca lizations , Mathematical Surveys and Monographs, 99 , American Mathematical So ciet y , Provide nce, RI, 2003, 457pp. [Ho] M. Hov ey , Sp e ct r a and symmetric sp e ctr a in gene r al mo del c ate gories , J. Pure Appl. Algebra 1 65 (2001) 63-127. [Ma] M. Mandell, Equivariant symmetric sp e ctr a , Conte mp. Math. 346 (2004), 399–452. [Sc] C. Sc hlich tkrull , Units of ring sp e ctr a and their tr ac es in algebr aic K- the ory , Ge om. T opol. 8 (2004) 645-673. [ScSh] S. Sch wede and B. Shipley , Algebr as and mo dules in monoidal mo del ca te gories , Pro c. London Math. Soc. 80 (2000), 491–511. [ScSt] S. Sch wede and N. Strickland, to appear. [S1] B. Shipley , H Z -algebr a sp ectr a ar e differ ential gr ade d algebr as , Amer. J. of Math. 129 (2007) 351–379. [S2] B. Shipl ey A c onvenient mo del c ate gory f or c ommutative ring sp e c tr a , Con temp. Math. 346 (2004), 473–483. [S3] B. Shipl ey , H Z -algebr a sp e c tr a ar e differ e ntial gr ade d algebr as , arXiv:math/0209215v2 [S4] S ymmetric sp e ctr a and t op olo gica l Ho chschild homolo gy , K -Theory 19 (2000), no. 2, 155–183. [St1] N. Stric kland, Priv ate comm unication. [St2] N. Stric kland, Chains on susp ension sp ectr a , arXi v:081 0.1747 Dep ar tment of Ma thema tics, 50 8 SEO m /c 249 , 851 S. Morgan Street, Chicago, IL 60607-704 5, USA E-mail addr ess : bshipley@math .uic.edu