Complete intersections and rational homotopy theory
aa r X i v : . [ m a t h . A T ] J un COMPLETE INTERSECTIONS IN RATIONAL HOMOTOPY THEORY.
J.P.C.GREENLEES, K.HESS, AND S.SHAMIR
Abstract.
We investigate various homotopy invariant formulations of commutative alge-bra in the context of rational homotopy theory. The main subject is the complete intersec-tion condition, where we show that a growth condition implies a structure theorem and thatmodules have multiply periodic resolutions.
Contents
1. Introduction 12. Conventions. 33. Rational homotopy theory. 44. The Morita context 85. Regular rings and spaces. 96. The centre of a triangulated category. 117. Complete intersection rings and spaces. 138. Standard form for sci spaces 159. Growth conditions. 1710. sci spaces are eci spaces 1811. Hochschild cohomology and pure Sullivan algebras 2112. Polynomial growth implies spherical extension. 2713. Examples. 3014. The nci condition 33Appendix A. Gorenstein rings and spaces 40References 441.
Introduction
Background.
It has been very fruitful to adapt the definitions of commutative algebraso that they apply in homotopy theory. The original motivation is that it is useful to study aspace X through a ring of functions, and for our purposes we will think of the ring C ∗ ( X ; k )of cochains on X . Of course, if the analogy is to be accurate, we need a commutative modelfor C ∗ ( X ; k ), and if it is to be effective we need to render the definitions homotopy invariant.The prime example of this is the connection between rational homotopy theory and rationaldifferential graded algebras (DGAs), but the availability of good models for ring spectra hasled to other useful examples in positive characteristic. The emphasis in classical rationalhomotopy theory has been on finite complexes and calculation, whereas one of the themesin characteristic p has been to consider classifying spaces of compact Lie groups where the This work was supported by EPSRC Grant number EP/E012957/1. atural finiteness condition is that the cohomology rings in question are Noetherian. Thepurpose of the present paper is to take the ideas developed for compact Lie groups andinvestigate them in the much more accessible context of rational homotopy theory. Fromone point of view this is the process of generalizing classical results [18] from the case when H ∗ ( X ; Q ) is finite dimensional to the case when it is Noetherian, and given the availabletools of rational homotopy theory, this is reasonably straightforward. From another point ofview this is an opportunity to give new and accessible examples of the theory, and to testexpectations in a context where complete calculation is often possible. Finally, the worksuggests a number of questions we may translate through the mirror [6] to local algebra, andwe plan to investigate these in future.1.B. Contents.
On the commutative algebra side we restrict attention to commutative,local, Noetherian rings. On the topological side, we restrict attention to simply connected,rational spaces X with H ∗ ( X ) Noetherian.We begin by considering analogues of regular and Gorenstein local rings. Both are alreadywell-known in rational homotopy theory, but it gives us an opportunity to introduce someterminology and to express things in a convenient language. For example, we emphasizethe importance of homotopy invariant finiteness conditions and Morita theory. The regularspaces X are precisely those which are finite products of even Eilenberg-MacLane spaces.There are enormous numbers of Gorenstein spaces, and they include manifolds and all finitePostnikov systems.In the classical literature of rational homotopy theory, Gorenstein duality does not seemto be a familiar phenomenon except in the zero-dimensional cases (Poincar´e Duality). Wetake the opportunity in Appendix A to explain how the Local Cohomology Theorem from[13] gives a Gorenstein duality statement in general. From the point of view of rationalhomotopy theory it shows (for example) that if X is any finite Postnikov system and H ∗ ( X )is Cohen-Macaulay, it is automatically Gorenstein. Furthermore, without any hypothesis onthe depth, H ∗ ( X ) is generically Gorenstein. From the point of view of homotopy invariantcommutative algebra, it gives an extremely rich and flexible source of examples.The main subject of the paper is a study of the complete intersection (ci) condition.We give a number of homotopy invariant definitions of ci spaces, corresponding to differentaspects of the ci condition. These have very different characters, so it is striking that we areable to show that in the rational context they are all equivalent. The structural conditionin commutative algebra is that a ci ring is a quotient of a regular local ring by a regularsequence. We say that a simply connected rational space X is sci if it is formed from a finiteproduct of even Eilenberg-MacLane spaces by iterated spherical fibrations (all definitionsare given precisely in Section 7). Secondly, Benson and the first author [9] introduced afiniteness condition (zci) on the category of modules analogous to requiring all modules tohave eventually multiply periodic resolutions. This is a strengthening of the condition in[14]. In this paper we needed to relax the zci condition to two new finiteness conditions, theeci and the nci conditions. Finally, in commutative algebra there is the growth conditionthat Ext ∗ R ( k, k ) has polynomial growth (equivalent to the structural condition by Gulliksen’stheorem); the condition on a rational simply connected space is the growth condition (gci)that H ∗ (Ω X ) has polynomial growth. ost remarkable of the equivalences, perhaps, is the fact that the growth condition impliesa structure theorem: X is gci if and only if there is a fibration F −→ X −→ KV, where KV is a finite product of even Eilenberg-MacLane spaces and π ∗ ( F ) is entirely in odddegrees. Amongst these spaces, those in which F has trivial k -invariants, so that F is aproduct of odd spheres, are the ones with pure Sullivan models.Another unexpected phenomenon is the importance of the Noetherian condition. On theone hand, an iterated spherical fibration over a product of even Eilenberg-MacLane spaces isobviously Noetherian. One might naively think that requiring H ∗ (Ω X ) to have polynomialgrowth would be enough without requiring H ∗ ( X ) to be Noetherian, but in fact the Milnor-Moore theorem shows that this just means π ∗ ( X ) is finite dimensional. It is very strikingthat the Noetherian condition is sufficient to give a structure theorem, and we are gratefulto N.P.Strickland for a timely remark. We also thank S.B.Iyengar for comments.1.C. The layout of the paper.
After summarizing conventions in Section 2, we begin inSection 3 by giving a brief summary of the results and terminology we need from rationalhomotopy theory. Next, in Section 4 we describe the Morita theory for moving between C ∗ ( X ) and C ∗ (Ω X ), and some results on cellularization from [13]. We are then in a positionto consider rational DGAs in parallel with rational spaces. In a series of sections we describethe definitions for rational DGAs and in particular for Sullivan models of rational spaces. InSection 5 we consider regular rings and spaces, and in Appendix A we discuss Gorensteinspaces and Gorenstein duality.From Section 6 onwards, our main concern is for complete intersections. First, Section 6discusses the centre of a derived category, and how bimodules and Hochschild cohomologygive elements of the centre. Section 7 introduces the definitions designed to capture variousaspects of hypersurface and ci spaces, which later sections show to be equivalent. Section 8takes the structural definition, and shows that any sci space has a standard form. Section 9gives the elementary argument that zci spaces satisfy the gci growth condition. Section 10shows that sci spaces all have eventually multiply periodic module theories. In Section 11we calculate the Hochschild cohomology of all pure sci spaces relative to their regular baseand use the result to show they are zci. Finally, and perhaps most interestingly, in Section12 we show that the growth condition alone is enough to show that a space has the standardsci form. Section 13 gives a number of explicit examples illustrating the phenomena we havestudied, and showing that the various classes of spaces are distinct. The final section exploresthe role of the Noetherian condition further, and gives a characterisation of the polynomialgrowth of H ∗ (Ω X ) when we do not require H ∗ ( X ) to be Noetherian in the same style as themultiply periodic resolution property for ci spaces.2. Conventions.
Terminology for triangulated categories.
Recall that an object X of a triangu-lated category T is called small if the natural map M i [ X, Y i ] −→ [ X, _ i Y i ]is an isomorphism for any set of objects Y i . thick subcategory of T is a full subcategory closed under completion of triangles andtaking retracts. We write thick( X ) for the smallest thick subcategory containing X , and if A ∈ thick( X ) we also say ‘ X finitely builds A ’ and write X | = A .A localizing subcategory of T is a thick subcategory which is also closed under takingarbitrary coproducts. We write loc( X ) for the smallest localizing subcategory containing X ,and if A ∈ loc( X ) we also say ‘ X builds A ’ and write X ⊢ A .Following [14] we say that X is virtually small if thick( X ) contains a non-trivial smallobject W , and we say that any such W is a witness for the fact that X is virtually small.2.B. Grading conventions.
We will have cause to discuss homological and cohomologicalgradings. Our experience is that this a frequent source of confusion, so we adopt the followingconventions. First, we refer to lower gradings as degrees and upper gradings as codegrees .As usual, one may convert gradings to cogradings via the rule M n = M − n . Thus both chaincomplexes and cochain complexes have differentials of degree − i )and cohomological suspensions (Σ i ): these are defined by(Σ i M ) n = M n − i and (Σ i M ) n = M n − i . Thus, for example, with reduced chains and cochains of a based space X , we have˜ C ∗ (Σ i X ) = Σ i ˜ C ∗ ( X ) and ˜ C ∗ (Σ i X ) = Σ i ˜ C ∗ ( X ) . Other conventions.
Unless explicitly stated to the contrary, all coefficients will be inthe rational numbers Q , and for a rational vector space V , we write V ∨ = Hom Q ( V, Q ) forthe dual vector space.For brevity we write CGA for commutative graded algebra (i.e., an algebra which is com-mutative in the graded sense that xy = ( − | x |·| y | yx ), DGA for differential graded algebra ,and CDGA for commutative differential graded algebra . When we refer to modules over aDGA, we intend differential graded modules unless otherwise stated.Finally, for a space X , we write C ∗ ( X ) for a CDGA model for the cochains on X .3. Rational homotopy theory.
Rational homotopy theory provides the ideal context to test ideas about homotopy in-variant commutative algebra. On the one hand many aspects of commutative algebra areespecially simple for Q -algebras and on the other we can appeal to the intuition and struc-tures of homotopy theory.3.A. Terminology for commutative differential graded algebras.
We will restrictattention to simply connected Q -algebras of finite type.If V is a graded rational vector space, we write Λ( V ) for the free CGA on V . This isa symmetric algebra on V ev tensored with an exterior algebra on V od . A Sullivan algebra is a CDGA which is free as a CGA on a simply connected graded vector space V of finitedimension in each degree, and whose differential has the property that if x ∈ V s then dx ∈ Λ( V
Any simply connected rationalCW-complex with cohomology finite in each degree is modelled by a simply connected ratio-nal CDGA (such as the CDGA of PL polynomial differential forms A P L ( X )). Furthermore,any such CDGA has a Sullivan minimal model, unique up to isomorphism. We write C ∗ ( X )for an unspecified CDGA model for the cochains on X . The process of building up a Sullivanalgebra degree by degree corresponds to building up a space using a Postnikov tower.If V is an evenly graded vector space, we write KV for the associated Eilenberg-MacLanespace. In principle we could use the same notation when V has an odd summand, but wewill not do so. Since odd spheres are rational Eilenberg-MacLane spaces, if W is a gradedvector space in odd degrees, we write S ( W ) for the corresponding Eilenberg-MacLane space.We say a space X is pure if X has pure Sullivan algebra model.A fibration E −→ B with fibre F can be modelled by a relative Sullivan algebra M ⋊ Λ( V ) ←− M where M models B , M ⋊ Λ( V ) models E and the fibre F is then modelled byΛ( V ).3.C. Homotopy Lie algebras and the Milnor-Moore theorem.
Recall that π ∗ (Ω X ) isa graded Lie algebra under the Samelson product. More precisely there is a natural bilinearproduct [ · , · ] : π i (Ω X ) × π j (Ω X ) −→ π i + j (Ω X )which is antisymmetric in the sense that[ x, y ] = − ( − | x |·| y | [ y, x ]and satisfies the graded Jacobi identity( − | x |·| z | [ x, [ y, z ]] + ( − | y |·| x | [ y, [ z, x ]] + ( − | z |·| y | [ z, [ x, y ]] = 0 . One way of forming a graded Lie algebra from an associative algebra A is to define [ x, y ] = xy − ( − | x |·| y | yx for homogeneous elements x, y ∈ A . Associated to a graded Lie algebra isa universal associative algebra U ( L ) = T L/I where
T L is the tensor algebra on L and I is the ideal generated by the relations [ x, y ] = x ⊗ y − ( − | x |·| y | y ⊗ x for x, y ∈ L . Henceforth we will generally omit the notation for thetensor product.The most important algebraic fact about the universal enveloping algebra of a Lie algebrais the Poincar´e-Birkoff-Witt theorem stating that if we filter U ( L ) by tensor length thenthere is an isomorphism Gr ( U ( L )) = Λ L. In particular, the growth rate of U ( L ) is the same as that of the symmetric algebra on L ev .The following theorem makes this relevant to topology. Theorem 3.1. (Milnor-Moore [26] ) If X is a simply connected rational space then H ∗ (Ω X ) = U ( π ∗ (Ω X )) . (cid:3) n particular, we see that H ∗ (Ω X ) has polynomial growth if and only if π ∗ (Ω X ) is finitedimensional, and in that case the growth is of degree one less thandim Q ( π ev (Ω X )) = dim Q ( π od ( X )) . Elliptic spaces.
Perhaps for historical reasons, classical rational homotopy theoryconcentrates on finite complexes, which is to say spaces with H ∗ ( X ) finite dimensional.These correspond to 0-dimensional local rings.A simply connected rational space X is called elliptic if H ∗ ( X ) and π ∗ ( X ) are both finitedimensional. It is called hyperbolic if π ∗ ( X ) has exponential growth.A major theorem of rational homotopy theory is the dichotomy theorem stating that asimply connected rational space with H ∗ ( X ) finite dimensional is either elliptic or hyperbolic.In a sense we will make precise, elliptic spaces correspond to 0-dimensionsional completeintersections.3.E. Noether normalization.
Polynomial rings on even degree generators play a specialrole in the theory. To start with, they are intrinsically formal : if P is a polynomial ringon even degree generators, then if A is any CDGA with H ∗ ( A ) ∼ = P , we have a quasi-isomorphism A ≃ P . Indeed, P has a useful universal property: for any CDGA A , and anymap θ : P −→ H ∗ ( A ) of CGAs, a choice of representative cycles for the polynomial generatorsallows us to realize θ by a map ˜ θ : P −→ A of CDGAs. Not only are they convenient, weshall see they have a structural role: polynomial rings on even degree generators provide theclass of CDGAs corresponding to regular local rings. We think of KV with V even and finitedimensional as a generalization of the rational classifying space of a compact connected Liegroup.Polynomial rings can then be used in the study of general Noetherian rings. Indeed, theNoether normalization theorem states that if R is a Noetherian connected CGA, it is finitelygenerated as a module over a polynomial subalgebra P on even degree generators. We willrepeatedly use the following counterpart of this statement. Proposition 3.2. If X is a 1-connected rational space with H ∗ ( X ) Noetherian, there is afibration F −→ X −→ KV of rational spaces where V is even and finite dimensional, and H ∗ ( F ) is finite dimensional. Proof :
By Noether normalization, H ∗ ( X ) is finite dimensional over a polynomial algebra P on even degree generators. Choosing representative cycles, we have a map P = KV −→ C ∗ ( X ) of CDGAs realizing this map in cohomology. This gives a fibration F −→ X −→ KV.
To see H ∗ ( F ) is finite dimensional, we note that H ∗ ( X ) is a finitely generated P -module,and therefore has a finite resolution by finitely generated free P -modules. (cid:3) We refer to this fibration as a Noether normalization of X , and to F as a Noether fibre of X . The long exact sequence in homotopy shows that the growth of π ∗ ( X ) is the same asthat of π ∗ ( F ). emma 3.3. (Dichotomy) For a space X with H ∗ ( X ) Noetherian, either π ∗ ( X ) is finitedimensional or it has exponential growth. The homotopy is finite dimensional if and only ifa Noether fibre is elliptic. (cid:3) This motivates the following extension of the notion of elliptic spaces to spaces withNoetherian cohomology.
Definition 3.4.
A space X is gci (or satisfies the growth condition for a complete intersec-tion) if H ∗ ( X ) is Noetherian and π ∗ ( X ) is finite dimensional.These spaces are the principal subject of the present paper, and we return to them inSection 7.3.F. Some analogies.
At the most basic level, cofibre sequences X −→ Y −→ Z of pointed spaces induce (additive) exact sequences C ∗ ( X ) ←− C ∗ ( Y ) ←− C ∗ ( Z )of reduced cochains. On the other hand, fibrations F −→ E −→ B of spaces induce (multiplicative) exact sequences C ∗ ( F ) EM ≃ C ∗ ( E ) ⊗ C ∗ ( B ) Q ←− C ∗ ( E ) ←− C ∗ ( B )provided C ∗ ( B ) is 1-connected so that an Eilenberg-Moore theorem (EM) holds, and C ∗ ( B ) −→ C ∗ ( E ) is a relative Sullivan model so that the tensor product is derived.More generally, a homotopy pullback square Z × X Y −→ Z ↓ ↓ Y −→ X induces a homotopy pushout square C ∗ ( Z × X Y ) ←− C ∗ ( Z ) ↑ ↑ C ∗ ( Y ) ←− C ∗ ( X )in the sense that C ∗ ( Z × X Y ) ≃ C ∗ ( Z ) ⊗ C ∗ ( X ) C ∗ ( Y )if X is 1-connected, and one of the maps C ∗ ( X ) −→ C ∗ ( Z ) or C ∗ ( X ) −→ C ∗ ( Y ) is a relativeSullivan algebra so that the tensor product is derived.We should also record the Rothenberg-Steenrod theorem stating that for a fibration F −→ E −→ B we have equivalences C ∗ ( E ) ≃ C ∗ ( F ) ⊗ C ∗ (Ω B ) k and C ∗ ( E ) ≃ Hom C ∗ (Ω B ) ( k, C ∗ ( F )) . . The Morita context
We have a simply connected rational space of finite type X , and we consider the CDGA C ∗ ( X ). We often wish to translate to statements about the DGA C ∗ (Ω X ). Throughoutwe work in derived categories of DG-modules such as D ( C ∗ ( X )) or D ( C ∗ (Ω X )), so tensorproducts and Homs are derived. As mentioned above, we usually refer simply to ‘modules’since the requirement that our modules respect the differentials is implicit in the categorywe work in. The material is adapted from [12, 13].4.A. The two algebras.
We need to see first that the C ∗ ( X ) (a commutative DGA) and C ∗ (Ω X ) (which will usually not be commutative) determine each other. Proposition 4.1. If X is 1-connected, there are equivalences C ∗ (Ω X ) ≃ Hom C ∗ ( X ) ( Q , Q ) and C ∗ ( X ) ≃ Hom C ∗ (Ω X ) ( Q , Q ) of DGAs. Proof:
The first of these is the Eilenberg-Moore theorem [15] and the second is the Rothenberg-Steenrod theorem [27]. (cid:3)
The adjunction.
The proposition shows that we have an adjoint pair of functorsHom C ∗ ( X ) ( Q , · ) : C ∗ ( X )-mod / / mod- C ∗ (Ω X ) : ( · ) ⊗ C ∗ (Ω X ) Q o o . This induces an equivalence between subcategories of the derived categories, but it will beenough for us to know we can move between the module categories and to understand onecomposite.4.C.
Cellularization.
An object in the derived category of C ∗ ( X )-modules is said to be Q -cellular if it is built from Q up to equivalence. A map M −→ N of C ∗ ( X )-modules is a Q -equivalence if Hom C ∗ ( X ) ( Q , M ) −→ Hom C ∗ ( X ) ( Q , N )is a homology isomorphism. A map M −→ N is Q -cellular approximation if it is a Q -equivalence and M is Q -cellular. By the usual formal argument, this is unique up to equiv-alence, and we write Cell Q ( N ) −→ N for it.We will give two models for Q -cellularization, and it will be valuable to know they areequivalent.4.D. The Morita model.
The first model comes from the Morita context.
Proposition 4.2. [12, 13] If H ∗ ( X ) is Noetherian, the counit Hom C ∗ ( X ) ( Q , M ) ⊗ C ∗ (Ω X ) Q −→ M of the adjunction is Q -cellularization. (cid:3) We need only observe that C ∗ ( X ) is proxy-regular in the sense of [13]. Since H ∗ ( X ) isNoetherian, the Koszul complex associated to a system of parameters provides a proof. .E. The stable Koszul model. If R is a commutative ring and I = ( x , x , . . . , x r ) is anideal, then Grothendieck defines the local cohomology of an R -module N by the formula H ∗ I ( R ; N ) = H ∗ (( R −→ [ 1 x ]) ⊗ R ( R −→ [ 1 x ]) ⊗ R · · · ⊗ R ( R −→ [ 1 x n ]) ⊗ R N ) , and shows it calculates the right derived functors of I -power torsion when R is Noetherian.We write H ∗ I ( R ) = H ∗ I ( R ; R ) for brevity.We now lift this to DGAs in the usual way. If x ∈ H ∗ ( A ), we write Γ x A = fibre( A −→ A [1 /x ]), and if I = ( x , x , . . . , x n ) is an ideal in H ∗ ( A ), for an A -module M we writeΓ I M = Γ x A ⊗ A Γ x A ⊗ A · · · ⊗ A Γ x n A ⊗ A M. It turns out that up to equivalence this depends only on the ideal I , and indeed, only on theradical of I . If I = m is the maximal ideal we abbreviate this Γ M = Γ m M .Note that Γ M has a filtration from its construction, and that we therefore have a spectralsequence for calculating its homology. Lemma 4.3.
There is a spectral sequence H ∗ I ( H ∗ ( A ); H ∗ ( M )) ⇒ H ∗ (Γ M ) . (cid:3) Finally, the relevance to us is that this gives another construction of cellularization.
Proposition 4.4. [13, 9.3]
The natural map Γ M −→ M is Q -cellularization. (cid:3) Now we specialize to the case A = C ∗ ( X ) to obtain the required equivalence from unique-ness of cellularization. Corollary 4.5.
There is a natural equivalence Γ M ≃ Hom C ∗ ( X ) ( Q , M ) ⊗ C ∗ (Ω X ) Q . (cid:3) Regular rings and spaces.
We shall show that the regular spaces are precisely the spaces KV where V is even andfinite dimensional. This is straightforward once we have established definitions.For all classical commutative algebra, we refer the reader to [25].5.A. Definitions.
In commutative algebra there are three styles for a definition of a regularlocal ring: ideal theoretic, in terms of the g rowth of the Ext algebra and a h omotopy invariantversion. Definition 5.1. (i) A local Noetherian ring R is regular if the maximal ideal is generatedby a regular sequence.(ii) A local Noetherian ring R is g-regular if Ext ∗ R ( k, k ) is finite dimensional.(iii) A local Noetherian ring R is h-regular if every finitely generated module is small in D ( R ). t is not hard to see that g-regularity is equivalent to h-regularity or that regularity impliesg-regularity. Serre proved that g-regularity implies regularity, so the three conditions areequivalent.It is not altogether clear what should play the role of finitely generated modules in themore general context. We would like it to include all small objects, and the object Q , and wewould like to know that if Q is small then all objects in the class are small. For the purposeof the present paper, we take F G := { M | H ∗ ( M ) is a finitely generated H ∗ ( X )-module } , and we will show that it has the properties we require. Definition 5.2. (i) A space X is s-regular if there are fibrations S n −→ X −→ X, S n −→ X −→ X , . . . , S n d −→ X d −→ X d − with X d ≃ ∗ .(ii) A space X is g-regular if H ∗ (Ω X ) is finite dimensional.(iii) A space X is h-regular if every object of F G is small in D ( C ∗ ( X )).If X is s-regular, we see Ω X d − ≃ S n d , and working back up the sequence of fibrations,we see that X is g-regular. Since Q ∈ F G it follows from Proposition 4.1 that an h-regularspace is g-regular. We will establish the reverse implication by classifying g-regular spaces. Remark 5.3.
The use of the classification is somewhat unsatisfactory, and suggests thatwe should seek a choice of class
F G that is appropriate even when we do not have such aclassification. One possibility is to consider all R -modules M which are small as Q -modules,for some map Q −→ R of algebras from a regular ring Q so that R is small as a Q -module.5.B. Classification of regular spaces.
In the rational context we can give a completeclassification of regular spaces.
Theorem 5.4.
A simply connected rational space X of finite type is g-regular if and onlyif π ∗ ( X ) is even and finite dimensional. It is therefore equivalent to the Eilenberg-MacLanespace K ( π ∗ ( X )) , and has polynomial cohomology Symm( π ∗ ( X )) . Proof:
Since Ω X is a product of Eilenberg-MacLane spaces, we need only remark that oddEilenberg-MacLane spaces are spheres, whereas even Eilenberg-MacLane spaces are infinitedimensional. (cid:3) Proposition 5.5. If X is g-regular and H ∗ ( M ) is finitely generated over H ∗ ( X ) then M issmall. Proof:
Suppose H ∗ ( M ) is a finitely generated H ∗ ( X )-module. Since H ∗ ( X ) is a polynomialring on even degree generators, there is a finite resolution by finitely generated free modules0 −→ P r d r −→ P r − d r − −→ · · · d −→ P d −→ P d −→ H ∗ ( M ) −→ . We proceed to realize this in the usual way. To start with we realize the free modules P i = ( H ∗ ( X )) ⊕ n by the C ∗ ( X )-modules P i = ( C ∗ ( X )) ⊕ n . Now take M = M and realize he algebraic resolution by constructing a diagram P (cid:15) (cid:15) Σ P (cid:15) (cid:15) Σ r P r (cid:15) (cid:15) M / / M / / · · · / / M r / / M r +1 ≃ i P i −→ M i −→ M i +1 are cofibre sequences, H ∗ (Σ − i M i ) = ker( d i − )and Σ i P i −→ M i realizes the map in the algebraic resolution. Reversing the process, we seethat M r , M r − , . . . , M and M = M are finitely built from C ∗ ( X ) and therefore small asrequired. (cid:3) This establishes the equivalence of the two definitions of regularity.
Corollary 5.6.
A space is g-regular if and only if it is h-regular. (cid:3)
Some small objects.
It is useful to identify some modules that are small rathergenerally.
Lemma 5.7. If f : Y −→ X is a map with homotopy fibre F ( f ) so that H ∗ ( F ( f )) is finitedimensional, then C ∗ ( Y ) is small in D ( C ∗ ( X )) . Proof:
By hypothesis, Q finitely builds C ∗ ( F ( f )) as a C ∗ (Ω X )-module. Applying Hom C ∗ (Ω X ) ( Q , · ),we deduce from the Eilenberg-Moore spectral sequence that C ∗ ( X ) finitely builds C ∗ ( Y ). Insymbols, Q | = C ∗ (Ω X ) C ∗ ( F ( f ))and hence C ∗ ( X ) ≃ Hom C ∗ (Ω X ) ( Q , Q ) | = Hom C ∗ (Ω X ) ( Q , C ∗ ( F ( f )) ≃ C ∗ ( Y ) . (cid:3) Lemma 5.8. If X is g-regular, and Y −→ X is a map with C ∗ ( F ( f )) finitely built from C ∗ (Ω X ) then C ∗ ( Y ) is small. Proof:
Suppose X is g-regular, so that H ∗ (Ω X ) is finite dimensional. Thus Q finitely builds C ∗ (Ω X ). It follows that if C ∗ (Ω X ) finitely builds C ∗ ( F ( f )) then Q finitely builds C ∗ ( F ( f ))and we may apply the argument of Lemma 5.7. (cid:3) The centre of a triangulated category.
It will be useful to recall certain constructions before turning to complete intersections. .A. Universal Koszul complexes.
To start with we suppose given a triangulated cate-gory T . The centre Z T of T is defined to be the graded ring of graded endomorphisms ofthe identity functor.Given χ ∈ Z T of degree a , for any object X , we may form the mapping cone X/χ of χ : Σ a X −→ X . This is well defined up to non-unique equivalence. Indeed, given a map f : X −→ Y , the axioms of a triangulated category give a map f : X/χ −→ Y /χ consistentwith the defining triangles, but this is not usually unique or compatible with composition.Now given a sequence of elements χ , χ , . . . , χ n we may iterate this construction, andform K ( X ; χ ) := X/χ /χ / · · · /χ n , which we refer to as the universal Koszul complex of the sequence. Once again, up toequivalence K ( X ; χ ) depends only on the sequence, and is independent of the order of theelements χ i .6.B. Bimodules and the centre.
Bimodules provide a useful source of elements of Z D ( R ).Indeed, if R is a flat l -algebra, and if X −→ Y is a map of R -bimodules over l (which is tosay, of modules over R e = R ⊗ l R ), then for any R -module M we obtain a map X ⊗ R M −→ Y ⊗ R M of R -modules, natural in M .It is sometimes convenient to package this in terms of the Hochschild cohomology ring HH ∗ ( R | l ) = Ext ∗ R e ( R, R ) . If l = Z , it is usual to omit it from the notation. Now a codegree d element of this cohomologyring can be viewed as a map R −→ Σ d R in the category of ( R, R )-bimodules, so that taking X = Y = R above, we obtain a ring homomorphism HH ∗ ( R | l ) −→ Z D ( R ) . If R is an l -algebra which is not flat, R e = R ⊗ l R is taken in the derived sense, and similarlyfor HH ∗ ( R | l ).Given maps l −→ Q −→ R , we obtain a map R ⊗ l R −→ R ⊗ Q R and hence a ring map HH ∗ ( R | Q ) −→ HH ∗ ( R | l ). In particular, we have maps R = HH ∗ ( R | R ) −→ HH ∗ ( R | Q ) −→ HH ∗ ( R | Z ) = HH ∗ ( R ) . If R = C ∗ ( X ), we may always take l = Q = C ∗ ( pt ), so that a bimodule is a module over R e = C ∗ ( X × X ), but it is usually more appropriate to work over Q = C ∗ ( K ) where we havea fibration X −→ K . In that case a bimodule over Q is a module over R e = C ∗ ( X × K X ).6.C. Hochschild cohomology transcended.
It seems natural to relax the role of Hochschildcohomology. For us it is really just a tool for building bimodules from R . We will supposegiven a map Q −→ R so that Q is regular and R is small over Q . This ensures that F G asdefined in Section 5 coincides with the R -modules which are small over Q .Now, if X is any R -bimodule finitely built from R , we can apply ⊗ R M to deduce X ⊗ R M is finitely built from M = R ⊗ R M : R | = R e X implies M = R ⊗ R M | = R X ⊗ R M. The important case for us is when X is a small R e -module. emma 6.1. If X is a small R e -module and M ∈ F G , then X ⊗ R M is a small R -module. Proof:
It suffices to consider the case X = R e . We then have X ⊗ R M = R ⊗ Q R ⊗ R M = R ⊗ Q M. By Proposition 5.5, M is small as a Q -module, so R = R ⊗ Q Q | = R ⊗ Q M as required. (cid:3) This comes close to saying that if R is virtually small as an R e -module then every M ∈ F G is virtually small as an R -module. The only obstacle is the need to show X ⊗ R M is non-trivial; in the context we need it, the non-zero degree of the maps constructing X will makeit clear. 7. Complete intersection rings and spaces.
We will give definitions of complete intersections as in the regular case. For commutativeNoetherian rings these were shown to be equivalent in [9]. We will show they are equivalentfor rational spaces.7.A.
The definition.
In commutative algebra there are three styles for a definition of acomplete intersection ring: ideal theoretic, in terms of the growth of the Ext algebra and aderived version.
Definition 7.1. (i) A local Noetherian ring R is a complete intersection (ci) ring if R = Q/ ( f , f , . . . , f c ) for some regular ring Q and some regular sequence f , f , . . . , f c . Theminimum such c (over all Q and regular sequences) is called the codimension of R .(ii) A local Noetherian ring R is gci if Ext ∗ R ( k, k ) has polynomial growth. The g-codimension of R is one more than the degree of the growth.(iii) A local Noetherian ring R is zci [9] if there are elements z , z , . . . z c ∈ Z D ( R ) ofnon-zero degree so that M/z /z / · · · /z c is small for all finitely generated modules M . Theminimum such c is called the z-codimension of R .The zci condition implies that every finitely generated module finitely builds a smallcomplex in a prescribed manner using elements in Z D ( R ). We can relax this by demandingonly that each step in the building of the small complex is the cone of an endomorphism ofthe previous step. This is the essence of the next definition.(iv) A local Noetherian ring R is eci if there is a regular ring Q , a map Q −→ R andhomotopy cofibration sequences of R e -modules, where R e = R ⊗ Q R , R = M g −→ Σ n M → M , . . . , M c − g c −→ Σ n c M c − → M c such that M c is small as an R e -module and the degree of each g i is not zero.Two variations are also useful.(v) A local Noetherian ring R is said to be bci if there is a regular ring Q and map Q −→ R so that R is virtually small as an R e -module, where R e = R ⊗ Q R .(vi) If R is a commutative ring or CDGA, it is said to be a quasi-complete intersection(qci) [14] if every finitely generated object is virtually small. heorem 7.2. [9] For a local Noetherian ring the conditions ci, gci and zci are all equivalent,and the corresponding codimensions are equal. These conditions imply the eci, bci and qciconditions.
It is a result of Gulliksen that if R is ci of codimension c , one may construct a resolutionof any finitely generated module growing like a polynomial of degree c −
1. A suitableconstruction of this resolution shows that R is zci. Considering the module k shows that thering Ext ∗ R ( k, k ) has polynomial g rowth. Perhaps the most striking result about ci rings isthe theorem of Gulliksen [20] which states that this characterises ci rings so that the ci andgci conditions are equivalent for local rings. Remark 7.3.
In commutative algebra, Avramov [3] proved Quillen’s conjectured character-ization of complete intersections by the fact that the Andr´e-Quillen cohomology is bounded.When k is of characteristic 0, the DG Andr´e-Quillen cohomology of C ∗ ( X ) gives the dualhomotopy groups of X , so the counterpart of Avramov’s characterization is the gci condition.On the other hand in positive characteristic, results of Mandell [24] show that the topo-logical Andr´e-Quillen cohomology of C ∗ ( X ) vanishes quite generally, so this does not givean appropriate counterpart of the ci condition.7.B. Definitions for spaces.
Adapting the above definitions for spaces is straightforward.
Definition 7.4. (i) A space X is spherically ci (sci) if it is formed from a regular space KV using a finite number of spherical fibrations. More precisely, we require that there is aregular space X = KV with V even and finite dimensional, and fibrations S n −→ X −→ X = KV, S n −→ X −→ X , . . . , S n c −→ X c −→ X c − with X = X c . The least such c is called the s-codimension of X .(ii) A space X is a gci space if H ∗ ( X ) is Noetherian and H ∗ (Ω X ) has polynomial growth.The g-codimension of X is one more than the degree of growth.(iii) A space X is a zci space if H ∗ ( X ) is Noetherian and there are elements z , z , . . . , z c ∈ Z D ( C ∗ ( X )) of non-zero degree so that C ∗ ( Y ) /z /z / · · · /z c is small for all C ∗ ( Y ) ∈ F G .(iv) A space X is an eci space if H ∗ ( X ) is Noetherian, there is a regular space K and fibra-tion X −→ K with C ∗ ( X ) small over C ∗ ( K ) and there are homotopy cofibration sequencesof C ∗ ( X × K X )-modules, C ∗ ( X ) = M g −→ Σ n M → M , . . . , M c − g c −→ Σ n c M c − → M c such that M c is small as an C ∗ ( X × K X )-module and the degree of each g i is not zero.(v) We say X is bci space if H ∗ ( X ) is Noetherian and C ∗ ( X ) is virtually small as a C ∗ ( X × K X )-module for some regular space K and fibration X −→ K with C ∗ ( X ) smallover C ∗ ( K ).(vi) We say X is qci space if H ∗ ( X ) is Noetherian and each C ∗ ( Y ) ∈ F G is virtually small.The main result of this paper is as follows. Theorem 7.5.
For a rational space X the sci, eci and gci conditions are equivalent. If inaddition X is pure, then the conditions above are equivalent to the zci condition. We will establish the implications sci A ⇒ eci B ⇒ gci C ⇒ sci. e establish A in Section 10, B in Section 9, and C in Section 12. The first two implicationsare fairly straightforward in the sense that they can also be proved in the non-rational context[10]. The implication C takes a growth condition and gives a structure theorem, and couldbe viewed as the main result of the present paper. In Section 11 we show that a pure scispace is zci, while in Section 9 we show that the zci condition implies gci. Remark 7.6. (i) If X is elliptic then H ∗ ( X ) and π ∗ (Ω X ) are both finite dimensional, so itis clear that every elliptic space is gci.(ii) It is also clear that zci implies qci, and that if the natural transformations giving thezci condition come from Hochschild cohomology then this implies eci, and eci clearly impliesbci.7.C. Hypersurface rings.
A hypersurface is a complete intersection of codimension 1. Thefirst four definitions adapt to define hypersurfaces, g-hypersurfaces, z-hypersurfaces and e-hypersurfaces. The notion of g-hypersurface (i.e., the dimension of the groups Ext iR ( k, k ) isbounded) may be strengthened to the notion of p-hypersurface where we require that theyare eventually periodic, given by multiplication with an element of the ring. All five of theseconditions are equivalent by results of Avramov.One possible formulation of b-hypersurface would be to require that the R builds a small R e -module in one step (or equivalently, that R is a z-hypersurface but z arises from HH ∗ ( R )).Both these definitions are equivalent to being an e-hypersurface.Finally, we may say that R is a q-hypersurface if every finitely generated module M hasa self map with non-trivial small mapping cone.7.D. Hypersurface spaces.
All six of these conditions have obvious formulations for spaces.A space X is an s-hypersurface if there is a fibration S n −→ X −→ KV with V even and finite dimensional. It is a z-hypersurface if there is an element z of non-zerodegree in Z D ( C ∗ ( X )) so that, for any M in F G , the mapping cone of z : M −→ M is small.It is a g-hypersurface if the dimensions of H i (Ω X ) are bounded, and a p-hypersurace if theyare eventually periodic given by multiplication by an element of the ring.The space X is an e-hypersurface if C ∗ ( X ) builds a small C ∗ ( X × K X )-module in one stepfor a regular space K . Finally, X is a q-hypersurface if every finitely module C ∗ ( Y ) in F G has a self map with non-trivial small mapping cone.8.
Standard form for sci spaces
We are eventually going to show that the sci, gci and eci conditions are equivalent forrational spaces. Of the conditions, the easiest to get a grip on is the sci condition, and itseems worthwhile to begin by anchoring it in reality by giving a structure theorem. In therational context, we may put sci spaces into a standard form.
Theorem 8.1.
A space X is sci if and only if there exists a fibration sequence F → X → KV, where KV is a regular space and π ∗ ( F ) is finite dimensional and entirely in odd degrees; inthis case codim( X ) = dim Q ( π ∗ ( F )) = dim Q ( π odd ( X )) . efore proceeding it is useful to note that all the spherical fibrations in the definition ofan sci space may be taken to be odd. Lemma 8.2. If X can be formed from B with an even spherical fibration S m −→ X −→ B ,then it can be formed from B × K ( Q , m ) by an odd spherical fibration S m − −→ X −→ B × K ( Q , m ) . Accordingly, an sci space of codimension c may be constructed in c steps from a regular spaceusing only odd dimensional spherical fibrations. Proof: If C ∗ ( X ) = C ∗ ( B ) ⋊ Λ( x m , y m − ), then if dy = x + ax + b we may change basis bytaking x ′ = x + a/ dx ′ = 0 , dy = ( x ′ ) + z , where z = b − a / ∈ C ∗ ( B ). Adjoining x ′ to the model of B , we get the base of the required fibration. (cid:3) Proof of Theorem 8.1: If X is sci, by Lemma 8.2 we may use only odd spheres inthe fibres. Now the composite function X −→ KV has fibre with only odd dimensionalhomotopy, giving a fibration of the stated form.We prove the converse statement by induction on the dimension of the odd homotopy.The result is trivial if the homotopy is entirely even. Suppose then that X lies in a fibration F −→ X −→ KV and that x ∈ π m ( F ) is an element of highest degree. Construct a fibration S m −→ F −→ F ′ by killing x , so that dim Q ( π ∗ ( F ′ )) = dim Q ( π ∗ ( F )) −
1. Thus, we may choose models so that C ∗ ( F ) = C ∗ ( F ′ ) ⋊ C ∗ ( S m ), and C ∗ ( X ) = C ∗ ( KV ) ⋊ [ C ∗ ( F ′ ) ⋊ C ∗ ( S m )] . Let X ′ be modelled by the subalgebra generated by C ∗ ( KV ) and C ∗ ( F ′ ). This gives fibra-tions S m −→ X −→ X ′ and F ′ −→ X ′ −→ KV.
By induction X ′ is sci, so that X is sci as required. The codimension is obviously boundedbelow by dim Q ( π odd ( X )), and we have described a procedure achieving this bound. (cid:3) The following rearrangement result will be useful later.
Corollary 8.3. If X occurs in a fibration X ′ −→ X −→ KV with X ′ sci of codimension c , then X is itself sci of codimension c . Proof :
By Theorem 8.1, X ′ has a model of the form X ′ = KV ′ ⋊ F ′ with π ∗ ( F ′ ) finitedimensional and in odd degrees and X = KV ⋊ X ′ with both V and V ′ even and finitedimensional. By parity there can be no differential from KV to KV ′ , so X = KV ′ ⋊ ( KV ⋊ F ′ ) ≃ ( KV ′ ⋊ KV ) ⋊ F ′ . Since any fibration with base KV ′ and fibre KV is a product, we obtain a fibration F ′ −→ X −→ K ( V ⊕ V ′ ) . y Theorem 8.1 again we deduce X is sci of codimension c . (cid:3) In terms of rational models we can restate the sci condition very simply. The result isimmediate from Theorem 8.1 by taking a Sullivan model of the fibration.
Corollary 8.4.
A space X is sci if and only if X has a cochain algebra model (Λ V, d ) where d ( V even ) = 0 . (cid:3) Growth conditions.
In this section we prove perhaps the simplest implication between the ci conditions: forsimply connected rational spaces of finite type, eci (and also zci) implies gci.9.A.
Polynomial growth.
Throughout algebra and topology it is common to use the rateof growth of homology groups as a measurement of complexity. We will be working over H ∗ ( X ), so it is natural to assume that our modules M are locally finite in the sense that H ∗ ( M ) is cohomologically bounded below and dim Q ( H i ( M )) is finite for all i . Definition 9.1.
We say that a locally finite module M has polynomial growth of degree ≤ d ,and write growth( M ) ≤ d , if there is a polynomial p ( x ) of degree d withdim Q ( H n ( M )) ≤ p ( n )for all n >> Remark 9.2. (i) In commutative algebra the usual terminology is that a module of growth d has complexity d + 1.(ii) Note that a complex with bounded homology has growth ≤ −
1. For complexes withgrowth ≤ d with d ≥
0, by adding a constant to the polynomial, we may insist that thebound applies for all n ≥ Mapping cones reduce degree by one.
We use the following estimate on growth.
Lemma 9.3.
Given cohomologically bounded below locally finite modules M and N in atriangle Σ n M χ −→ M −→ N with n = 0 , then growth( M ) ≤ growth( N ) + 1 . Proof:
The homology long exact sequence of the triangle includes · · · −→ H i − n ( M ) χ −→ H i ( M ) −→ H i ( N ) −→ · · · . This shows dim Q ( H i ( M )) ≤ dim Q ( H i ( N )) + dim Q ( χH i − n ( M )) . Iterating s times, we finddim Q ( H i ( M )) ≤ dim Q ( H i ( N )) + dim Q ( H i − n ( N )) + · · ·· · · + dim Q ( H i − ( s − n ( N )) + dim Q ( χ s H i − sn ( M )) . o obtain growth estimates, it is convenient to collect the dimensions of the homogeneousparts into the Hilbert series h M ( t ) = P n dim Q ( H i ( M )) t i . An inequality between such formalseries means that it holds between all coefficients.First suppose that n >
0. Since H ∗ ( M ) is bounded below, if h M ( t ) is the Hilbert series of H ∗ ( M ) then we have h M ( t ) ≤ h N ( t )(1 + t n + t n + · · · ) = h N ( t )1 − t n , giving the required growth estimate.If n = − n ′ < N ′ −→ M −→ Σ n ′ M where N ′ = Σ n ′ − N and argue precisely similarly. (cid:3) Growth of eci spaces.
The implication we require is now straightforward.
Theorem 9.4. If X is eci then it is also gci, and if X has e-codimension c it has g-codimension ≤ c . Proof:
It is sufficient to show C ∗ (Ω X ) ≃ Q ⊗ C ∗ ( X ) Q has polynomial growth.By hypothesis there is an appropriate regular space K and self maps γ : M → Σ | γ | M , γ : M → Σ | γ | M , . . . , γ c : M c − → Σ | γ c | M c − of non-zero degree in D ( C ∗ ( X × K X )), so that M i is the cone of γ i and M c , which is thecone of γ c , is small. Thus, applying Q ⊗ C ∗ ( X ) ( · ) to M c we obtain a complex with growth ≤ −
1. By the lemma if we apply Q ⊗ C ∗ ( X ) ( · ) to M c − we obtain a complex of growth ≤ Q ⊗ C ∗ ( X ) ( · ) to Q itself we obtain acomplex with growth ≤ c − (cid:3) The proof above, with minor changes, also yields the following Theorem.
Theorem 9.5. If X is zci then it is also gci, and if X has z-codimension c it has g-codimension ≤ c . sci spaces are eci spaces In this section we show that sci spaces (defined by a particular construction) have aperiodic module theory in the sense that they are eci. This may not be too surprising, butthe particular way in which bimodules and fibrations are used may be of some interest.
Theorem 10.1. If X is an sci space of codimension c , then it is eci of codimension c . Remark 10.2.
The construction will show that all the maps building the small bimoduleare of positive degree, so that Lemma 6.1 shows that if X is sci then all C ∗ ( X )-modules in F G are virtually small.We will upgrade the conclusion to show that if X is a pure sci space then X is zci inSection 11. Fibration lemmas.
We will repeatedly use two elementary lemmas. The first is verywell known.
Lemma 10.3. If F −→ E p −→ B is a fibration with a section s , then there is a fibration Ω F −→ B s −→ E. Proof:
We start from the square B / / = (cid:15) (cid:15) E p (cid:15) (cid:15) B = / / B and take iterated fibres. (cid:3) The second lemma is a Third Isomorphism Theorem for fibrations.
Lemma 10.4.
Given fibrations Y −→ B −→ C , if F = fibre( B −→ C ) there is a fibration Ω F −→ Y × B Y −→ Y × C Y. Proof:
We start from the cube Y × C Y / / (cid:15) (cid:15) Y (cid:15) (cid:15) Y × B Y / / (cid:15) (cid:15) qqqqqqqqqq Y (cid:15) (cid:15) ? ? (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) Y / / CY / / qqqqqqqqqqqqq B ? ? ~~~~~~~ and take iterated fibres. (cid:3) Building bimodules.
It is worth isolating the process that we use repeatedly tobuild bimodules. Abstracted from its context, the proof is extremely simple. The strengthof the result is that the cofibre sequence is one of C ∗ ( Y )-modules. Proposition 10.5.
Suppose given a fibration Ω S m −→ X f −→ Y. (i) If m is odd, then there is a cofibre sequence of C ∗ ( Y ) -modules Σ m − C ∗ ( X ) ←− C ∗ ( X ) f ∗ ←− C ∗ ( Y ) . (ii) If m is even, then there is a cofibre sequence of C ∗ ( Y ) -modules Σ m − C ∗ ( X ) ←− C ∗ ( X ) ←− F ith F small. More precisely, F is built from two copies of C ∗ ( Y ) in the sense that there isa cofibre sequence Σ m − C ∗ ( Y ) ←− F ←− C ∗ ( Y ) of C ∗ ( Y ) -modules. Proof:
We take a relative Sullivan model for f ∗ . This has the form C ∗ ( Y ) ⋊ C ∗ (Ω S m ).If m is odd, the relative Sullivan model is of the form C ∗ ( X ) ≃ C ∗ ( Y ) ⋊ Λ( z m − ), where z is a polynomial generator. The quotient of this by the DG- C ∗ ( Y )-submodule C ∗ ( Y ) · z isagain a C ∗ ( Y )-module, and the compositeΣ m − C ∗ ( X ) z −→ C ∗ ( X ) −→ C ∗ ( X ) /C ∗ ( Y )is an isomorphism as required.If m is even, the relative Sullivan model is of the form C ∗ ( X ) ≃ C ∗ ( Y ) ⋊ Λ( z m − , t m − ) . This time t is a polynomial generator, and z is an exterior generator. Accordingly we let F bethe DG- C ∗ ( Y )-submodule generated by t and z , so that C ∗ ( Y ) /F is again a C ∗ ( Y )-module.It is isomorphic to Σ m − C ∗ ( X ) in the sense that the compositeΣ m − C ∗ ( X ) t −→ C ∗ ( X ) −→ C ∗ ( X ) /F is an isomorphism. (cid:3) Remark 10.6.
As an example, we observe that this shows that an s-hypersurface is a z-hypersurface. Indeed, by hypothesis, we have a fibration S m −→ X −→ KV , and hence bypullback a fibration S m −→ X × KV X −→ X with a section ∆ : X −→ X × KV X . By Lemma 10.3, we obtain a fibrationΩ S m −→ X ∆ −→ X × KV X so we may apply Proposition 10.5 with Y = X × KV X , noting that a C ∗ ( Y )-module is thena C ∗ ( X )-bimodule. If m is odd we then get a cofibre sequenceΣ m − C ∗ ( X ) ˜ χ ←− C ∗ ( X ) ←− C ∗ ( X × KV X )of bimodules. As in Subsection 6.C note that ˜ χ gives an element of Z D ( C ∗ ( X )) by tensoringdown, in the sense that for any C ∗ ( X )-module M we apply M ⊗ C ∗ ( X ) ( · ) to get a cofibresequence Σ m − M χ ←− M ←− C ∗ ( X ) ⊗ C ∗ ( KV ) M. If M is finitely generated, then it is small as a C ∗ ( KV )-module by Proposition 5.5 showingthat the fibre of χ is small as required.The argument when m is even is precisely similar. The proof.
We now have the necessary ingredients for proving Theorem 10.1.We suppose X is sci of codimension c , so that we may form X = X c in c steps from X = KV using fibrations S n i −→ X i −→ X i − . It will simplify the argument to assume all the spheres are odd dimensional, as we may doby Lemma 8.2.We must show that C ∗ ( X ) builds C ∗ ( X × KV X ) as a bimodule (i.e., as a C ∗ ( X × KV X )-module) using s cofibre sequences. It is convenient to write X ei = X × X i X , and X e = X e ,so that we want to work with C ∗ ( X e )-modules. However, since we have maps X = X s −→ X s − −→ · · · −→ X = KV, we have maps X s = X es −→ X es − −→ · · · −→ X e = X e , so we may view C ∗ ( X ei )-modules as C ∗ ( X e )-modules by restriction.We are ready to apply our fibration lemmas.Pulling back the fibration along X i −→ X i − we obtain a fibration S n i −→ X i × X i − X i π −→ X i with a section given by the diagonal ∆. Applying Lemma 10.3 we obtain a fibrationΩ S n i −→ X i ∆ −→ X i × X i − X i . Similarly, applying Lemma 10.4 to X s −→ X i −→ X i − where s ≥ i , we obtain a fibrationΩ S n i −→ X ei −→ X ei − . Now using the first of these, Proposition 10.5 gives a cofibrationΣ n s C ∗ ( X ) ←− C ∗ ( X ) ←− C ∗ ( X es − ) , of C ∗ ( X es − )-modules, which we view as a cofibration of C ∗ ( X e )-modules by pullback. Suc-cessive fibrations give Σ n i C ∗ ( X ei ) ←− C ∗ ( X ei ) ←− C ∗ ( X ei − ) , until we reach Σ n C ∗ ( X e ) ←− C ∗ ( X e ) ←− C ∗ ( X e ) , so that C ∗ ( X ) = C ∗ ( X es ) | = C ∗ ( X e ) = C ∗ ( X e )as required. (cid:3) Hochschild cohomology and pure Sullivan algebras
In this section we calculate the Hochschild cohomology of a pure sci space X and upgradethe conclusion of Section 10 to give the required conclusion that any pure sci space is alsozci. Theorem 11.1. If X is a pure sci space of codimension c , then it is zci of codimension c . In view of Theorem 10.1 and Remark 10.2 we need only show that the maps of bimodulesused in the constructions of Section 10 all lift to elements of Z D ( R ). Specifically, this isCorollary 11.7. Hochschild cohomology.
It is convenient to adapt the algebraic notation for Hochschildcohomology to cochain algebras.
Notation 11.2. If X → Y is a fibration of spaces, set HH ∗ ( X | Y ) := Ext ∗ C ∗ ( X × Y X ) ( C ∗ ( X ) , C ∗ ( X )) . Note that we consider C ∗ ( X ) as a C ∗ ( X × Y X )-module via the diagonal map X → X × Y X .We will be applying this to sci spaces, and use the notation of Section 10: X = X s is ansci space of codimension s , so that for 1 ≤ i ≤ s we have fibrations S n i → X i → X i − with n i odd and X = KV , where V is finite dimensional and even. Theorem 11.3. If X is a pure sci space as above, the Hochschild cohomology is given by HH ∗ ( X | KV ) = H ∗ ( X )[[ ζ , ..., ζ s ]] , where the degree of ζ i is n i − . Remark 11.4. (i) The completion involved in forming the power series ring is homogeneous,so that if X is finite dimensional the ring is a polynomial ring. Otherwise the formula is tobe interpreted as formed by successive adjunction of the variables in the stated order.(ii) One would expect it to follow from Theorem 11.3 that the construction of Section 10can be upgraded to show X is zci. In any case, this upgrading of the construction is aningredient in the Hochschild cohomology calculation.The theorem evidently follows by repeated application of the following general result aboutfibrations with fibre an odd sphere. Proposition 11.5.
Suppose given a fibration sequence S n +1 −→ Y −→ Z , and a fibration X −→ Y . Assuming (i) π ∗ ( Y ) −→ π ∗ ( Z ) is surjective, (ii) π ∗ ( X ) −→ π ∗ ( Y ) is surjective,with kernel concentrated in odd degrees, and (iii) π ∗ ( X ) is finite dimensional and X is apure space, we have HH ∗ ( X | Z ) ∼ = HH ∗ ( X | Y )[[ ζ ]] , where ζ is of degree n . This will be proved in Subsection 11.F below.11.B.
Upgrading bimodule maps.
Using the notation from Proposition 11.5, we supposegiven a fibration S n +1 −→ Y −→ Z , and a map X −→ Y . We write A = C ∗ ( X ), B = C ∗ ( Y )and C = C ∗ ( Z ).It is shown in Section 10 (see Lemma 10.4 and Proposition 10.5 ) that there is a cofibresequence of A ⊗ C A -modules A ⊗ C A → A ⊗ B A ϕ −→ Σ n A ⊗ B A. To obtain an element of Z D ( R ) we proceed as follows. For each A -module M , we apply − ⊗ A M to obtain the cofibre sequence A ⊗ C M → A ⊗ B M ϕ ⊗ A M −−−−→ Σ n A ⊗ B M. To establish the zci condition, we must check it is natural for maps of A -modules. roposition 11.6. There is a morphism ζ : A → Σ n A of A ⊗ C A -modules (i.e., an element ζ ∈ HH n ( A | C ) ) such that ζ ⊗ B A ≃ ϕ. We will prove the proposition in Subsection 11.E below. For the present we just observethat it has the desired consequence.
Corollary 11.7.
There is a natural transformation z of the identity functor on A -modulessuch that for every A -module M z ( A ⊗ B M ) ≃ ϕ ⊗ A M. Proof :
The natural transformation z that ζ induces on A -modules is given by z ( M ) = ζ ⊗ A M . We easily verify this has the required property: z ( A ⊗ B M ) = ζ ⊗ A ( A ⊗ B M )= ζ ⊗ B M = ( ζ ⊗ B A ) ⊗ A M = ϕ ⊗ B M. (cid:3) This completes the proof of Theorem 11.1.11.C.
Models for spaces.
Using the notation of Proposition 11.5, we work with a fibrationsequence S n +1 → Y → Z and a fibration X −→ Y , and we let A , B and C be minimalSullivan models for X , Y and Z respectively. More explicitly, we take models as follows.(1) C = (Λ W, d ) with W finite dimensional.(2) B = (Λ( W ⊕ Q x n +1 ) , d ), containing C as a sub-algebra. We denote W ⊕ Q x by V .(3) A = (Λ( V ⊕ U ) , d ), containing B as a sub-algebra, with U concentrated in odd degreesand finite dimensional. We also assume that d ( U ) ⊂ Λ W , this is possible because X has a pure Sullivan model.Let X eY = X × Y X and let X eZ = X × Z X . The cochain algebras A eB = A ⊗ B A and A eC = A ⊗ C A are minimal Sullivan models for X eY and X eZ . We can write these cochainalgebras explicitly as well: • A eB = (Λ( V ⊕ U l ⊕ U r ) , d ) where U l = { u l | u ∈ U } and U r = { u r | u ∈ U } . Thedifferential d is the obvious one satisfying d ( u l ) = d ( u r ) = d ( u ) ∈ Λ W and so u l − u r is always a cocycle. • A eC = (Λ( W ⊕ Q { x l , x r } ⊕ U l ⊕ U r ) , d ). Remark 11.8.
There are three important morphisms for X eY :(1) l : X eY → X which is mapping to the left component,(2) r : X eY → X which is mapping to the right component and(3) ∆ : X → X eY which is the diagonal.The algebraic counterparts of these maps are l : A → A eB , r : A → A eB and ∆ : A eB → A .The morphism l is the map to the left component of A ⊗ B A = A eB , it is defined by l ( u ) = u l for u ∈ U and l ( v ) = v for v ∈ V . The description of r : A → A eB is precisely similar.The diagonal map ∆ : A eB → A is defined by ∆( u l ) = ∆( u r ) = u and ∆ is the identity on x and W . There are similar maps for X eZ and A eC . Note that A is a A eC -module via themorphism ∆ : A eC −→ A .11.D. Some useful fibrations.
The following fibrations will be central to the proof.
Lemma 11.9.
There are the following fibration sequences, (1) X eY ∆ → X eZ → S ( Q x ) . (2) X ∆ → X eY → S ( U ) . (3) X ∆ → X eZ → S ( U ⊕ Q x ) .where the bases are products of odd spheres with the indicated homotopy groups. Proof:
We reformulate the lemma algebraically: there are the following cofibration sequencesof Sullivan algebras:(1) L = (Λ Q x ′ , → A eC ∆ −→ A eB , where x ′ x l − x r .(2) L = (Λ U ′ , → A eB ∆ −→ A , where U ′ ∼ = U and u ′ u l − u r .(3) L ⊗ Q L = (Λ( Q x ′ ⊕ U ′ ) , → A eC ∆ −→ A .Since the composite (Λ Q x ′ , → A eC ∆ −→ A eB is the trivial morphism, there is a natural mor-phism ǫ : A eC ⊗ L Q → A eB . It is easy to see that ǫ is an isomorphism on homotopy groups.The proof for the two other cofibration sequences is similar. (cid:3) We shall make two uses of these fibrations. The first use is to build relative cofibrantmodels for our cochain algebras. An A eC -cofibrant model for A eB is˜ A eB = (Λ( W ⊕ Q { x l , x r , z n } ⊕ U l ⊕ U r ) , d ) where dz = x l − x r .It is easy to see that the obvious morphism ˜ A eB → A eB is indeed a weak equivalence. Similarlya cofibrant model for A over ˜ A eB (and therefore also over A eC ) is given by the formula˜ A = (Λ( W ⊕ Q { x l , x r , z n } ⊕ U l ⊕ U r ⊕ ˜ U ) , d ) where ˜ U = Σ U and d (˜ u ) = u l − u r .The second use of Lemma 11.9 is in defining a strange and useful space T . Lemma 11.10.
Let T be the homotopy fibre of the map X eZ → S ( U ′ ) . (1) There is a fibration sequence X → T → S ( Q x ) . (2) The following is a homotopy pullback square X ∆ / / (cid:15) (cid:15) X eY (cid:15) (cid:15) T / / X eZ . Proof: An A eC -cofibrant cochain model for T is given by the formula F = (Λ( W ⊕ Q { x l , x r } ⊕ U l ⊕ U r ⊕ ˜ U ) , d ) where ˜ U = Σ U and d (˜ u ) = u l − u r .Note that u l − u r is a cocycle because d ( U ) ⊂ Λ W . From this model the fibration sequence X → T → S ( Q x ) is evident. et X ′ be the homotopy pullback of the diagram: X eY (cid:15) (cid:15) T / / X eZ . A cochain algebra model for X ′ is F ⊗ A eC ˜ A eB = (Λ( W ⊕ Q { x l , x r } ⊕ U l ⊕ U r ⊕ ˜ U ⊕ Q z n ) , d )which is clearly isomorphic to ˜ A . Moreover, the morphism ˜ A eB → F ⊗ A eC ˜ A eB ∼ = ˜ A is indeedthe diagonal ∆. (cid:3) Lifting the map of bimodules.
We can now prove Proposition 11.6. Recall thecochain model F for T given in the proof of Lemma 11.10. The fibration X → T → S ( Q x )induces an exact sequence of F -modules (and therefore also of A eC -modules): F → ˜ A ζ −→ Σ n ˜ A. We will require an explicit description of ζ . It is defined by • ζ ( f z q ) = f z q − and • ζ ( f ) = 0 if f is not divisible by z .Similarly, the fibration sequence X eY → X eZ → S ( Q x ) gives rise to an exact sequence A eC → ˜ A eB ϕ −→ Σ n ˜ A eB of A eC -modules. An explicit description of ϕ is given by • ϕ ( f z q ) = f z q − and • ϕ ( f ) = 0 if f is not divisible by z .As a cofibrant replacement of B over B ⊗ C B , we take ˜ B = (Λ( W ⊕ Q { x l , x r , z n } ) , d )with dz = x l − x r . We can now prove the following proposition, which is an explicit cochainlevel version of Proposition 11.6. Proposition 11.11.
There is a natural equivalence ϕ ≃ ζ ⊗ ˜ B ˜ A Proof :
We shall define a cochain algebra model ˆ A for X , which is cofibrant over ˜ B . Letˆ A = (Λ( W ⊕ Q { x l , x r , z n } ⊕ U l ) , d ) where dz = x l − x r . The morphism ζ is equivalent tothe obvious morphism ˆ ζ : ˆ A → Σ n ˆ A . It is now easy to see there is an equality of morphismsof A eC -modules: ϕ = ˆ ζ ⊗ ˜ B ˆ A (cid:3) Remark 11.12.
Proposition 10.5 shows that the fibration S n +1 → Y → Z yields an exactsequence: B ⊗ C B → ˜ B ψ −→ Σ n ˜ B. It is easy to see that ϕ = A ⊗ B ψ ⊗ B A (note that ψ is a morphism of B ⊗ C B -modules,which justifies tensoring over B on the left and right). Proof of Proposition 11.5.
Using our cofibrant models, we have explicit complexesfor calculating Hochschild cohomology: HH ∗ ( A | C ) = H ∗ (End A eC ( ˜ A )) and HH ∗ ( A | B ) = H ∗ (End ˜ A eB ( ˜ A )) . In these terms, we may state Proposition 11.5 more explicitly as follows.
Proposition 11.13. H ∗ (End A eC ( ˜ A )) = H ∗ (End ˜ A eB ( ˜ A ))[[ ζ ]] Proof:
Let R = H ∗ (End A eC ( ˜ A )) and let Q = H ∗ (End ˜ A eB ( ˜ A )). Note that both R and Q aregraded-commutative, because Hochschild cohomology is always graded-commutative. Toprove the proposition we need several ingredients. The first ingredient is a short exactsequence Σ n R ζ −→ R → Q of R -modules.Consider the morphism F p −→ ˜ A of Lemma 11.10. Applying the functor Hom A eC ( − , ˜ A ) yieldsa morphism End A eC ( ˜ A ) p ∗ −→ Hom A eC ( F, ˜ A ). Since ˜ A is a ˜ A eB -module there is an adjunction:Hom A eC ( F, ˜ A ) ∼ = Hom ˜ A eB ( ˜ A eB ⊗ A eC F, ˜ A ) ∼ = End ˜ A eB ( ˜ A )(the isomorphism ˜ A eB ⊗ A eC F ∼ = ˜ A is Part 2 of Lemma 11.10). Thus p ∗ is a map End A eC ( ˜ A ) → End ˜ A eB ( ˜ A ). On the other hand, we have the natural multiplicative change of rings map ι : End ˜ A eB ( ˜ A ) → End A eC ( ˜ A ). Using the explicit construction of internal Hom of DG-modulesover a CDGA, it is straightforward to verify that p ∗ is left inverse to ι .Next, consider the short exact sequence F → ˜ A ζ −→ Σ n ˜ A . Applying Hom A eC ( − , ˜ A ) to thissequence yields a distinguished triangle of left End A eC ( ˜ A )-modulesΣ n End A eC ( ˜ A ) ζ ∗ −→ End A eC ( ˜ A ) p ∗ −→ End ˜ A eB ( ˜ A ) . The morphism ζ ∗ : Σ n End A eC ( ˜ A ) → End A eC ( ˜ A ) is just composition with ζ , i.e., right mul-tiplication by ζ ∈ End A eC ( ˜ A ). This distinguished triangle yields a long exact sequence ofhomology groups. Since H ∗ ( p ∗ ) is an epimorphism, we have a short exact sequence of gradedleft R -modules: Σ n R ζ −→ R H ∗ ( p ∗ ) −−−−→ Q. Note that there are two multiplicative structure on Q : the usual one and the one comingfrom Q being a quotient of the graded ring R by the ideal ( ζ ). These structures mustcoincide, because H ∗ ( ι ) is multiplicative and has a left inverse.The second ingredient is that the homotopy inverse limit of the tower T := h R z ←− R z ←− · · · i is zero. Because A is zero in negative codegrees, the homotopy colimit ˜ A ∞ of the telescope˜ A ζ −→ Σ n ˜ A ζ −→ · · · is zero. Applying Hom A eC ( − , ˜ A ) to this telescope gives a towerEnd A eC ( ˜ A ) ζ ∗ ←− Σ n End A eC ( ˜ A ) ζ ∗ ←− · · · ts homotopy inverse limit, Hom A eC ( ˜ A ∞ , ˜ A ) is therefore also zero. By the Milnor exactsequence, lim ← T = 0 and R lim ← T = 0.Next, consider the two towers:(1) U = h R = ←− R = ←− · · · i , and(2) V = [ Q = R/ ( ζ ) ← R/ ( ζ ) ← R/ ( ζ ) ← · · · ].There is a short exact sequence of towers 0 → T → U → V →
0, given by: R ζ (cid:15) (cid:15) R ζ o o ζ (cid:15) (cid:15) R ζ o o ζ (cid:15) (cid:15) · · · ζ o o R (cid:15) (cid:15) R (cid:15) (cid:15) = o o R (cid:15) (cid:15) = o o · · · = o o Q R/ ( ζ ) o o R/ ( ζ ) o o · · · o o The six term exact sequence:0 → lim ← T → lim ← U → lim ← V → R lim ← T → R lim ← U → R lim ← V → R (which is isomorphic to lim ← U ) is isomorphic to lim ← V .To complete the proof we need to show that R/ ( ζ n ) is isomorphic to the truncated poly-nomial ring Q [ ζ ] / ( ζ n ). Observe that ( ζ n ) / ( ζ n +1 ) is isomorphic to Q as an R -module, andthat there is a subalgebra Q [ ζ ] ⊆ R . These facts yield a morphism of short exact sequences: Q ∼ = (cid:15) (cid:15) / / Q [ ζ ] / ( ζ n +1 ) / / (cid:15) (cid:15) Q [ ζ ] / ( ζ n ) (cid:15) (cid:15) ( ζ n ) / ( ζ n +1 ) / / R/ ( ζ n +1 ) / / R/ ( ζ n )Since R/ ( ζ ) ∼ = Q we get an inductive argument showing that R/ ( ζ n ) ∼ = Q [ ζ ] / ( ζ n ) as rings.Therefore R ∼ = lim ← Q [ ζ ] / ( ζ n ) = Q [[ ζ ]] , as required. (cid:3) Polynomial growth implies spherical extension.
The purpose of the present section is to complete the loop of implications and prove thefollowing theorem.
Theorem 12.1. If X is a gci space, it is also sci. This states that a finiteness condition (Noetherian cohomology and finite homotopy) im-plies that a space has a particular form (fibration F −→ X −→ KV , where π ∗ ( F ) is in odddegrees) and is therefore perhaps the most interesting step. Strategy.
Assume X is gci. By the Milnor-Moore theorem, π ∗ (Ω X ) is finite dimen-sional, and in particular its even part is finite dimensional.We argue by induction on dim Q ( π ∗ ( X )). The result is trivial if π ∗ ( X ) = 0, and theinductive step will be to remove the top homotopy group and retain the Noetherian condition.Suppose then that the top non-zero homotopy is in degree s and that 0 = x ∈ π ∨ s ( X ).If s = 2 n − S n − −→ X −→ X ′ , with dim Q ( π ∗ ( X ′ )) = dim Q ( π ∗ ( X )) −
1. In the rational setting, the fibration is principal, sothere is also a fibration X −→ X ′ −→ K ( Q , n − , and we may use the Serre spectral sequence to deduce that H ∗ ( X ′ ) is Noetherian. Thus X ′ is gci, and by induction we conclude it is also sci. The fibration displays X as being sci asrequired.If s = 2 n is even, killing homotopy groups gives a fibration, K ( Q , n ) −→ X −→ Y , butthis is not of use to us. We will argue, heavily using the fact that H ∗ ( X ) is Noetherian, thatin fact the element x is in the image of the dual Hurewicz map. Accordingly there is anotherfibration X ′ −→ X −→ K ( Q , n )where dim Q ( π ∗ ( X ′ )) = dim Q ( π ∗ ( X )) −
1. Applying the Serre spectral sequence to the fibra-tion S n − −→ X ′ −→ X, we see that H ∗ ( X ′ ) is Noetherian. By induction we conclude X ′ is sci, and from Corollary8.3 it follows that X is sci.12.B. The dual Hurewicz map.
In rational homotopy it is natural to dualize the Hurewiczmap h : π n ( X ) −→ H n ( X )and concentrate on the dual Hurewicz map h ∨ : H ∗ ( X ) = H ∗ ( X ) ∨ → π ∗ ( X ) ∨ . In fact, the dual Hurewicz map h ∨ must be zero on decomposable elements of H ∗ X , and soit yields a map from the indecomposable quotient of H ∗ ( X ) to π ∗ ( X ) ∨ .If (Λ V, d ) is a minimal Sullivan model for X then this dual Hurewicz map is the linearmap h ∨ : H ∗ (Λ V, d ) → V that comes from dividing (Λ V, d ) by the sub-cochain complex (Λ ≥ V, d ). An element x ∈ V is in the image of h ∨ if and only if there is a g ∈ Λ ≥ V such that d ( x + g ) = 0.12.C. The dual Hurewicz map and the Noetherian condition.
It is immediate fromTheorem 8.1 that if X is sci then h ∨ is an epimorphism in even degrees. The critical stepin showing that gci implies sci is to prove a special case of this surjectivity holds for gcispaces. We are grateful to S.Iyengar for pointing out that a corresponding result with a verydifferent proof appears as a crucial lemma in [11]. roposition 12.2. Suppose X is a rational space with finite dimensional homotopy and thatthe top degree in which homotopy is nonzero is n . If H ∗ ( X ) is Noetherian then the dualHurewicz map h ∨ : H n ( X ) −→ π ∨ n ( X ) is surjective. Proof:
By killing homotopy groups, there is a fibration sequence Y ← X Φ ← K ( Q , n )so that Φ a monomorphism in homotopy. We suppose S = (Λ W, d ) is a minimal Sullivanmodel for Y , and R = (Λ V, d ) is a minimal Sullivan model for X , where V = W ⊕ Q x . Wecan take Q = (Λ x,
0) as a Sullivan model for K ( Q , n ) with Φ : R → Q being the obviousmap, so that S → R Φ → Q ≃ R ⊗ S Q . provides an algebraic model of the fibration.To conclude, we apply the following lemma. Lemma 12.3. If H ∗ (Φ) is non-trivial then x is in the image of the dual Hurewicz map. Proof:
Suppose x n ∈ im( H ∗ ( X ) → H ∗ ( K ( Q , n )) = Q [ x ]) . This implies there is a cocycle in R of the form: x n + f x n − + f x n − + · · · + f n with f i ∈ Λ W . The differential of this cocycle is( ndx + df ) x n − + [ terms of lower degree in x ] = 0 , which implies ndx + df = 0. Hence there is an element g ∈ R such that x + g is a cocyclein R . The element x + g cannot be a coboundary because R is minimal, and hence x is inthe image of the dual Hurewicz map. (cid:3) If H ∗ ( X ) is Noetherian, then by [13, 9.3], the stable Koszul complex can be used toconstruct the Q -cellularization and there is a local cohomology spectral sequence H − pI ( H ∗ ( Q )) q ⇒ H p + q (Cell R Q ( Q )) , where Q is considered as an R -module via Φ.We now suppose x is not in the image of the dual Hurewicz map and deduce two con-tradictory statements about Cell Q Q ( Q ). First, Lemma 12.3 implies that H ∗ (Φ) is trivial andhence the spectral sequence collapses at the E -page to showCell R Q ( Q ) ≃ Q. In particular Cell R Q ( Q ) has cohomology only in codegrees ≥ x is not in the image of the dual Hurewicz map, we willsee that the cohomology of Cell R Q ( Q ) must be quite different. Lemma 12.4. If K ( Q , n ) −→ X −→ Y is a fibration sequence killing the top homotopygroup of X then C ∗ (Ω Y ) is Q -cellular as a module over C ∗ ( K ( Q , n )) . roof: Since the homotopy groups of Ω Y are concentrated in degrees less than 2 n − Y is a product of Eilenberg-MacLane spaces, the connecting map Ω Y −→ K ( Q , n ) is null.It follows that the induced map in cohomology Q [ x ] = H ∗ ( K ( Q , n )) −→ H ∗ (Ω Y ) factorsthrough Q , and hence that C ∗ (Ω Y ) is Q -cellular as required. (cid:3) The third author has identified two elementary but very useful base change results forcellularization.
Lemma 12.5. (Independence of base [28, 3.1] ) Suppose R −→ S is a map of rings.(i) (Strong) If B is an S -module and S ⊗ R B is B -cellular over S , then for any S -module Y Cell RB Y ≃ Cell SB Y. (ii) (Weak) If A is an R -module and S ⊗ R A is A -cellular over R , then for any S -module Y Cell RA Y ≃ Cell SS ⊗ R A Y. (cid:3) It follows from Lemma 12.4 that Q ⊗ R Q is Q -cellular as an Q -module. We may thereforeapply the Strong Independence of Base property to the map R → Q with B = Q andconclude Cell R Q ( Q ) ≃ Cell Q Q ( Q ) . Since H ∗ ( Q ) = Q [ x ], we can easily compute H ∗ (Cell Q Q ( Q )) = Σ H I ( Q [ x ]) = Σ Q [ x ] ∨ using the spectral sequence above. In particular Cell Q Q Q has cohomology in negative code-grees. It is therefore not equivalent to Q , and the assumption that x is not in the image ofthe dual of the Hurewicz map leads to a contradiction.This completes the proof of the proposition. (cid:3) This completes the proof of Theorem 12.1. (cid:3)
Examples.
It is quite easy to construct examples in rational homotopy theory, so we can see thatvarious classes are distinct.13.A.
Homotopy invariant notions and cohomology rings.
We may impose a homo-topy invariant condition on a space X or a conventional condition on the cohomology ring H ∗ ( X ).In the regular case there is no distinction by Theorem 5.4, since rational graded connectedcommutative rings are regular if and only if they are polynomial on even degree generators.In the ci case the homotopy invariant notion is strictly weaker than the notion for coho-mology rings. Indeed, we show in Proposition 13.1 that if H ∗ ( X ) is ci then X is sci. Onthe other hand, Example A.6 gives a pure sci space whose cohomology ring is not evenGorenstein. roposition 13.1. If H ∗ ( X ) is a complete intersection, then X is formal, and there is afibration S m × · · · × S m c −→ X −→ KV with m , m , . . . , m c odd. In particular, X is also sci. Remark 13.2.
By Theorem 8.1 a general sci space X has a similar fibration with fibrean arbitrary space with finite dimensional odd homotopy. Those with H ∗ ( X ) ci also havezero Postnikov invariants: there are vastly more sci spaces than those with ci cohomology.However Example A.6 shows that even when the fibre is a product of odd spheres, thecohomology ring need not be ci. Proof:
We may suppose H ∗ ( X ) = k [ x , . . . , f n ] / ( f , . . . , f c ) for suitable even degree gener-ators x , . . . , x n and regular sequence f , . . . , f c . Now let V be a graded vector space withbasis x ′ , . . . , x ′ n , where the degree of x ′ i is the same as that of x i , and let W be a gradedvector space with basis φ ′ , . . . , φ ′ c where the codegree of φ ′ i is one less than that of f i . We nowtake X ′ to have model M ( X ′ ) = (Λ( V ) ⋊ Λ( W ) , dφ ′ = f ′ , . . . , dφ ′ c = f ′ c ). Since f , . . . , f c isa regular sequence, H ∗ ( X ′ ) ∼ = H ∗ ( X ).Now construct a map g : M ( X ′ ) −→ C ∗ ( X )by taking g ( x ′ i ) to be a representative cycle for x i ∈ H ∗ ( X ). Since f i is trivial in H ∗ ( X ), wemay choose φ i ∈ C ∗ ( X ) so that dφ i = f i , and define g ( φ ′ i ) = φ i . The resulting map g is acohomology isomorphism and therefore an equivalence.The structure of M ( X ′ ) gives a fibration as claimed. (cid:3) Similarly, in the Gorenstein case the homotopy invariant notion is strictly weaker. On theone hand, Corollary A.3 shows that if H ∗ ( X ) is Gorenstein, then X is h-Gorenstein. If X ish-Gorenstein and H ∗ ( X ) is Cohen-Macaulay then H ∗ ( X ) is Gorenstein. However ExampleA.6 gives an h-Gorenstein space whose cohomology ring is not Cohen-Macaulay.13.B. Separating the hierarchy.
Since h-regular spaces are of the form KV , it is easy tosee that there are sci spaces that are not regular. To give an example of an h-Gorensteinspace that is not gci, we may use connected sums of manifolds, as in Example 13.3. Finally,there are many spaces with Noetherian cohomology that are not h-Gorenstein: two easysources of examples are either finite dimensional spaces whose cohomology ring does notsatisfy Poincar´e duality, or Cohen-Macaulay rings which are not Gorenstein. Example 13.3.
We provide a space X with H ∗ ( X ) so that X is h-Gorenstein but not gci.Almost any non-trivial connected sum of manifolds will do, but we give an explicit example.First, note that if M and N are manifolds, their connected sum M N = ( M ′ ∨ N ′ ) ∪ e n where M ′ is M with a small disc removed, and similarly for N ′ . By considering Lie modelsas in [18, 24.7], we obtain π ∗ (Ω( M N )) = ( π ∗ (Ω M ′ ) ∗ π ∗ (Ω N ′ )) / ( α + β ) , where ∗ is the coproduct of graded Lie algebras, and where α and β are the attaching mapsfor the top cells in M and N . erhaps the simplest thing to try is M = N = C P . Here we obtain π ∗ (Ω( C P C P )) = Lie( u , v ) / ([ u , u ] + [ v , v ]) , where Lie( V ) denotes the free graded Lie algebra on V . This shows π ∗ (Ω( C P C P )) isfinite, so C P C P is gci, which also follows from the fact that its cohomology ring H ∗ ( C P C P ) = Q [ a , b ] / ( a , b )is a complete intersection.However, once we take three copies, we obtain π ∗ (Ω( C P C P C P )) = Lie( u , v , w ) / ([ u , u ] + [ v , v ] + [ w , w ]) , which is not finite, so that C P C P C P is not gci, giving the required example. Example 13.4.
Here is an example of a space which is gci but not zci. Let X be the spacewith model R = (Λ( u , v , w ) , dw = uv ). There is a map of graded rings Ψ : Z D ( R ) → H ∗ (Ω X ) given by Ψ( ζ ) = ( ζ Q : Q → Σ n Q ) ∈ H n (Ω X )Clearly the image of Ψ is contained in the center of H ∗ (Ω X ). The graded ring H ∗ (Ω X ) isthe enveloping algebra of the graded Lie algebra L , where L is generated by three elements U , V and W and a single relation U V = W . The center of H ∗ (Ω X ) is therefore the set { Q W n | n ≥ } . Now suppose X was zci, then we would have had appropriate elements ζ , ..., ζ n ∈ Z D ( R ). Since the degree of ζ i is non zero, Ψ( ζ i ) is either zero or a i W n i for some n i > M be the R -module that is the cone of the map Q W −→ Σ Q . The module M/ζ / · · · /ζ n must be a small R -module. Now consider the C ∗ (Ω X )-module ¯ M = Hom R ( Q , M ). By theYoneda lemma, the map ¯ ζ i = Ext ∗ R ( Q , ζ i ) : H ∗ ( ¯ M ) → H ∗ + | ζ i | ( ¯ M )of H ∗ (Ω X )-modules is simply multiplication by Ψ( ζ i ). It is easy to see that H ∗ ( ¯ M ) = H ∗ (Ω X ) / ( W ) = Q [ U, V ]and therefore the induced map ¯ ζ i is zero on the homology of ¯ M . We conclude that ¯ M / ¯ ζ / · · · / ¯ ζ n has infinitely many nonzero homology groups. However,¯ M / ¯ ζ / · · · / ¯ ζ n ≃ Hom R ( Q, M/ζ / · · · /ζ n )and since M/ζ / · · · .../ζ n is small and C ∗ ( X ) is h-Gorenstein we see that the C ∗ (Ω X )-moduleHom R ( Q, M/ζ / · · · /ζ n ) has only finitely many nonzero homology groups, in contradiction.13.C. Miscellaneous examples.
The following example shows that the Noetherian condi-tion is essential in the definition of gci.
Example 13.5.
The space X with model (Λ( v a , x b +1 , w a +2 b ) , dw = vx ) is not sci. Fordefiniteness, we work with a = b = 1, so that we have the model (Λ( v , x , w ) , dw = vx ).Indeed, if X is sci it must be in a fibration S −→ X −→ KV where V = Q { v, w } . Butthen in homotopy we have a short exact sequence0 −→ π ∗ (Ω S ) −→ π ∗ (Ω X ) −→ π ∗ (Ω KV ) −→ f graded Lie algebras, which implies that π ∗ (Ω S ) is an ideal of π ∗ (Ω X ). On the otherhand, since dw = vx , the corresponding elements v , x and w in the Lie algebra π ∗ (Ω X )satisfy w = [ v, x ], and we have a contradiction since π ∗ (Ω S ) is generated by x .This is consistent with our general results since the cohomology ring is not Noetherian.Indeed, the cohomology ring H ∗ ( X ) is Q [ v ] in even degrees, whilst all products of the odddegree elements x, wx, w x, . . . are zero.14. The nci condition
In this section we conclude by giving a condition in the style of the zci condition whichcaptures polynomial growth in the non-Noetherian situation. From another point of view,since Example 13.5 shows that the Noetherian condition is essential, the nci condition intro-duced here is genuinely weaker than both the zci and the eci conditions. The letter ‘n’ innci stands for nilpotent.14.A.
The condition.
We suppose that R is a CDGA and continue to write D ( R ) for thederived category of dg- R -modules. Definition 14.1.
We say that R is nci of length ≤ n if there are(1) a sequence of triangulated subcategories D ( R ) = D ⊇ D ⊇ · · · ⊇ D n (not neces-sarily full) and(2) natural transformations ζ i : 1 D i → Σ | ζ i | D i for i = 0 , ..., n − D i is closed under coproducts.(2) Every ζ i is central among natural transformations of 1 D i .(3) For every X ∈ D i there exists an object X/ζ i ∈ D i +1 and a distinguished triangle X ζ i −→ Σ | ζ i | X → X/ζ i .(4) If 0 = X ∈ D i is in the thick subcategory generated by Q then X/ζ i is non-zero.(5) If 0 = X ∈ D ( R ) is in the thick subcategory generated by Q then X/ζ /ζ / · · · /ζ n − is non-zero and small as an object of D ( R ). Remark 14.2. (i) Note first that if R is zci or eci of codimension c , it is clearly nci oflength c .(ii) On the other hand, if R is nci and the natural transformations ζ , ..., ζ n − can beextended to central natural transformations of 1 D ( R ) , then R is almost zci (the only additionalcondition required is that the cohomology be Noetherian). Remark 14.11 below provides anexplicit example where it is not possible to extend one of these natural transformations.In the context of rational homotopy theory there is a straightforward characterization ofnci CDGAs. Theorem 14.3.
Let R = (Λ V, d ) be a minimal Sullivan algebra. Then R is nci if and onlyif V is finite dimensional. Lemma 14.5 below shows that if V is infinite dimensional, then R is not nci. The converseis proved in Subsection 14.G below. Remark 14.4.
The contrast with eci spaces, where Theorem 8.1 shows the structure ismuch more constrained (the differential on even generators is zero) is very striking. n the other hand, the only difference is the Noetherian condition. Indeed, if X is nci and H ∗ ( X ) is Noetherian, then Theorem 14.3 shows that X is gci and therefore (by Theorems12.1 and 11.1) also eci.14.B. Exponential Growth.
The hard work in this section is in dealing with the case ofa natural transformation of degree zero. This may be a useful counterpart to the approachto the Jacobson radical in [9].
Lemma 14.5.
Let X be a simply-connected finite CW-complex. If C ∗ ( X ) is nci, then H ∗ (Ω X ) has polynomial growth. Proof:
Since R = C ∗ ( X ) is nci there is a sequence of R -modules M , M , ..., M n such that(1) M = Q and M i ∈ D i .(2) There is a distinguished triangle Σ | ζ i | M i ζ i −→ M i → M i +1 .(3) M n is a small R -module.Suppose, by way of contradiction, that H ∗ (Ω X ) has exponential growth. As in Section 9,if all the degrees | ζ | , ..., | ζ n − | are non-zero, then we have a contradiction, since by Lemma9.3 H ∗ ( M n ⊗ R Q ) must have exponential growth because H ∗ ( M ⊗ R Q ) ∼ = H ∗ (Ω X ) hasexponential growth.We are left with showing that if | ζ i | = 0 for some i , the exponential growth still propagates.So suppose that | ζ i | = 0 for some i and M i ⊗ R Q has exponential growth. Consider thehomotopy colimit M ∞ i of the telescope M i ζ i −→ M i ζ i −→ · · · . Since this homotopy colimit ispart of a distinguished triangle: ⊕ ∞ n =0 M i − ζ i −−→ ⊕ ∞ n =0 M i → M ∞ i where the first map is in D i , M ∞ i is also in D i .Next we show that M ∞ i is finitely built from Q . By construction H ∗ ( M i ) is non-zero onlyin finitely many degrees. Each cohomology group H j ( M i ) is a finite dimensional vector spaceon which ζ i acts as a linear transformation. For large enough m , the kernel of ζ mi stabilizes.Denote this kernel by K ⊆ H j ( M i ), so that we can write H j ( M i ) ∼ = K ⊕ V, where ζ mi is zero on K and is an isomorphism on V . We see that H j ( M ∞ i ) ∼ = V is also finitedimensional, and M ∞ i itself is finitely built from Q .The morphism M ∞ i ζ i −→ M ∞ i is an equivalence, so Condition (4) of the definition of ncishows M ∞ i ≃
0. As M ∞ i ⊗ R Q ≃ H j ( M i ⊗ R Q ) is finite dimensional for each j , itfollows that H j ( ζ i ⊗ R Q ) is nilpotent on H j ( M i ⊗ R Q ) for each j , i.e., for each j there existsan n such that ζ ni ⊗ R Q induces the zero map on H j ( M i ⊗ R Q ).First observe that [ M i , M i ] R (i.e., the ring of degree 0 homotopy endomorphisms) is afinite dimensional algebra. Indeed, we start by observing that [ M , M ] ∗ R = [ Q , Q ] ∗ R is finitedimensional in each degree, and deduce the same for [ M , M ] ∗ R , [ M , M ] ∗ R , . . . , [ M i , M i ] ∗ R using the defining triangles.Now let z be the morphism M i ⊗ R Q ζ i ⊗ R Q −−−−→ M i ⊗ R Q , and note that the span of { z, z , z ... } inside [ M i ⊗ R Q , M i ⊗ R Q ] C ∗ (Ω X ) ∼ = [ M i , M i ] R is finite dimensional. We now resort to a classicaltrick to show that z is nilpotent on H ∗ ( M i ⊗ R Q ), i.e., that there is some N for which z N induces the zero map on H ∗ ( M i ⊗ R Q ).Suppose { z, z , ..., z n } is a basis for the span of { z, z , z , ... } . Let x ∈ H j ( M i ⊗ R Q )for some j and let { z ( x ) , z ( x ) , ..., z k ( x ) } be a basis for the span of { z ( x ) , z ( x ) , z ( x ) , ... } . learly k ≤ n . Suppose that z m ( x ) = 0 and z m +1 ( x ) = 0, so that m ≥ k . We will show that m = k so z k +1 ( x ) = 0.Write z m ( x ) = a z ( x ) + a z ( x ) + · · · + a j z j ( x ) , where a j = 0 and j ≤ k . First, we note that j = k . Indeed, applying z we find 0 = z m +1 ( x ) = P ji =1 a i z i +1 ( x ), so that if j < k we get a linear dependence. Thus, for some t ≤ k we have z m ( x ) = a t z t ( x ) + a t +1 z t +1 ( x ) + · · · + a k z k ( x ) , where a t = 0. It suffices to show t = k , so we suppose t < k and deduce a contradiction. If t < k we apply z to our equation and deduce z m +1 ( x ) = 0 = a t z t +1 ( x ) + a t +1 z t +2 ( x ) + · · · + a n − z k ( x ) + a k z k +1 ( x )which means that z k +1 ( x ) is in the span of { z t +1 ( x ) , ..., z k ( x ) } . Applying z repeatedly, wededuce z k + s is also in the span of { z t +1 ( x ) , ..., z k ( x ) } for all s ≥
1. Now either m = k andwe are done, or m = k + s for some s ≥ t = k and z k +1 ( x ) = 0 as required.Since H ∗ ( M i ⊗ R k ) has exponential growth, it follows that the kernel of the map H ∗ ( z ) : H ∗ ( M i ⊗ R k ) → H ∗ ( M i ⊗ R k ) has exponential growth. In particular, this implies that thecone of z = ζ i ⊗ Q : M i ⊗ R Q → M i ⊗ R Q has exponential growth. (cid:3) The first unravelling move.
We now describe three constructions we may use tobuild a new nci space X from a given nci space X ′ . In practice we are given X , and weunravel the process to obtain X ′ in such a way that if X ′ is nci, so too is X . Only the lastof these three was necessary in the eci case (it was the critical role of Proposition 12.2 toshow this). We work entirely algebraically, so that R is a model for X and R ′ is a model for X ′ .The first move is eliminating an even generator that is also a cocycle. Lemma 14.6.
Let R = (Λ V, d ) be a minimal Sullivan algebra and let x ∈ V be an elementof even degree such that dx = 0 . Then multiplication by x yields a natural transformationon D ( R ) : M x · −→ Σ −| x | M whose cone is R/ ( x ) ⊗ R M . If R/ ( x ) is nci of length ≤ n then R is nci of length ≤ n + 1 . We record the topological counterpart of this lemma.
Lemma 14.7.
Let x ∈ π ∨ n ( X ) be an element that is in the image of the dual Hurewicz map H ∗ ( X ) → π ∨∗ ( X ) . Then there is a fibration sequence: X ′ → X → K ( Q x ) such that x is in the image of π ∨ n ( K ( Q x )) . If X ′ is nci of length ≤ n then X is is nci oflength ≤ n + 1 . Proof of Lemma 14.6:
In this case there is a short exact sequence of dg- R -modules:Σ | x | R ֒ → R ։ R/ ( x ) . The leftmost map is given by multiplication by x : a a · x . We write V = W ⊕ Q x so that R/ ( x ) ∼ = (Λ W, d ), isomorphic to a sub-DGA of R . efine a natural transformation ζ : 1 D ( R ) → Σ −| x | D ( R ) as multiplication by x . If M is acofibrant dg- R -module, then applying − ⊗ R M to the short exact sequence above yields adistinguished triangle Σ | x | M ζ −→ M → R/ ( x ) ⊗ R M in D ( R ).Now suppose that R/ ( x ) is nci of length ≤ n , so that there are subcategories D R/ ( x ) = D ⊇ D ⊇ · · · ⊇ D n and appropriate natural transformations ζ , ..., ζ n − . The map p : R → R/ ( x ) of CDGAs induces an obvious functor p ∗ : D ( R/ ( x )) → D ( R ). Define D ′ i = p ∗ D i andsimilarly ζ ′ i = p ∗ ζ i . Set D ′− = D ( R ) and let ζ ′− be the natural transformation ζ definedabove.We may check that the subcategories D ′− , ..., D ′ n − and natural transformations ζ ′− , ..., ζ ′ n − satisfy the conditions for being nci. One need only note three things. First, if M is a smalldg- R/ ( x )-module then p ∗ M is a small dg- R -module, because R/ ( x ) is a small R -module.Second, if N is a dg- R -module finitely built by Q , then so is R/ ( x ) ⊗ R N . The reason isthat N is finitely built by Q over R if and only if H ∗ ( N ) is finite dimensional. Third, if N = 0 is finitely built by Q , then N/ζ is not zero, because | ζ | 6 = 0 and so ζ cannot inducean isomorphism of H ∗ ( N ). (cid:3) The second unravelling move.
The second move eliminates an even cocycle byadding an odd generator. The argument is essentially the same as the previous one (exceptthat the cocycle is not a generator), so we omit the proof.
Lemma 14.8.
Let R = (Λ V, d ) be a minimal Sullivan algebra and let f ∈ R be an evencocycle. Let R ′ = (Λ( V ⊕ Q y ) , d ′ ) , where d ′ y = f and d ′ v = dv for all v ∈ V . Thenmultiplication by f yields a natural transformation on D ( R ) : M f · −→ Σ −| f | M whose cone is R ′ ⊗ R M . If R ′ is nci of length ≤ n then R is is nci of length ≤ n + 1 . The topological counterpart of this lemma is again a fibration: X ′ → X → K ( Q f ) , only this time we just require that X → K ( Q f ) represent a nontrivial element in H ∗ ( X ).14.E. The third unravelling move.
Finally, the third move is passing to a subalgebra.This is the precise counterpart of the argument of Subsection 12.A, and the conclusion isanalogous to a spherical fibration S | x |− −→ X −→ X ′ . Nevertheless, we describe how this move fits within the nci context.
Lemma 14.9.
Let R = (Λ V, d ) be a minimal Sullivan algebra. Let x ∈ V be an element ofodd degree such that dv / ∈ x Λ V for all v ∈ V . Then there is a sub Sullivan algebra Q ⊂ R and a natural transformation ζ : 1 D ( R ) → Σ | x | +1 D ( R ) such that for any M ∈ D ( R ) there isa distinguished triangle in D ( R ) : M ζ −→ Σ | x | +1 M → Σ R ⊗ Q M . If Q is nci of length ≤ n then R is nci of length ≤ n + 1 . roof : Write V = W ⊕ Q x , and let Q be the Sullivan algebra (Λ W, d ). Clearly Q is aminimal Sullivan algebra and there is an obvious inclusion ι : Q ֒ → R of Sullivan algebras.We have seen earlier that there is a natural transformation ζ on D ( R ) such that M ζ −→ Σ | x | +1 M → Σ R ⊗ Q M is a distinguished triangle for all M ∈ D ( R ) .Suppose that Q is nci of length ≤ n , so there are subcategories D Q = D ⊇ D ⊇ · · · ⊇ D n and appropriate natural transformations ζ , ..., ζ n − . The map ι : Q → R of CDGAs inducesa functor ι ∗ : D ( Q ) → D ( R ), where ι ∗ ( N ) is the induced dg- R -module R ⊗ Q N . Define D ′ i = ι ∗ D i and similarly ζ ′ i = ι ∗ ζ i . Set D ′− = D ( R ) and let ζ ′− be the natural transformation ζ on D ( R ) defined above.We may check that the subcategories D ′− , ..., D ′ n − and natural transformations ζ ′− , ..., ζ ′ n − satisfy the conditions for being nci. One need only note three things. First, if N is a smallDG- Q -module then ι ∗ N is a small DG- R -module. Second, if M is a dg- R -module finitelybuilt by Q , then M is finitely built by Q also as a dg- Q -module. Third, | ζ | 6 = 0 and therefore M/ζ = 0 for every non-zero dg- R -module M that is finitely built by Q . (cid:3) Two examples.
We discuss some examples of minimal Sullivan algebras which arenci but not eci.
Example 14.10.
Consider the minimal Sullivan algebra R = (Λ( x , y , z , a ) , dx = dy = dz = 0 , da = xyz ) . First, we see that it is nci by showing explicitly how to unravel it. Indeed, we may applyLemma 14.8 to the cocycle xy to yield R ′ = (Λ( x, y, z, w, a ) , da = xyz, dw = xy ) . Now, d ( wz ) = da , so by a change of variables a ′ = a − wz we see that R ′ ∼ = (Λ( x, y, z, w, a ′ ) , da ′ = 0 , dw = xy )Now R ′ is eci and from its homotopy we see it is of codimension 4. It is therefore also nciof length 4, and therefore R is nci of length ≤ R is not eci. Most explicitly, one mayidentify the cohomology ring explicitly and observe that it is not Noetherian: it has a basisof monomials x i y j z k a l where i, j, k ∈ { , } omitting the monomials xyza l for l ≥ a l for l ≥
1. All elements (except those in codegree zero) are nilpotent.Note also that the dual Hurewicz map is not surjective in codegree 8, so that the methodof Subsection 12.A cannot be applied.
Remark 14.11.
Finally, we can see explicitly why the natural transformation in the ncidefinition cannot always be extended as we would require for the zci definition. In theprevious example, multiplication by the cocycle a ′ = a − wz defines a natural transformationon the D ( R ′ ) and therefore also on the image of D ( R ′ ) under restriction. This naturaltransformation cannot be extended to a central natural transformation of 1 D ( R ) , since we ould then have a commutative diagramΣ | xy | + | a ′ | R xy / / a ′ (cid:15) (cid:15) Σ | a ′ | R / / a ′ (cid:15) (cid:15) Σ | a ′ | R ′ a ′ (cid:15) (cid:15) Σ | xy | R xy / / (cid:15) (cid:15) R / / (cid:15) (cid:15) R a ′ (cid:15) (cid:15) Σ | xy | B xy / / B / / B ′ of distinguished triangles in D ( R ). The natural transformation a ′ must be the zero map on R , hence B ∼ = R ⊕ Σ | a | R . This shows that B ′ is isomorphic to R ′ ⊕ Σ | a | R ′ , and therefore a ′ acts as a regular element. However this contradicts the fact that ( a ′ ) = 0 from the longexact sequence of the triangle. Example 14.12.
Consider the minimal Sullivan algebra R = (Λ( x , y , z , y ′ , z ′ , a ) , dx = yz + y ′ z ′ , da = xyy ′ ) . First, we see that it is nci by showing explicitly how to unravel it. We apply Lemma 14.8 tothe cocycle yy ′ , yielding: R ′ = (Λ( x , y , z , y ′ , z ′ , a , w ) , dx = yz + y ′ z ′ , da = xyy ′ , dw = yy ′ )This yields d ( a + xw ) = ( dx ) w . We apply Lemma 14.8 twice more, for the cocycles wy and wy ′ , yielding: R ′′ = (Λ( x , y , z , y ′ , z ′ , a , w , t , t ′ ) , dx = yz + y ′ z ′ , da = xyy ′ , dw = yy ′ , dt = wy, dt ′ = wy ′ )Finally we have d ( a + xw − zt − zt ′ ) = 0. So, as in the previous example, we can do a changeof variables a ′ = a + xw − zt − zt ′ and see that R ′′ is eci of codimension 8, and hence nci oflength 8. It follows that R is nci of length ≤ R is not eci. Indeed, since the differentialis non-zero on the top even class a , the dual Hurewicz map is not surjective in codegree10, so that the method of Subsection 12.A cannot be applied. By Proposition 12.2, H ∗ ( R )is not Noetherian and so R is not eci.14.G. Proof of Theorem 14.3.
Let R = (Λ V, d ) be a simply-connected minimal Sullivanalgebra where V is finite dimensional. We will show that R is nci. The proof proceeds byinduction on the dimension of V even . The induction starts since if V even = 0, successiveapplications of the third unravelling move (Lemma 14.9) will reduce to a Sullivan algebrawith trivial differential. This is then the model of a product of odd spheres, which is obviouslynci.If V even = 0, then we apply Lemma 14.14 below, which says we may repeatedly apply thesecond unravelling move (i.e., add a finite number of odd generators) until we reach a CDGA R ′ with an even generator a ∈ V even such that da = 0. Now use the first unravelling move(Lemma 14.6) on a . The minimal Sullivan algebra R ′ / ( a ) is nci by the inductive hypothesis,so that R ′ is nci by Lemma 14.6, and R is nci by Lemma 14.8. (cid:3) The key ingredient is the following technical result. Note that for a minimal Sullivanalgebra R = (Λ V, d ) with V of finite type, the image of an element [ f ] ∈ H n ( R ) under the ual Hurewicz h ∨ map is non-zero if and only if there is an isomorphism of minimal Sullivanalgebras ρ : R ∼ = −→ (Λ V ′ , d ′ ) such that ρ ( f ) ∈ V ′ . Lemma 14.13.
Let R = (Λ V, d ) be a minimal (simply connected) Sullivan algebra such that V is finite dimensional and concentrated only in odd degrees. Let = [ f ] ∈ H i +1 ( R ) be anodd element of the cohomology of R . If h ∨ ([ f ]) = 0 , then there is an element = [ g ] ∈ H j ( R ) such that < j ≤ i . Proof:
Suppose the image of [ f ] under the dual Hurewicz map is zero. The proof goes byinduction on the dimension of V . Let x ∈ V be an element of minimal codegree, therefore dx = 0 and h ∨ ( x ) = 0. Eliminate x by adding an even generator S = (Λ( V ⊕ Q a ) , da = x )( d being defined on V as before). There is a distinguished triangle in D ( R ):Σ | x | S → R ϕ −→ S ψ −→ Σ | x | +1 S. Since S is equivalent to the minimal Sullivan algebra R/ ( x ), the induction assumptionholds for S . There are two possible cases.In the first case H ∗ ϕ [ f ] = 0. The image of ϕ [ f ] under the dual Hurewicz map is zerobecause π ∨∗ R → π ∨∗ S is an epimorphism with kernel Q x . By the induction assumption thereis a even degree element in the cohomology of S whose codegree is smaller than | f | . Let[ g ] be such an element of minimal degree. If [ g ] is in the image of ϕ , then we are done. Ifnot, then ψ [ g ] = 0. But ψ [ g ] ∈ H | g |−| x | +1 ( S ), which contradicts the minimality of | g | (notethat | g | − | x | + 1 >
0, since otherwise g is one degree below the minimal generators, whichis impossible).The remaining option is that ϕ [ f ] = 0. Hence f = dw for some w ∈ S . Write w as w = a n A n + a n − A n − + · · · + aA + A , where A i ∈ Λ V . Clearly f is homologous to f − dA , so without loss of generality we canassume that A = 0. Also note that all the A i are of even codegrees smaller than the codegreeof f (in fact | A i | ≤ | f | − | x | ). Calculating dw gives dw = a n d ( A n ) + n X i =1 a i − ( dA i − + ixA i ) . Since f ∈ Λ V we see that: f = xA dA i − = − ixA i for i ≥ dA n = 0Thus A n is a cocycle. If A n is not a coboundary in R , then we are done. Otherwise thereis a B n so that A n = dB n Now dA n − = − nxA n = − nxdB n = d ( nxB n ), whence A n − − nxB n is an even codegreecocycle. Again, if it is not a coboundary then we are done. Otherwise there is a B n − sothat A n − − nxB n = dB n − . ow dA n − = − ( n − xA n − = − ( n − x ( nxB n + dB n − )= − ( n − xdB n − = ( n − d ( xB n − ) . So we see that A n − − ( n − xB n − is an even codegree cocycle. We continue in this manneruntil either we get the desired even codegree element in the cohomology of R , or we end with A − xB = dB . But now f = xA = x (2 xB + dB ) = xdB = − d ( xB ), i.e. [ f ] = 0, which is a contradic-tion. (cid:3) Using the previous lemma we can now show that the second unravelling move may beused to make an even generator become a cycle.
Lemma 14.14.
Let R = (Λ V, d ) be a minimal (simply connected) Sullivan algebra suchthat V is finite dimensional. Then there is a sequence R = R → R → · · · → R n ofunravelling moves of the second type (i.e., moves to which Lemma 14.8 applies) such that R n is isomorphic to a minimal Sullivan algebra: R ′ n = (Λ V ′ , δ ) , where there is a minimaleven element a ∈ V ′ such that δa = 0 . Proof:
If there is a minimal even codegree element a ∈ V such that da = 0 we are done.Otherwise, choose some minimal even codegree element a ∈ V . Let V s ⊂ V be the subspaceof codegrees smaller than || a || . Minimality of the Sullivan algebra implies that da ∈ Λ V s .Apply the second unravelling move (Lemma 14.8) to the even elements of H ∗ ( R ) startingfrom the bottom and going up. Hence, let g ∈ R be a minimal even degree cocycle that isnot a coboundary. Applying Lemma 14.8 to g yields R = (Λ( V ⊕ Q x g ) , dx g = g )and a map R → R . We continue adding odd generators to R in this way until there is nomore homology in even codegrees less than or equal to || a || . Note that we need only killcocycles composed of odd generators only, even in codegree a .We end with a minimal Sullivan algebra R n = (Λ U, d ), where R n has no even degreecohomology in dimension || a || or lower. Let U s ⊂ U be the subspace of codegrees smaller than || a || . Clearly U s has only odd degree elements. Define T = (Λ U s , d ) to be the appropriatesub-CDGA of R n . Then T also has no cohomology in even codegrees || a || or lower. Because da is a cocyle in T , h ∨ ( da ) = 0 and H j ( T ) = 0 for 2 j ≤ || a || , it follows from Lemma 14.13that da is a coboundary in T . Thus da = du for some u ∈ Λ U s .Now a − u is a cocycle in R n . So a change of variables: a ′ = a − u yields a new minimalSullivan algebra R ′ n = (Λ U ′ , δ ) where a minimal codegree even generator a ′ is a cocycle. (cid:3) Appendix A. Gorenstein rings and spaces
This appendix discusses h-Gorenstein spaces, emphasizing the duality this gives. Thematerial comes from [17], [13] and [19], but the results have not been brought togetherexplicitly before. .A. Contents.
Recall that a commutative local Noetherian ring ( R, m , k ) is Cohen-Macaulay if its depth is equal to its dimension, and that it is
Gorenstein if R is of finite injective di-mension as a module. Furthermore, if R is Gorenstein of dimension r Ext ∗ R ( k, R ) = Ext rR ( k, R ) = k. Despite the definition, the important content of the Gorenstein condition is a duality property(this will be a special case of one in the CDGA case below).F´elix-Halperin-Thomas [17] have considered the analogue for spaces (which we call theh-Gorenstein condition) at length. We recall the definition below. It transpires that forspaces with finite dimensional cohomology (or finite category) X is h-Gorenstein if and onlyif H ∗ ( X ) is Gorenstein. Our contribution is to make explicit the duality statements inthe positive dimensional case following [13]. There is a structural duality statement at thelevel of derived categories even when H ∗ ( X ) is not Gorenstein. Thus if X is h-Gorenstein,there are consequences for the cohomology ring [19]: the cohomology ring H ∗ ( X ) is alwaysgenerically Gorenstein, if it is Cohen-Macaulay, it is automatically Gorenstein (and henceits Hilbert series satisfies a functional equation), and if it has Cohen-Macaulay defect 1, itsHilbert series satisfies a suitable pair of functional equations.A.B. The definition.
We recall the definition from [17] in the language of [13].
Definition A.1.
We say that a DGA A is h-Gorenstein of shift a if Hom A ( Q , A ) ≃ Σ a Q .We say that a space X is h-Gorenstein if C ∗ ( X ) is h-Gorenstein.We begin with the remark that the definition is an invariant of quasi-isomorphism, so thatany particular rational model of the space X may be used.A.C. Gorenstein duality.
The purpose of the Gorenstein condition is to capture a dualityproperty. This takes some work to extract. Since the argument is in [13] we will be brief.Since we are now mixing two sorts of duality, it is essential to emphasize that the suspensionΣ a is homological : it increases degrees by a (i.e., it reduces codegrees by a ). Proposition A.2. [13] If A is h-Gorenstein of shift a and H ∗ ( A ) is 1-connected and Noe-therian, then there is an equivalence Cell Q A ≃ Σ a A ∨ , and hence a spectral sequence H ∗ I ( H ∗ ( A )) ⇒ Σ a H ∗ ( A ) ∨ . Proof:
By definition, we have an equivalence Hom A ( Q , A ) ≃ Σ a Q of A -modules. To proceedwe need to apply Morita theory, so we consider the endomorphism ring E = Hom A ( Q , Q ).There is a natural right E -module structure on Hom A ( Q , M ) for any M , so the Gorensteincondition gives an E -action on Q . However, since A is 1-connected, there is a unique E -module structure on Q . Thus the Gorenstein condition gives an equivalenceHom A ( Q , A ) ≃ Hom A ( Q , Σ a A ∨ )of E -modules. Now apply ⊗ E Q . Since H ∗ ( A ) is Noetherian, Q is proxy-small in the sense of[13], and we may use Morita theory to deduceCell Q ( A ) ≃ Cell Q (Σ a A ∨ ) . ince A ∨ is Q -cellular, the Cell Q on the right may be omitted.This proves the first statement. For the second, by Corollary 4.5 the stable Koszul com-plex Γ A provides a model for Cell Q ( A ). Using the natural filtration, we obtain the spectralsequence. (cid:3) We remark that the spectral sequence collapses if H ∗ ( A ) is Cohen-Macaulay to show H rI ( H ∗ ( A )) ∼ = Σ a + r H ∗ ( A ) ∨ (where r is the Krull dimension of H ∗ ( A )). Thus H ∗ ( A ) is alsoGorenstein, and a is the classical a -invariant.The spectral sequence also collapses if H ∗ ( A ) is of Cohen-Macaulay defect 1, to give anexact sequence0 −→ H rI ( H ∗ ( A )) −→ Σ a + r H ∗ ( A ) ∨ −→ Σ H r − I ( H ∗ ( A )) −→ . This is discussed in more structural terms in [19, 5.4].A.D.
Functional equations.
It may be helpful to record the functional equations satisfiedby the Hilbert series p A ( t ) of an h-Gorenstein algebra A when H ∗ ( A ) is a Noetherian ringof Cohen-Macaulay defect 0 or 1. The equations are deduced from the existence of a localcohomology theorem in [19, Section 6]. Since H ∗ ( A ) is cograded, we take t to be of codegree1 (i.e., of degree − H ∗ ( A ) is Cohen-Macaulay we have p A (1 /t ) = ( − t ) r t a p A ( t ) . If H ∗ ( A ) is of Cohen-Macaulay defect 1, we have a pair of functional equations (introducedin the group theoretic context by Benson and Carlson [8]) p A (1 /t ) − ( − t ) r t a p A ( t ) = ( − r − (1 + t ) δ A ( t )and δ A (1 /t ) = ( − t ) r − t a δ A ( t ) , and in fact δ A ( t ) is the Hilbert series of H r − I ( H ∗ ( A )) ∨ .A.E. Examples.
First we show that there are many familiar examples of h-GorensteinDGAs.
Corollary A.3. [17, 3.2(ii)] If H ∗ ( A ) is Gorenstein then A is h-Gorenstein. Proof: If H ∗ ( A ) is Gorenstein then the E -term of the spectral sequenceExt ∗ , ∗ H ∗ ( A ) ( Q , H ∗ ( A )) ⇒ H ∗ (Hom A ( Q , A ))degenerates to an isomorphismExt rH ∗ ( A ) ( Q , H ∗ ( A )) = Σ r + a Q , where a is the conventional a -invariant. The spectral sequence therefore collapses to show A is h-Gorenstein with shift a . (cid:3) Corollary A.4. [17, 3.6] If H ∗ ( A ) is finite dimensional then A is h-Gorenstein if and onlyif H ∗ ( A ) is a Poincar´e duality algebra. roof : If H ∗ ( A ) is a Poincar´e duality algebra of formal dimension n then it is a zero di-mensional Gorenstein ring with a -invariant − n , so A is h-Gorenstein with shift − n by theprevious corollary.Conversely, if A is h-Gorenstein of shift a , we have a Gorenstein duality spectral sequence.Since H ∗ ( A ) is finite dimensional, it is all torsion. Accordingly, H ∗ I ( H ∗ ( A )) = H ∗ ( A ), andthe spectral sequence reads H ∗ ( A ) = Σ a H ∗ ( A ) ∨ and H ∗ ( A ) is a Poincar´e duality algebra of formal dimension − a . (cid:3) One may use these to construct other examples which are h-Gorenstein but not Gorenstein.
Proposition A.5. [17, 4.3]
Suppose we have a fibration F −→ E −→ B with F finite. If F and B are h-Gorenstein with shifts f and b then E is h-Gorenstein with shift e = f + b . (cid:3) This allows us to construct innumerable examples. For example any finite Postnikovsystem is h-Gorenstein [17, 3.4], so that in particular any sci space is h-Gorenstein. A simpleexample will illustrate the duality.
Example A.6.
We construct a rational space X in a fibration S × S −→ X −→ C P ∞ × C P ∞ , so that X is h-Gorenstein. We will calculate H ∗ ( X ) and observe that it is not Gorenstein.Let V be a graded vector space with two generators u, v in degree 2, and let W be agraded vector space with two generators in degree 4. The two 4-dimensional cohomologyclasses u , uv in H ∗ ( KV ) = Q [ u, v ] define a map KV −→ KW , and we let X be the fibre,so we have a fibration S × S −→ X −→ KV as required. By [13], this is h-Gorenstein with shift − − S × S and the shift (viz 2) of KV ).It is amusing to calculate the cohomology ring of X . It is Q [ u, v, p ] / ( u , uv, up, p ) where u, v and p have degrees 2 , , , , , , , , , , . . . (i.e., its Hilbert series is p X ( t ) = (1 + t ) / (1 − t ) + t , where t is of codegree 1).In calculating local cohomology it is useful to note that m = p ( v ). 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E-mail address : [email protected] School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK.
E-mail address : [email protected]@sheffield.ac.uk