ON SIMPLICIAL COMMUTATIVE ALGEBRAS WITH NOETHERIAN HOMOTOPY JAMES M. TURNER Abstract.In this paper, we introduce a strategy for studying simplicial * *commuta- tive algebras over general commutative rings R. Given such a simplicial * *algebra A, this strategy involves replacing A with a connected simplicial commutative k(* *")-algebra A("), for each " 2 Spec(ß0A), which we call the connected component of A* * at ". These components retain most of the Andr'e-Quillen homology of A when th* *e coeffi- cients are k(")-modules (k(") = residue field of " in ß0A). Thus these c* *omponents should carry quite a bit of the homotopy theoretic information for A. Ou* *r aim will be to apply this strategy to those simplicial algebras which possess Noethe* *rian homo- topy. This allows us to have sophisticated techniques from commutative a* *lgebra at our disposal. One consequence of our efforts will be to resolve a more g* *eneral form of a conjecture of Quillen that was posed in [13]. Overview Our focus, in this paper, is to take the view that the study of Noetherian ri* *ngs and algebras through homological methods is a special case of the study of simp* *licial commutative algebras having Noetherian homotopy type. Our goal is to show that * *such simplicial algebras can be given a suitably rigid structure in the homotopy cat* *egory, which then allows us to bring in methods from commutative algebra. Such methods should enable more facile techniques from homological algebra to be ferried in * *for the purpose of elaborating the global structure of such simplicial algebras. To begin, we define for a simplicial commutative algebra A to have Noetherian* * homo- topy provided: 1. ß0A is a Noetherian ring, and 2. each ßm A is a finite ß0A-module. If, more strongly, ß*A is a finite graded ß0A-module, we that A has finite No* *etherian homotopy. In order to achieve a more systematic study of simplicial algebras with Noeth* *erian homotopy, particularly to allow us a straighter path to proving our main result* *, Theorem B below, we first seek to rigidify the action of ß0 from the homotopy groups to* * the simplicial algebra. This is accomplished by the following: ___________ Date: February 11, 2002. 1991 Mathematics Subject Classification. Primary: 13D03, 18G30, 18G55; Second* *ary: 13D40. Key words and phrases. simplicial commutative algebras, Andr'e-Quillen homolo* *gy, Noetherian homotopy. Research was partially supported by a grant from the National Science Foundat* *ion (USA). 1 2 JAMES M. TURNER Theorem A: Any simplicial commutative algebra A is weakly equivalent to a conn* *ected simplicial supplemented ß0A-algebra. Theorem A provides the means to import in methods from commutative algebra, m* *ost notably localizations and completions. In particular, we use these methods as a* * means to provide a proof of a conjecture posed in [13] which generalizes a conjecture* * of Quillen regarding the vanishing of Andr'e-Quillen homology. Our larger interests lie in* * providing an understanding of the of the homotopy type of a simplicial commutative algebr* *a A with Noetherian homotopy over a Noetherian ring R through its Andr'e-Quillen ho* *mology D(A|R; -). Here we shall view this homology as a functor of ß0A-modules. This e* *nables us to be specific about the homology's rigidity properties. Before stating our result, we first need a homotopy invariant notion of compl* *ete in- tersection. To obtain one, we first define a map A ! B of simplicial commutati* *ve R-algebras, augmented over a field `, to be virtually acyclic provided D 1(B|A;* * `) = 0. Also, if W is a graded `-module, define the simplicial `-algebra So(W ) by O So(W ) = S(Wn, n) n where S(V, n) is the free commutative `-algebra generated by the Eilenberg-MacL* *ane space K(V, n). Define a simplicial commutative R-algebra A over ` to be a homotopy n-interse* *ction, for n 1, provided there is a commutative diagram R -! R0 j # # j0 A -! A0 # # ` -=! ` with the horizontal maps being virtually acyclic over ` and in the homotopy cat* *egory there is an isomorphism A0 LR0` ~=So(W ) with W a graded `-module satisfying W>n = 0. We call a general simplicial commu* *ta- tive R-algebra A a locally homotopy n-intersection if, for each " 2 Spec(ß0A), * *A is a homotopy n-intersection over the residue field k(") Recall that the flat dimension of an R-module M to be the positive integer fd* *RM such that (0.1) fdRM m () TorRi(M, -) = 0 for i > m. Theorem B: Let A be a simplicial commutative R-algebra with finite Noetherian h* *o- motopy, char(ß0A) 6= 0, and fdR(ß*A) is finite. Then Ds(A|R; -) = 0 for s 0 i* *f and only if A is a locally homotopy 1-intersection. This resolves a conjecture posed in [13] generalizing a conjecture of Quillen* * [11, 5.7]. ON SIMPLICIAL ALGEBRAS WITH NOETHERIAN HOMOTOPY 3 Notes: 1. Theorem B fails when char(ß0A) = 0, as shown in [13]. 2. Theorem B fails for general simplicial algebras having Noetherian homotopy* *. The case of the simplicial algebras S(V, n) over a field of non-zero character* *istic provide counterexamples, by computations of Cartan [5]. 3. A homomorphism between Noetherian rings is a locally complete intersection* * if and only if it is a locally homotopy 1-intersection, as shown in [2, 13]. Quillen further conjectured a more general result [11, 5.6] which drops the f* *inite flat dimension condition. We would like to indicate a possible simplicial version o* *f this conjecture of Quillen. To formulate it, we first indicate a special vanishing * *result for Andr'e-Quillen homology that we will prove. Theorem C: Let A be a simplicial commutative R-algebra with Noetherian homotop* *y. Then Ds(A|R; -) = 0 for s 3 if and only if A is a locally homotopy 2-intersec* *tion. This now leads us to pose the following: Conjecture: Let A have finite Noetherian homotopy with char(ß0A) 6= 0. Then Ds(A|R; -) = 0 for s 0 implies that A is a locally homotopy 2-intersection. The strategy for proving Theorem B is to show that Ds(A|R; k(")) = 0 for s * *2 for each " 2 Spec(ß0A). This is sufficient by a result of Andr'e [1, S.30]. Followi* *ng a strat- egy of Avramov [2], we use Theorem A coupled with commutative algebra techniques developed in [3] to replace A with A("), its connected component at ", which ha* *s the following properties: 1. A(") is a connected simplicial supplemented k(")-algebra; 2. fdR(ß*A) < 1 implies that A(") has finite Noetherian homotopy; 3. Ds(A|R; k(")) ~=Ds(A(")|k("); k(")) for s 2. Theorem B now follows from the algebraic version of a theorem of Serre establis* *hed in [13]. Acknowledgements. The author wishes to thank Lucho Avramov for sharing his ex- pertise on commutative algebra and to Paul Goerss for sharing his expertise on * *Postnikov systems. 1. Postnikov Systems and Theorem A Throughout this paper, we fix a commutative ring with unit and let Alg be * *the category of (unitary) commutative rings augmented over . Finally, we denote by* * Alg the category of -algebras in Alg . We will also be assuming the reader has an acquaintance with closed (simplici* *al) model category theory. Our main resource is [10]. We will further need specific resul* *ts on the model category structure for simplicial commutative rings and algebras. Our pri* *mary sources are [10, 12, 6]. 4 JAMES M. TURNER 1.1. Postnikov Systems. Let A be an object in the category sAlg of simplicial * *com- mutative rings over . We review the construction of a Postnikov tower for A de* *rived from [4, 7] which we will be use in the proof of Theorem A. Following [7, x5], define the nth Postnikov section of A as follows: for fix* *ed k, let In,k! Ak be the kernel of the map Y d : Ak ! An ffi:[m]![k] where OE runs over all injectionsQin the ordinal number category with m n, d * *is induced by the maps OE* : Ak ! Am , and denotes the product in the category of algeb* *ras augmented over . Define (1.2) A(n)k = Ak=In,k Notice that there is a quotient map in sAlg , A ! A(n), and that if k n, A(n* *)k = Ak. There are also quotient maps (1.3) qn : A(n) ! A(n - 1) and A ~=limA(n). Let F (n) be the fibre of qn, i.e. (1.4) F (n) = ker(qn : A(n) ! A(n - 1)). qn Note that F (n) ! A(n) ! A(n - 1) forgets to a fibration sequence as simplicial* * abelian groups. As such, the following can be proved just as in [7, 5.5]. Lemma 1.1. The homotopy groups of F (n) are computed as follows: ( ßnA k = n; ßkF (n) = 0 k 6= n. 1.2. Eilenberg-MacLane objects. Following [4, x5], define an object A of sAlg * *to be of type K if ß0A ~= and the higher homotopy groups of A are trivial. Suppo* *se M is a -module. We say that a map A ! B is of type K (M, n) n 1, if A is of ty* *pe K , ß0B ~= , ßnB ~=M (as a -module), all other homotopy groups of B are trivia* *l, and the map A ! B is a ß0-isomorphism. For a general map f : A ! B in sAlg , let C be the pushout of the diagram B0* * A0! A(0)0obtained by using a functorial construction to replace A by a cofibran* *t object and the two maps A ! B and A ! A(0) by cofibrations. There is then a commutative diagram f A - ! B ~" "~ f0 0 (1.5) A0 - ! B # # n(f) A(0)0 - ! C(n + 1) ON SIMPLICIAL ALGEBRAS WITH NOETHERIAN HOMOTOPY 5 The bottom map n(f) is called the difference construction of f. The following * *can be proved just as in [4, 6.3]. Proposition 1.2. Suppose that A ! B is a map of simplicial commutative algebras which is a ß0-isomorphism and whose homotopy fibre F is (n-1)-connected. Let M* * = ßnF . Then M is naturally a -module for = ß0B and n(f) is a map of type K (M, n + 1). If ßkF vanishes except for k = n, then the right-hand square in 1* *.5 is a homotopy fibre square. 1.3. Differentials functor. For an object A in Alg , define its -differentials* * to be the -module D A = J=J2 A where J is the kernel of the product A A ! A. As a functor to the category of -modules, D posseses a right adjoint - the functor (-)+ : Mod ! Alg defined by M+ = M with the usual twisted product (x, a) . (y, b) = (bx + ay, ab). An equivalent identification of the differentials functor (1.6) D ~=I=I2 A , where I is the augmentation ideal of A, which can be seen to follow from Yoneda* *'s lemma. The next proposition is proved in [10, xII.5]. Proposition 1.3. The prolonged adjoint pair of functors D : sAlg () sMod : (-)+ induces an adjoint pair on the homotopy categories LD : Ho(sAlg ) () Ho(sMod ) : R(-)+. Finally, the following useful property of the derived functor of differential* *s follows from [12, 7.3]. Proposition 1.4. If f : A ! B is a ß n-isomorphism, then LD (f) is a ß n-isomor* *phism. 1.4. Characterizing K (M, n)-type. Fix a -module M. In sMod , the fibration pn* * : E(M, n) ! K(M, n) is determined by the Dold-Kan correspondence by to correspond* * to the map of normalized chain complexes {M !1 M} ! {M} with the source concentrat* *ed in degrees n and n-1, the target concentrated in degree n, and the map being th* *e identity in degree n and trivial otherwise. Applying (-)+ to pn gives a K (M, n)-type fibration in sAlg (pn)+ : E (M, n) ! K (M, n) which we call the canonical map of type K (M, n). 6 JAMES M. TURNER Proposition 1.5. Let A ! B be of type K (M, n) between cofibrant objects in sAl* *g . Then there is a commuting diagram in sAlg A -~! E (M, n) # # pn B -~! K (M, n) with the horizontal maps being weak equivalences. Proof. To begin, note that the canonical map B ! is (n-1)-connected. Thus t* *he induced map D B ! 0 is (n-1)-connected by Proposition 1.4. Let I = ker(B ! ). Filtering B by powers of I we note that B cofibrant implies that Iq=Iq+1 = Sq(I=I2) ~=Sq(D B) where the last identity always holds when the augmentation is surjective, by (1* *.6). Thus there is a convergent spectral sequence E1p,q= Hp+q[Sq(D B)] =) ßp+qB. From the connectivity indicated above and [12, 7.40], E1p,q= 0 for 0 < p+q 2(* *q-2)+n. Thus we obtain M ~=ßnB ~=ßnD B. Thus there is an n-connected map D B ! K(M, n) and its adjoint B ! K (M, n) will be a weak equivalence by the computations above and the assumption that A ! B i* *s of type K (M, n). Finally, A ! is a weak equivalence, hence D A ! 0 is a weak equivalence by Proposition 1.4. Since A, and hence D A, are cofibrant, the composite D A ! D B* * ! K(M, n) lifts to a map D A ! E(M, n), whose adjoint A ! E (M, n) is necessarily* * a weak equivalence. 2 1.5. Proof of Theorem A. Fix an object A in sAlg . We will show, by induction, that there is a map X ! Y in s Alg and a commutative diagram in Ho(sAlg ) A(n) -~! X (1.7) qn # # A(n - 1) -~! Y with the horizontal maps being equivalences. It is clear for n = 0 as A(0) ! * *is a weak equivalence. Using 1.5, some closed model category theory and induction, we may assume that there is a trivial fibration oe : A(n - 1)0 ! Y with the target Y a cofibrant* * object in s Alg . Lemma 1.6. Let M = ßnA. Then there is a commuting diagram in Ho (sAlg ) of t* *he form A(n - 1)0 - ! C(n + 1) ~# oe #~ Y - ! K (M, n + 1) ON SIMPLICIAL ALGEBRAS WITH NOETHERIAN HOMOTOPY 7 with the top arrow from 1.5. Proof. First, note that since oe : A(n - 1)0! Y is a trivial fibration between * *suitably cofibrant objects (see above) it follows from that and from 1.6 that D oe : D A(n - 1)0! D Y is a trivial fibration between cofibrant objects in sMod . By [10, I.1.7], D o* *e has a homotopy left inverse i (i O D oe ' IdD A(n-1)). Next, utilizing Lemma 1.5, let t : A(n - 1)0 ! K (M, n + 1) be the composite * *of A(n - 1)0! C(n + 1) ! K (M, n + 1). Let w : D Y ! K(M, n + 1) be the composite (D t) O i. Then w O D oe ' D t and the result now follows from Proposition 1.3.* * 2 From the previous lemma, we may form the homotopy pullback diagram in s Alg X -! E (M, n + 1) (1.8) # # (pn)+ Y -! K (M, n + 1). By Proposition 1.2, the diagram below is also a homotopy pullback in sAlg A(n)0 - ! A(0)0 (1.9) q0n# # [qn] A(n - 1)0 - ! C(n + 1). By Proposition 1.5 and Lemma 1.6, there is an induced map of diagrams 1.9 to 1.8 in the category Ho (sAlg ). Since fibrations and pullbacks in sAlg are fibrat* *ions and pullbacks as simplicial groups, a computation of homotopy groups can be perform* *ed utilizing Lemma 1.1 to show that the induced map A(n)0! X is a weak equivalence. This completes the induction step. 2.Andr'e-Quillen homology and Theorems B and C 2.1. Base change property of Andr'e-Quillen homology. Recall that the cotangent complex of a simplicial R-algebra A is defined to be the object of Ho(ModA) (2.10) L(A|R) := P|R P A where the T -module T|S= J=J2, J = ker(T ST ! T ), denotes the Kahler differe* *ntials of an S-algebra T , and P ! A is a cofibrant replacement of A as a simplicial R* *-algebra. Note: As in x1.3, T|Sis left adjoint to the functor M 7! M T where the image* * has a T -algebra structure with M2 = 0. Also recall that given another simplicial R-algebra B, the derived tensor pro* *duct of A and B to be the object of Ho(sModR) A LRB := P R Q where Q ! B is a cofibrant replacement of B. We now derive a base change property for the cotangent complex following [12]. 8 JAMES M. TURNER Lemma 2.1. If TorRq(Ak, Bk) = 0 for all k 0 and all q > 0 then A LRB ' A * *R B. Proof. This follows immediately from the spectral sequence [10, xII.6] E2p,q= ßpTorRq(A, B) =) ßp+q(A LRB). 2 Lemma 2.2. A RB|B ~= A|R R B Proof. Let A0= A R B and fix an A0-module M. Then hom A0( A0|B, M) ~=hom BAlgA0(A0, M A0) ~=hom RAlgA(A, M A) ~=hom A( A|R, M) ~=hom A0( A|R R B, M). The result now follows from Yoneda's lemma. 2 Proposition 2.3. L(A LRB|B) ' L(A|R) LRB Proof. Fix cofibrant replacements P and Q for A and B, respectively. Then (2.11) L(A LRB|B) = P RQ|Q ~= P|R R Q by Lemma 2.2. Since P is projective as a simplicial R-module then P|R is a pro* *jective P -module. Thus, by Lemma 2.1, the map P|R ~! P|R P A is a weak equivalence. Since Q is projective, Lemma 2.1 further tells us that (2.12) P|R R Q ~!( P|R P A) R Q ~=L(A|R) LRB is a weak equivalence. The result now follows by combining 2.11 with 2.12. * * 2 Corollary 2.4. As a functor of A R B-modules, D*(A LRB|B; -) ~=D*(A|R; -). Proof. This follows from Proposition 2.3 and the identity D*(T |S; M) := ß*[L(T |S) T M]. 2 2.2. Proof of Theorem B. We first recall the main result of [13]. Theorem 2.5. Let A be a homotopy connected simplicial supplemented commutative algebra over a field ` of non-zero characteristic. Then Ds(A|`; `) = 0 for s * *0 implies that there is an equivalence S`(D1(A|`; `), 1) ~=A in the homotopy category. We now begin by establishing a special case of Theorem A. To that end let A b* *e a simplicial commutative R-algebra and assume that the unit R ! ß0A = is a surj* *ection. For " 2 Spec , define the connected component of A at " to be the connected sim* *plicial supplemented k(")-algebra A(") = A LRk("). Lemma 2.6. Let A be as above. Then 1. D*(A|R; k(")) ~=D*(A(")|k("); k(")), and ON SIMPLICIAL ALGEBRAS WITH NOETHERIAN HOMOTOPY 9 2. if A also has finite Noetherian homotopy and fdR(ß*A) < 1 it follows that * *A(") has finite Noetherian homotopy. Proof. 1. follows from Corollary 2.4. For 2., [10, xII.6] gives a spectral sequ* *ence E2s,t= TorRs(ßtA, k(")) =) ßs+t(A LRk(")). From the finiteness conditions, each E2s,tis a finite k(")-module and vanishes * *for s, t 0. Thus A LRk(")) has finite Noetherian homotopy. 2 Corollary 2.7. Let A be as in Lemma 2.6.2 and further assume that char(k(")) 6=* * 0. Then Ds(A|R; k(")) = 0 for s 0 implies that Ds(A|R; k(")) = 0 for s 2. Proof. This follows from Lemma 2.6 and Theorem 2.5. 2 Now assume that the simplicial algebra A in question is a homotopy connected * *simpli- cial supplemented -algebra, by Theorem A. We further assume that A has Noether* *ian homotopy. Fix " 2 Spec and let d(-)denote the completion functor on R-modules at ". De* *fine the homotopy connected simplicial supplemented b-algebra A0by A0= A L b. Proposition 2.8. Suppose A is a simplicial commutative R-algebra, with R a Noet* *her- ian ring. Then ß*A0 ~=dß*Aand there exists a (complete) Noetherian R0 that fit* *s into the following commutative diagram in Ho(sRAlg ) '' R -! A OE # # _ ''0 0 R0 -! A with the following properties: 1. OE is a flat map and its closed fibre R0="R0is weakly regular; 2. _ is a D*(-|R; k("))-isomomorphism; 3. j0 induces a surjection j0*: R0! ß0A0; 4. fdR(ß*A) finite implies that fdR0(ß*A0) is finite Proof: First, Quillen's spectral sequence [10, II.6] Tor*(ß*A, b) =) ß*A0 colla* *pses to give the first result since b is flat over and each ßm A is finite over [9,* * 8.7 and 8.8]. ffi0 ''0* Next, by [3, 1.1], the unit ring homomorphism R ! b factors as R ! R ! b wi* *th OE having the properties described in 1. and j0*is a surjection. Thus the induc* *ed map j0: R0! A0induces a surjection on ß0, giving 3., and the desired diagram commut* *es. Now, by the transitivity sequence [12, 4.12] applied to R ! A ! A0, 2. follow* *s from the isomorphism D*(A0|A; k(")) ~=D*(b | ; k(")) ~=0 which follows from Corollary 2.4. Finally, 4. follows from [3, 3.2], as A has Noetherian homotopy. * * 2 10 JAMES M. TURNER Now, let A have finite Noetherian homotopy with Ds(A|R; -) = 0 for s 0. From Proposition 2.8, Theorem 2.5, Corollary 2.7, and [1, xS.30], if fdR(ß*A) < 1 th* *en A(") ~=Sk(")(D1(A|R; k("), 1), for each " 2 Spec(ß0A), if and only if D(A|R; -)* * = 0. Thus Theorem B follows from the definition of locally homotopy complete interse* *ction (see introduction) and a transitivity sequence argument. 2.3. Proof of Theorem C. Let A be a simplicial commutative R-algebra with Noe- therian homotopy. It follows from Lemma 2.6.1, Proposition 2.8, and [1, xS.30]* *, that D 3(A|R; -) = 0 if and only if D 3(A(")|k("); k(")) = 0, for all " 2 Spec(ß0A).* * From the definition of locally virtual homotopy complete intersection (see introduct* *ion), Theo- rem C will follow if we can show that, for each prime ideal ", A(") ~=So(D 2(A|* *R; k("))) in the homotopy category. But this in turn follows from [13, (2.2)]. References [1]M. Andr'e, Homologie des alg`ebres commutatives, Die Grundlehren der Mathem* *atischen Wis- senschaften 206, Springer-Verlag, 1974. [2]L. Avramov, öL cally complete intersection homomorphisms and a conjecture o* *f Quillen on the vanishing of cotangent homology,Ä nnals of Math. (2) 150 (1999), 455-487. [3]L. L. Avramov, H.-B. Foxby, and B. Herzog, "Structure of local homomorphism* *s," J. Algebra 164 (1994), 124-145. [4]D. Blanc, W. Dwyer, and P. Goerss, "The realization space of a -algebra: a* * moduli problem in algebraic topology," preprint, Northwestern University (2001) [5]H. Cartan, Ä lg`ebres d'Eilenberg-MacLane et homotopie," Expos'es 2 `a 11, * *S'em. H. Cartan, 'Ec. Normale Sup. (1954-1955), Sect'etariat Math., Paris, 1956; [reprinted * *in:] OEvres, vol. III, Springer, Berlin, 1979; pp. 1309-1394. [6]P. Goerss, Ä Hilton-Milnor theorem for categories of simplicial algebras,Ä* * mer. J. Math,. 111 (1989), 927-971. [7]P. Goerss and J. Turner, öH motopy theory of simplicial abelian Hopf algebr* *as," J Pure Appl. Alg. 135 (2) (1999), 167-206. [8]S. MacLane, Homology, Classics in Mathematics, Springer-Verlag, 1995. [9]H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Math. * *8, Cambridge University Press, 1996. [10]D. Quillen, Homotopical algebra, Lecture Notes in Mathematics 43, Springer-* *Verlag, 1967. [11]__________, Ö n the (co)homology of commutative rings," Proc. Symp. Pure Ma* *th. 17(1970), 65-87. [12]__________, Ö n the homology of commutative rings," Mimeographed Notes, M.I* *.T. [13]J. M. Turner, Ö n simplicial commutative algebras with vanishing Andr'e-Qui* *llen homology," In- vent. Math. 142 (3) (2000) 547-558. Department of Mathematics, Calvin College, 3201 Burton Street, S.E., Grand Ra* *pids, MI 49546 E-mail address: jturner@calvin.edu