Contemporary Mathematics Characterizing Simplicial Commutative Algebras with Vanishing Andr'e-Quillen Homology James M. Turner Abstract.The use of homological and homotopical devices, such as Torand * *Andr'e- Quillen homology, have found substantial use in characterizing commutati* *ve algebras. The primary category setting has been differentially graded algebras and* * modules, but recently simplicial categories have also proved to be useful settings. * *In this paper, we take this point of view up a notch by extending some recent uses of h* *omological algebra in characterizing Noetherian commutative algebras to characteriz* *ing simplicial commutative algebras having finite Noetherian homotopy through the use o* *f simplicial homotopy theory. These characterizations involve extending the notions o* *f locally com- plete intersections and locally Gorenstein algebras to the simplicial ho* *motopy setting. Overview Following a program set forth by Grothendieck (see [6]), a major research ef* *fort has been underway to characterize ring homomorphisms f : R ! S of Noetherian rings. Use of homological devices have been pivotal to making such characterizations. * * For example, in [17 ] D. Quillen, in the process of developing a homology of commut* *ative algebras, conjectured that the higner vanishing of this homology of S over R, i* *n the case f is essentially of finite type and of finite flat dimension, charactertizes f * *as a locally complete intersection homomorphism. Motivated by this conjecture, L. Avramov [* *4] defined locally complete intersection homomorphisms more generally and establis* *hed their properties, including providing a proof of Quillen's conjecture by a tour* * de force use of differentially graded and simplicial techniques. This fit into a larger* * program of Avramov with various collaborators to fulfill Grothendieck's program for rin* *gs and their homomorphisms. See, for example, [4, 5, 8]. In particular, characteriza* *tions of homomorphisms to be locally regular, complete intersection, Grothendieck, and C* *ohen- Macaulay were achieved. ___________ 2000 Mathematics Subject Classification. Primary: 13D03, 18G30, 18G55; Secon* *dary: 13D40. Key words and phrases. simplicial commutative algebras, Andr'e-Quillen homol* *ogy, Noetherian ho- motopy, homotopy complete intersections, homotopy Gorenstein algebras. Partially supported by National Science Foundation (USA) grant DMS-0206647 a* *nd a Calvin Re- search Fellowship. He thanks the Lord for making his work possible. cO0000 (copyrigh* *t holder) 1 2 JAMES M. TURNER Also motivated by Quillen's conjecture and a perspective of commutative alge* *bra from a strictly simplicial viewpoint (see, for example, [12 , 14]), a version o* *f Quillen's conjecture was formulated and proved for simplicial commutative algebras with f* *inite Noetherian homotopy [19 , 20 ], relying on techniques developed by analogy from* * the homotopy of spaces for the hootopy of simplicial commutative algebras. In the p* *rocess, the notion of homotopy complete intersection was formulated and shown to be cha* *rac- terized by the vanishing of higher Andr'e-Quillen homology. The drawback was th* *at the extension was valid only when the ß0 has non-zero characteristic. Since such re* *strictions is not needed in the constant simplicial case, as the main result of [4] clearl* *y implies, then there is a gap in a full characterization of locally complete intersection* *s through purely simplicial techniques. The aim of this paper is twofold. The first is to begin to describe a progra* *m which extends the notions of complete intersection, Gorenstein, and Cohen-Macaulay to* * simpli- cial commutative algebras having Noetherian homotopy. This will involve summari* *zing the results of [19 , 20] and extending some of the notions in [5, 8]. In parti* *cular, we will define the notions of homotopy Gorenstein and homotopy Cohen-Macaulay alge* *bras. The second aim of this paper will be to characterize, using these notions, simp* *licial algebras with finite Noetherian homotopy, finite flat dimensional homotopy, and* * finite Andr'e-Quillen homology free of conditions on the characteristic of ß0. Specifi* *cally, we will prove the following (see x2 for definitions): Homology Characterization Theorem. Let A be a simplicial commutative R-algebra (R Noetherian) having finite Noetherian homotopy and fdR(ß*A) < 1. If Ds(A|R; -* *) = 0 for s 0, as a functor of ß0A-modules, then: (a) A is locally homotopy Gorenstein; (b) If char(ß0A) 6= 0 then A is a locally homotopy complete intersection; (c)If A is locally regularly 2-degenerate then R ! A is a locally complete* * inter- section homomorphism; that is, for each " 2 Spec(ß0A) there is a factor* *ization (2.1) such that ker(j0) is generated by a regular sequence. This paper is organized as folowed: x1 reviews the simplicial model structur* *e for sim- plicial commutative algebras, as well as the tensor, hom, and differential stru* *ctures and their derived functors. We also review relationships between certain simplicial* * categories and differentially graded categories. x2 reviews the definition of homotopy co* *mplete intersections and introduces the notions of homotopy Gorenstein and homotopy Co* *hen- Macaulay algebras. We end with a proof of the Homology Characterization Theorem. Acknowledgements: The author would like to thank Lucho Avramov for several dis- cussions over the years which greatly assisted him in the development of this p* *aper's content and for giving him access to [7]. He would also like to thank Brooke Sh* *ipley for alerting him to the paper [18 ]. CHARACTERIZING SIMPLICIAL ALGEBRAS WITH VANISHING HOMOLOGY 3 1. Homotopy theory of simplicial commutative algebras and simplicial modules 1.1. Simplicial model category structures. Fix a commutative ring R with unit and let A denote the category of commutative R-algebras with unit. If B 2 A, le* *t AB denote the subcategory of objects (A, ffl) with ffl : A ! B an R-algebra map. * *Given any category C, we let sC denote the category of simplicial objects over C. Fin* *ally, for A 2 sA, let MA denote the category of simplicial A-modules. Let f : A ! B be a map in sA. Recall from [15 ] that f is defined to be a: (1) weak equivalence provided f is a weak equivalence of simplicial groups; (2) fibration provided f is a fibration of simplicial groups; (3) cofibration if and only if f is a retract of an almost free map, i.e. * *a map f0 : A0! B0such that f0 makes B0free as an almost simplicial A0-algebra* * (that is, without d0). By [15 , xII.3], this structure makes sAS a closed model category for any fi* *xed S 2 sA. Fixing A 2 sA, let f : K ! L be a map in MA. Following [15 , 18], we will sa* *y that f is defined to be a: (1) weak equivalence provided Nf is a weak equivalence of chain complexes; (2) fibration provided Nf is a level-wise surjection in positive degrees; (3) cofibration if and only if Nf is a level-wise monomorphism whose cokern* *el in each degree k 0 is a projective NkA-module. Here N : MA ! ModNA is the normalized chain functor. By [15 , xII.3], th* *is structure makes MA into a closed model category. Furthermore, part of the main theorem of [18 ] states Schwede-Shipley Theorem: The Dold-Kan correspondence N : MA () Mod+NA: K is a Quillen equivalence of symmetric monoidal closed model categories. 1.2. Tensor and Tor-modules. For A 2 sA, level-wise tensor product gives a functor A : MA x MA ! MA. In turn, this tensor product induces the derived tensor product LR: Ho(MA) x Ho(MA) ! Ho(MA) on the homotopy category, which is defined by K LAL := X A Y, where X and Y are cofibrant replacements for K and L in MA, respectively. Note also that A induces a functor on sA which descends to a functor LAon * *sAB , for any fixed B. Next, define the Tor-modules for K, L 2 MA to be TorAs(K, L) := ßs(K LAL), s 0. 4 JAMES M. TURNER For each such s, TorAsis a ß0A-module. A key method for computing TorA*is the following device found in [15 , II.6]. Kunneth Spectral Sequence: For K, L 2 MA, there is a first quadrant spectral sequence E2*,*= Tori*A*(ß*K, ß*L) =) TorA*(K, L). 1.3. Ext-modules. As noted above, in [18 ] it is shown that the normalization functor induces a functor N : MA ! Mod+NA which is an equivalence of categories. We will therefore use N to define the Ext-modules for K, L 2 MA. First, if T i* *s a (commutative) DG ring and U, V are DG T-modules, let ExtsT(U, V ) := H-s(Hom T (P*, I*)) where P* ! U is a DG projective replacement and V ! I* is a DG injective replac* *ement for NL in NA - Mod. Here the Hom-complex is defined as in [9, x1]. We then defi* *ne, for K, L 2 MA, Ext sA(K, L) := ExtsNA(NK, NL). A basic property of Ext is then given by: Lemma 1.1. [9, (1.8)] If A !~ B is a weak equivalence of simplicial algebras* * aug- mented over a field `, then Ext*A(`, A) ~=Ext*B(`, B). Now assume that ` is a field of characteristic 0. Let <` denote the categor* *y of commutative `-algebras over `, henceforth refered to as rational `-algebras. Le* *t s+<`and ch+<` denote the category of connected simplicial rational `-algebras and the c* *ategory of connected differentially graded rational `-algebras, respectively. Quillen's Theorem. [16 , p. 223] Normalization induces a functor N : s+<` ! ch* *+<` which is an equivalence of closed model categories. Our aim now is to prove the following variation of [9, (4.1)]. Proposition 1.2. Let _ : `[x] ! T be a map of augmented DG rings over ` from* * a free commutative graded `-algebra on one generator x, |x| = n > 0, and let M be* * a DG `[x]-module. Assume furthermore that (1) H0(T ) is Noetherian and each Hi(T ) is a finitely generated H0(T )-mod* *ule for i 2 Z; (2) H(M) is bounded above. Letting F (_) := T L`[x]`, there is an isomorphism of graded `-modules Ext *T(`, M L `[x]T ) ~=Ext*`[x](`, M) `Ext *F(_)(`, F (_)). CHARACTERIZING SIMPLICIAL ALGEBRAS WITH VANISHING HOMOLOGY 5 Proof. By inspection of the statement of [9, 4.1], the key difference involves * *replacing an augmented DG ring R, for which H*(R) ~=H0(R) is Noetherian, with N(S`(n)). In their proof of (4.1), the condition on R insures that: (1) There is a factorization R ! X '! ` with X a free R-algebra such that X !' X0 with X0 finite type over R wi* *th bounded below generators. (2) There is a DG injective R-resolution M ! I with I bounded above. (1) and (2) insures that [9, (1.10)] can be applied to establish [9, (4.9)]. To show (1) in our context, let `[x, y] be the free DG `-algebra such that |* *y| = n + 1 and @y = x. Then there is a factorization `[x] ! `[x, y] ~!` with the required properties. Finally, (2) can be found in [7, (9.3.2.1)]. * * 2 Let S`(n) be the free commutative `-algebra generated by the Eilenberg-MacLa* *ne object K(`, n). Let A 2 s<` and let OE : S`(n) ! A be a map in s<`. OE determin* *es a cofibration sequence in Ho(s<`) ffi S`(n) ! A ! Fffi where Fffi:= A LS`(n)`. Corollary 1.3. If A 2 s+<` with ß*A a finite graded `-module, then Ext*A(`, A) ~=Ext*S`(n)(`, S`(n)) `Ext *FOE(`, Fffi). Proof. Note first that N(S`(n)) ' `[x] with |x| = n. Since the result follows i* *mmediately from Proposition 1.2 for n odd (`[x] is exterior on x), we assume that n is eve* *n. Using Quillen's Theorem, let L(k) 2 s<` satisfy NL(k) ' `[x]=(xk). Thus, by Propositi* *on 1.2: Ext*A(`, L(k) LS`(n)A) ~=Ext*S`(n)(`, L(k)) `Ext *FOE(`, Fffi). (Note that the Schwede-Shipley Theorem implies that NFffi' F (N(OE)).) Now NS`(n) ! NL(k) is equivalent to a map that is an isomomorphism in degrees < nk. By a Kunneth spectral sequence argument, since ß*A is bounded the map A ! L(k) LS`(n)A induces a ß*-injection which is a ß*-isomorphism through degree n* *k, for k 0. In particular, as ß*A-modules: TorS`(n)*(L(k), A) ~=ß*A x nk+1ß*A. Thus the map A ! limk(L(k) LS`(n)A) is a weak equivalence, whose normalization* * is equivalent to an isomorphism. Furthermore, using Quillen's Theorem and the Schw* *ede- Shipley Theorem, the induced map NA ! (`[x]=(xk)) L`[x]NA can be shown to be equivalent to a split injection, for k 0, as NA-modules. Thus the result now * *follows from an argument using Lemma 1.1 and the Milnor sequence [21 , x3.5]. * * 2 Note: Corollary 1.3 holds over fields ` of arbitrary characteristic when n = 1. 6 JAMES M. TURNER 1.4. Differentials and Andr'e-Quillen homology. Let A be a commutative ring and B a commutative A-algebra. If M is a B-module, recall that an A-module map f : B ! M is a derivation provided f(xy) = xf(y) + yf(x). Let DerA(B, M) be the A-module of derivations. The functor M 7! DerA (B, M) is representable: there* * is a canonically defined B-module B|A, called the differentials of B over A, such t* *hat there is a natural isomorphism: DerA(B, M) ~=Hom B( B|A, M). From the differentials, the cotangent complex of a simplicial commutative R-* *algebra A is defined by L(A|R) := X|R X A, where X ~! A is a cofibrant replacement of A in Ho(sA). The Andr'e-Quillen homo* *logy of A over R with coefficients in the A-module M is then defined to be D*(A|R; M) := ß*(L(A|R) A M). We now recall two important properties of Andr'e-Quillen homology: (1) (Transitivity Sequence) Given maps A ! B ! C in sA and a C-module M, there is a long exact sequence: . .!.Ds+1(C|B; M) ! Ds(B|A; M) ! Ds(C|A; M) !! Ds(C|B; M) ! . .;. (2) (Flat Base Change) For A, B 2 sA and M an A LR B-module D*(A LR B|B; M) ~=D*(A|R; M). A further useful relationship between homotopy and Andr'e-Quillen homology is g* *iven by the following: Hurewicz Theorem: For a connected simplicial R-algebra A over a field `, there * *is a homomorphism h* : TorR*(A, `) ! D*(A|R; `) for which h* is an isomorphism in degrees n provided A is (n-1)-connected. Let ffl : A ! ` be a commutative algebra over a field. Let I = ker(ffl). D* *efine the indecomposables of A to be the `-module QA := I=I2. A well known result [12 , 1* *4] for A a supplemented `-algebra is A|` A ` ~=QA. We thus define the Andr'e-Quillen homology of a simplicial supplemented `-algeb* *ra A by HQ*(A) := ß*QX, where X !~ A is a cofibrant resolution of A as simplicial supplemented algebras* *. Thus we have HQ*(A) ~=D*(A|`; `). We now return to inspecting maps OE : S`(n) ! A of simplicial supplemented `- algebra. Our aim is to prove CHARACTERIZING SIMPLICIAL ALGEBRAS WITH VANISHING HOMOLOGY 7 Proposition 1.4. Let OE : S`(n) ! A be a map of (n-1)-connected simplicial s* *up- plemented `-algebras such that HQn(OE) is an injection. If ß*A is a finite grad* *ed `-module and ß*(Fffi) is unbounded then HQ*A is unbounded. (Recall that a positively graded module M is unbounded provided Mt6= 0 for infi* *nitely many t.) To prove this proposition, we adapt the proof of [10 , (4.2)]. Assume char`* * = 0. Recall for a commutative DG `-algebra T that a minimal model for T is a free DG- algebra (`[X], @) such that @X I2, where I is the augmentation ideal of `[X],* * together with a weak equivalence `[X] ~!T . For existence and properties of minimal mode* *ls, see [11 ]. Note that from Quillen's Theorem (see also [17 , Thm. 9.5], if A is a * *simplicial supplemented `-algebra and `[X] is a minimal model for NA, then HQ*(A) ~=QX. Proof of Proposition 1.4. First, assume that char` 6= 0. If ß*A and HQ*ANare * *both finite graded `-modules, then, by the AlgebraicNSerre Theorem [19 ], A ~= IS`* *(1) in the homotopy category. Thus n = 1 and Fffi~= I S`(1) with |J| = |I| - 1. It fol* *lows that ß*Fffiis bounded. Now assume that char` = 0. Let (`[X], @) be a minimal model for NA and let `* *[x] be a minimal model for N(S`(n)). By the assumption that HQn(OE) is injective, i* *t follows that we may assume that x 2 X and that the map `[x] ! `[X] induced by the inclu* *sionS {x} X is equivalent to N(OE). It follows from Quillen's Theorem that if X = {* *x} Y , then `[Y ] is a minimal model for NFffi. Thus writing `[X] ~=`[x] `[Y ], we c* *an express, for u 2 `[Y ], @(1 u) = 1 ~@u + x Ou. It follows that O : (`[Y ], ~@) ! (`[Y ], ~@) is a derivation of degree -n-1. Let J be the augmentation ideal of `[Y ]. Let O = i 1Oi with Oi(J) Ji. Th* *us O1 : J ! J is a derivation and, hence, induces O1 : J=J2 ! J=J2. Claim: There exists an element u in the `-dual (J=J2)* such that (O*1)nu 6= 0 f* *or all n 0. It follows from this claim that HQ*(Fffi) and, hence, HQ*(A) are unbounded. To establish the claim, assume for each u 2 (J=J2)* there is an n 1 such t* *hat (O*1)nu = 0. Following the same argument as in [10 , p.181], this implies that,* * for each u 2 J*, (O*)nu = 0 for n 1. Now, consider the exact sequence 0 ! J !fiI ! J ! 0, 8 JAMES M. TURNER where I is the augmentation ideal of `[X] and ø(w) = z w. Dualizing and apply* *ing cohomology, we obtain a long exact sequence * i * . .!.Hi-1(J*) !ffiHi+n(J*) ! (Hi+n(I))* fi!H (J ) ! . . . An easy computation shows that ffi = H(O*). By our assumption on the finitenes* *s of ß*A, Hi(I) = 0 for i N, N 0. Thus ffi is injective for i N. Since we are * *assuming ß*Fffiis unbounded, this contradicts our local nilpotency condition on (O)*. Th* *us our claim is established. 2 2.Characterizing Simplicial Commutative Algebras We focus on extending the characterizations of homomorphisms R ! S of Noethe* *rian rings achieved in [4, 5, 8] to simplicial commutative R-algebras. To set in wha* *t direction this extension is to take, we view S as a constant simplicial R-algebra. To ac* *hieve a suitable type of extension, the notion of Noetherian needs to be spelled out fo* *r simplicial algebras. Such a notion was delineated and explored in [20 ], motivated by an a* *nalogous for concept for DG rings described in [5], which we now describe. A simplicial commutative algebra A is said to have Noetherian homotopy provi* *ded (1) ß0A is a Noetherian ring; (2) each ßm A is a finite ß0A-module. We furthermore say that A has finite Noetherian homotopy provided that (2) is r* *eplaced by (2)0ß*A is a finite graded ß0A-module. The key to characterizing simplicial commutative R-algebras with Noetherian * *ho- motopy through homotopical/homological methods is to locally reduce to connected simplicial algebras over a field. Such objects yield more information under hom* *ological scrutinity. This approach was pioneered by L. Avramov [3] through the notion of* * DG fibre and used with great effect in [5, 4]. To adopt this approach in the simpl* *icial setting, the following extension of the main theorem of [8] is needed Factorization Theorem. [20 , (2.8)] Suppose A is a simplicial commutative R-alg* *ebra with Noetherian homotopy, R a Noetherian ring, and " 2 Spec(ß0A). Then there is a simplicial commutative algebra A0 with Noetherian homotopy, such that ß*A0~= * *dß*A, and there exists a (complete local) Noetherian R0that fits into the following c* *ommutative diagram in Ho(sAk(")) '' R -! A (2.1) OE # # _ ''0 0 R0 -! A with the following properties: (1) OE is a flat map and its closed fibre R0="R0is regular; (2) _ is a flat D*(-|R; k("))-isomomorphism; (3) j0 induces a surjection j0*: R0! ß0A0; CHARACTERIZING SIMPLICIAL ALGEBRAS WITH VANISHING HOMOLOGY 9 (4) fdR(ß*A) finite implies that fdR0(ß*A0) is finite We will call a choice of diagram (2.1) a factorization for a simplicial R-algeb* *ra A with Noetherian homotopy. Also, recall that, for an R-module M, fdRM is the flat dim* *ension of M. We now describe extensions of the notions of locally complete intersection, * *locally Gorenstein, and locally CM (i.e. Cohen-Macaulay) for homomorphisms of Noetherian rings, as described in [4, 5, 8], to simplicial algebras with Noetherian homoto* *py. To begin, let A be a connected simplicial supplemented `-algebra, ` a field.* * We then declare that A is (1) aNhomotopy complete intersection provided there is a finite set I with * *A ~= IS`(1) in Ho(sA); (2) homotopy CM provided there exists an n 2 Z such that ExtiA(`, A) = 0 for i 6= n; (3) homotopy Gorenstein provided A is homotopy CM and dim`Ext nA(`, A) = 1. To get a sense of how these notions fit together and apply to basic examples* * of simplicial supplemented algebras, we prove the following: Proposition 2.1. (a) S`(n) is a homotopy complete intersection when n = 1 and homotopy Gorenstein in general when char` = 0. (b) We have the string of implications homotopy complete intersection=) homotopy Gorenstein=) homotopy CM . Proof. To prove (a), we note that NS`(n) ' `[x], a free commutative graded alge* *bra, on one generator x with |x| = n, and zero differential. Since `[x] is Gorenste* *in it is therefore homotopy Gorenstein (see [13 , (18.1) & (21.3)]). The result now foll* *ows from [9, (1.8)]. To prove (b), it is clearly enough to prove that a homotopy complete interse* *ctionNis homotopy Gorenstein. But for this, noteNthat an inclusion OE : S`(1) ,! IS`(1* *) onto one factor has homotopy fibre Fffi~= J S`(1) with |J| = |I| - 1. The result now* * follows from an induction using (a) and Proposition 1.3. * * 2 Consider now a simplicial commutative R-algebra A over a field ` such that t* *he unit map R ! ß0A is surjective. We will then say that A is a homotopy complete intersection (resp. homotopy Gorenstein, homotopy CM) over ` provided A LR ` i* *s a homotopy complete intersection (resp. homotopy Gorenstein, homotopy CM). Finally, for a Noetherian ring R and a simplicial commutative R-algebra A wi* *th Noe- therian homotopy, we say that A is a locally homotopy complete intersection (re* *sp. locally homotopy Gorenstein, locally homotopy CM) provided that for each " 2 Spec(ß0A) * *there is a factorization (2.1) such that the simplicial R0-algebra A0 is a homotopy c* *omplete intersection (resp. homotopy Gorenstein, homotopy CM) over the residue field k(* *"). Now we proceed to proving our main result. Before we do so, we need a techni* *cal definition and result. First, given a simplicial R-algebra A over a field `, wi* *th (R, m) 10 JAMES M. TURNER local Noetherian and j : R ! ß0A a surjection, let x1, . .,.xn 2 m be a maximal* * regular sequence in ker(j) which extends to a minimal generating set for ker(j). We the* *n say that A is regularly r-degenerate provided the Kunneth spectral sequence TorR=(x1,...,xn)*(ß*A, `) =) TorR=(x1,...,xn)*(A, `) degenerates at the Er-term. For a general simplicial commutative R-algebra A w* *ith Noetherian homotopy, we declare that A is locally regularly r-degenerate if for* * each " 2 Spec(ß0A), there is a factorization such that the simplicial R0-algebra A0i* *s regularly r-degenerate over k("). We will also need the following result. Lemma 2.2. For a simplicial R-algebra A over a field `, let x 2 ker(j) be a * *non-zero divisor in R. Then the sequence R=(x) LR ` ! A LR ` ! A LR=(x)` is a cofibration sequence of simplicial supplemented `-algebras. Proof. It is enough to check that (A LR `) L(R LR=(x)`)` ' A LR=(x)` but this is a straightforward computation. * * 2 We now have reached the main goal of this paper. Proof of the Homology Characterization Theorem. (a) For " 2 Spec(ß0A), choose a factorization (2.1) of A. Under the hypotheses on A, if chark(") 6= 0 then A0 L* * R0` is a homotopy complete intersection, by Theorem B of [20 ]. Thus it is homotopy Gore* *nstein by Proposition 2.1 (b). So assume that chark(") = 0. Since X := A0 L R0` is connected, we may assume that X is (n-1)-connected for some n 1. Let OE : S`(n) ! X be a map so that HQn(OE)(x) is a basis member of HQn(X) ~=Dn(A|R; k(")) (by Flat Base Change and the Factorization Theorem). Here we write HQn(S`(n)) ~=`. Now, by Flat Base Change and the Transitivity Sequence, ( HQs(X) s 6= n; HQs(Fffi) ~= HQn(X)=` s = n. Also, from the finite flat dimension condition and the Kunneth spectral sequenc* *e, ß*X is a finite graded `-module. It follows from Proposition 1.4 that ß*Fffiis a fi* *nite graded `-module. Thus, by an induction, using Proposition 2.1 (a), we may assume that* * Fffi is homotopy Gorenstein. But now combining Proposition 2.1 (a) with Corollary 1.* *3 it follows that X is homotopy Gorenstein. (b) This is just Theorem B of [20 ]. (c) We adapt another argument of L. Avramov and S. Halperin [10 , (4.1)]. Again* *, choose a factorization (2.1) for A at " 2 Spec(ß0A). For the unit map j0 : (R0, m0) ! * *A0, let x1, . .,.xn 2 m0be a maximally regular sequence which extends to a minimal gene* *rating CHARACTERIZING SIMPLICIAL ALGEBRAS WITH VANISHING HOMOLOGY 11 set for ker(j0). Note that if x1, . .,.xn fails to generate this kernel, then,* * by the main 0=(x ,...,xn)0 result of [2], TorR* 1 (ß*A , k(")) is unbounded. Since we are assuming A i* *s locally 0=(x ,...,xn)0 regularly 2-degenerate, we may assume that TorR* 1 (A , k(")) is unbounded. Now, for x 2 ker(j0) a non-zero divisor in R0, we have HQ*(R0=(x) LR0`) ~=D*(R0=(x)|R0; k(")) ~=k("), by [1, (6.25)]. Thus R0=(x) LR0k(") ~= Sk(")(1) in the homotopy category, by * *[19 , (2.1.3)]. Thus, by an induction, using Lemma 2.2, there is a cofibration sequen* *ce S`(1) ! A0 L (R0=(x1,...,xs-1))k(") ! A0 L (R0=(x1,...,xs))k(") for each 1 s n. By the Factorization Theorem and our discussion above, ther* *e exists an 1 s n such that 0=(x ,...,x ) (1) TorR* 1 (As-1, k(")) is bounded; 0=(x ,...,xs) (2) TorR* 1 (A, k(")) is unbounded. Furthermore, by Flat Base Change, the Transitivity Sequence, and our finiteness* * as- sumptions, we also have (3) D*(A0|R0=(x1, . .,.xt); k(")) is bounded for all 1 t n. Applying Proposition 1.4, (1) and (2) together imply that D*(A0|R0=(x1, . .,.xs* *-1); k(")) is unbounded, contradicting (3). We therefore conclude that ker(j0) = (x1, . .,* *.xn). 2 Remark: Proposition 2.1 and the Homology Characterization Theorem shows that the standard stratified characterization of Noetherian rings Complete Intersection Gorenstein extends to our present homotopy setting, yet sensitivity to differences in char* *acteristic appear. 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Gugenheim, On PL De Rham theory and rational ho* *motopy type, Memoirs of the A.M.S. 179 (1976). [12]P. Goerss, Ä Hilton-Milnor theorem for categories of simplicial algebras,Ä* * mer. J. Math,. 111 (1989), 927-971. [13]H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Math. * *8, Cambridge University Press, 1996. [14]H. Miller, "The Sullivan conjecture on maps from classifying spaces,Ä nnal* *s of Math. 120 (1984), 39-87. [15]D. Quillen, Homotopical algebra, Lecture Notes in Mathematics 43, Springer-* *Verlag, 1967. [16]__________, äR tional homotopy theory,Ä nnals of Math. 90 (1968), 205-295. [17]__________, Ö n the (co)homology of commutative rings," Proc. Symp. Pure Ma* *th. 17 (1970), 65-87. [18]B. Shipley and S. Schwede, "Equivalences of monoidal model categories,Ä lg* *ebraic and Geometric Topology 3 (2003), 287-334. [19]J. M. Turner, Ö n simplicial commutative algebras with vanishing Andr'e-Qui* *llen homology," In- vent. Math. 142 (3) (2000) 547-558. [20]__________, Ö n simplicial commutative algebras with Noetherian homotopy," * *J. Pure Appl. Alg. 174 (2002) pp 207-220. [21]C. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Adv* *anced Mathematics 38, Cambridge University Press, 1995. Department of Mathematics, Calvin College, 3201 Burton Street, S.E., Grand Rapids, MI 49546 E-mail address: jturner@calvin.edu