A new lower bound for LScategory
A new lower bound for LScategory
Mathematics Subject Classification
Let X be a simply connected CWcomplex of finite type and K an arbitrary field. In this paper, we use the EilenbergMoore spectral sequence of C∗( (X ), K) to introduce a new homotopical invariant r(X, K). If X is a Gorenstein space with nonzero evaluation map, then r(X, K) turns out to interpolate depth(H∗( (X ), K)) and eK(X ). We also define for any minimal Sullivan algebra ( V , d) a new spectral sequence and make use of it to associate to any 1connected commutative differential graded algebra ( A, d) a similar invariant r( A, d). When ( V , d) is a minimal Sullivan model of X , this invariant fulfills the relation r(X, K) = r( V , d).

1 Introduction
Lusternik–Schnirelman category (LScategory for short), originally introduced in [
19
], is an integer that gives
a numerical measure of possible dynamics on a smooth closed manifold. In fact, it provides a lower bound on
the number of critical points admitted by any smooth function on any closed manifold. If X is a topological
space, cat(X ) is the least integer n such that X is covered by n + 1 open subsets Ui , 0 ≤ i ≤ n, each of
which is contractible in X . It is an invariant of homotopy type of the space. Though its definition seems easy,
its computation turns out to be a tough task. In [
23
], Toomer has introduced eK(X ) (K being an arbitrary field)
which proves to be a lower bound of cat(X ). Subsequently, Félix et al. proved the famous depth theorem which
states that depth(H∗( (X ), K)) ≤ cat(X ) [
6
]. Recall that when X is a simply connected CWcomplex such
that each Hi (X, K) is finite dimensional, then H∗( (X ), K) is a positively graded connected finite type and
cocommutative Hopf algebra. It is worthwhile reminding that the enveloping algebra U L of any homotopy
Lie algebra L associated to a finite type minimal Sullivan algebra ( V , d) is a further example of a positively
graded connected finite type and cocommutative Hopf algebra. If depth(U L) < ∞, then U L is left Noetherian
if and only if L is finite dimensional [8, Example 1.3 and Theorem C]. In this context, Bisiaux improved the
depth theorem by proving that depth(H∗( (X ), K)) ≤ eK(X ) if in addition evC∗(X,K) = 0 [
2
]. Recall from
[
6
] that for any graded Kalgebra G, depth(G) (possibly ∞) is the largest integer n such that ExtiG,∗(K, G) = 0,
∀i < n. Equivalently, depth(G) = inf{ p, Ext Gp,∗(K, G) = 0} with the convention that G has infinite depth
when Ext∗G,∗(K, G) ≡ 0. Here Ext p,∗ stands for the ( p, ∗) component of the graded functor Ext (see §20 in
[
9
] for instance).
The object of this paper is twofold. First, inspired by the definition of Toomer’s invariant in terms of the
Milnor–Moore spectral sequence, we use the Eilenberg–Moore spectral sequence
E2p,q = Ext Hp,∗q( (X),K)(K, H∗( (X ), K)) ⇒ E x tCp∗+(q (X),K)(K, C∗( (X ), K))
(1.0.1)
to define the integer
r(X, K) = sup{ p ∈ N  E ∞p,∗
= 0}
for any simply connected CWcomplex X of finite type and any field K. If there is no such integer, we put
r(X, K) = ∞. Here E x t is the differentialExt of Eilenberg–Moore (see Sect. 2 for more details). Our first
main theorem proves the following improvement of Bisiaux result:
Theorem 1.1 Let X be a simply connected CWcomplex such that each Hi (X, K) is finite dimensional over
an arbitrary field K. If X is Gorenstein and evC∗(X,K) is nonzero, then
depth(H∗( (X ), K)) ≤ r(X, K) ≤ eK(X ).
On the other hand, in order to make an algebraic study of r(X, K), our second goal is to associate to any
minimal Sullivan algebra ( V , d) the following spectral sequence:
E x t(p,qV,dk )(K, ( V , dk )) ⇒ E x t(p+Vq,d)(K, ( V , d))
(1.0.2)
which we call the Eilenberg–Moore spectral sequence of ( V , d). Throughout, the differential dk (cf. Sect. 3.1)
is the first nonzero homogeneous part of the differential d. Now, given a 1connected commutative differential
graded algebra ( A, d) and ( V , d) its minimal Sullivan model, we set
r( A, d) := sup{ p ∈ N  E ∞p,∗[( V , d)] = 0},
where E ∞∗,∗[( V , d)] is the ∞ term of (1.0.2). Similarly, if there is no such integer, we put r( A, d) = ∞.
In this perspective, our second main result reads the following:
Theorem 1.2 Let K be a field whose char(K) = 2. If X is a simply connected CWcomplex of finite type (or
else in the range of Anick) and ( V , d) its minimal Sullivan model, then
1. the two Eilenberg–Moore spectral sequence (1.0.1) and (1.0.2) are isomorphic.
2. r(X, K) = r( V , d) ≤ r( V , dk ) and the equality holds if dim(V ) < ∞.
2 Preliminaries
This section provides the tools and notions which are useful in the sequel. All graduations are written either as
superscripts (for cohomology) or as subscripts (for homology) with the convention V k = V−K . A commutative
differential graded algebra (resp., differential graded algebra, resp., differential graded Lie algebra) will be
abbreviated by cdga (resp., dga, resp., dgl). The suspension (resp., desuspension, resp., dual) of a graded
Kvector space V is defined by (s V ) p = V p+1 (resp., (s−1V ) p = V p−1, resp., V ∨ = HomK(V , K)).
2.1 A minimal Sullivan model
Let K be a field of characteristic p = 2. A Sullivan algebra is a free cdga ( V , d), where
V = Exterior(V odd) ⊗ Symmetric(V even),
generated by the graded Kvector space V = ⊕ii==0∞V i which has a well ordered basis {xα}α∈I such that
d xα ∈ V<α (V<α = span{vγ , γ < α}). Such algebra is said to be minimal if
deg(xα) < deg(xβ )
⇒ α < β, ∀α, β ∈ I.
If V 0 = K and V 1 = 0, this is equivalent to say that d(V ) ⊆ ⊕ii==2∞ i V , where i V designates the subspace
of V spanned by elements of wordlength i . A minimal Sullivan model for a cdga ( A, d) is a minimal
Sullivan algebra ( V , d) equipped with a quasiisomorphism ( V , d) → ( A, d). By [12, Theorem 7.1], if
H 0( A, d) = K, H 1( A, d) = 0 and dim(H i ( A, d)) < ∞ for all i ≥ 0, then ( A, d) always has a minimal
Sullivan model.
To define such a notion for spaces, we distinguish two cases. First, suppose that K has characteristic zero
and let X be a finite type simply connected CWcomplex. The minimal Sullivan model ( V , d) of X is by
definition that of the cdga AP L (X ) of polynomial differential forms on X with coefficients in K [
22
]. It is
unique (up to quasiisomorphism) and its generator satisfies the isomorphism V ∼= HomZ(π∗(X ), Q). Now,
assume that char(K) = p > 2 and let X be an r connected CWcomplex with dim(X ) < r p (some r ≥ 1).
For such a space, said to be in the range of Anick [
1
], the chain algebra (C∗( (X ), K) is quasiisomorphic to
the enveloping algebra U L of an appropriate finite type dgl L = L≥1. Denote by C∗(L) = ( (s L)∨, d) the
Cartan–Chevalley–Eilenberg complex of L [
12
]. This is a cdga which is related to C∗(X, K) by a sequence of
chain homotopies [
1
]. We still call its minimal Sullivan model, the minimal Sullivan model of X .
2.2 Eilenberg–Moore functors
Given ( A, d) a connected augmented Kdga on an arbitrary field K and denote by A its underlying graded
algebra. Following [
7
] (see also §6. in [
9
]), an ( A, d)module (P, d) is called free if it is free as an A module on
a basis of cocycles. It is said ( A, d)semifree if it is the union of ( A, d)submodules 0 = F−1 ⊂ F0 ⊂ F1 ⊂ · · ·
such that each Fi /Fi−1 is free. If in addition K ⊗A P is zero, (P, d) is called an ( A, d)semifree minimal
module.
Now, let (M, d) be an ( A, d)module. A morphism (P, d)→(M, d) of degree zero of ( A, d)modules
inducing an isomorphism in cohomology is called a quasiisomorphism of Amodules. We denote it by
(P, d) → (M, d) and call it an ( A, d)semifree resolution (resp., minimal semifree resolution) of (M, d)
if (P, d) is ( A, d)semifree (resp., ( A, d)semifree and minimal). If A1 = 0 and M = M≥r (some r ∈ Z)
such a resolution exists and may be chosen to be minimal [7, Lemma A.3].
We are now in a position to recall the definition of EilenbergMoore functors called also differential graded
Tor and Ext, since they are introduced in the context of differential graded homological algebra, where semifree
resolutions replace ordinary free resolutions.
Given another ( A, d)module (N , d) and let (P, d) → (M, d) an ( A, d)semifree resolution of
(M, d). The ( A, d)module Hom A(P, N ) = ⊕p≥0Hom Ap,∗(P, N ) where the pcomponent Hom Ap,∗(P, N ) =
⊕i≥0Hom A(Pi , N i+p), is provided with the differential defined by:
D( f ) = d ◦ f − (−1)p f ◦ d, ∀ f ∈ Hom Ap,∗(P, N ).
The Eilenberg–Moore functor E xt is defined as follows:
E xt(A,d)((M, d), (N , d)) = H ∗(Hom A(P, N ), D).
Analogously, if P ⊗A N = ⊕p≥0(P ⊗A N )p,∗ with (P ⊗A N )p,∗ = ⊕i≥0(Pi ⊗A Ni−p) is endowed with the
following differential
D( p ⊗ n) = d( p) ⊗ n + (−1)p p ⊗ d(n), ∀ p ⊗ n ∈ (P ⊗A N )p,∗,
we obtain the Eilenberg–Moore functor T or defined by
T or(A,d)((M, d), (N , d)) = H∗(P ⊗A N , D).
Remark 2.1 Now assume that char(K) = 2. Consider a dga ( A, d) over K endowed with an augmentation
ε : A → K and denote by A¯ = K er (ε) its ideal of augmentation. Recall that the reduced barconstruction
( A⊗ B( A), d) with coefficients in A, where B( A) = n≥0 T n(s A¯), is a semifree resolution of K as an ( A,
d)module (cf. §19 in [
9
] or §2.2 in [
18
]). On the other hand, H ( A, d) ⊗ B(H ( A, d)) being a free resolution
K of
of K as an H ( A, d)module [9, Proposition 20.11], there is an Eilenberg–Moore resolution ( P, d) →
K as an ( A, d)module. Therefore, there is an equivalence of ( A, d)modules: ( P, d) → ( A ⊗ B( A), d) [9,
Proposition 6.6]. Roughly speaking, one can suppose the E1 term of the spectral sequence induced by the
filtration (Fq = ⊕n≤q A ⊗ T n(s A¯))q≥0 on A ⊗ B( A) to be an H ( A, d)semifree resolution of K.
2.3 Evaluation map and Gorenstein spaces
Let ( A, d) be an augmented dga over an arbitrary field K and ρ : ( P, d) → (K, 0) any minimal semifree
resolution of K. A chain map:
(cf. [
23
] or [
14
], Prop. 1.6 (iii)) by
or ∞ if such p doesn’t exists.
2.4 The Toomer invariant
We assume that char(K) = 2 and consider the projection
pn :
V →
V / ≥n+1V
of a minimal Sullivan algebra ( V , d) onto the quotient dga obtained by factoring out by the differential graded
ideal generated by monomials of length at least n + 1. Thus, we define The Toomer invariant eK( V , d) to be
the smallest integer n (possibly ∞) such that pn induces an injection in cohomology. By [
4
], if char(K) = 0
and ( V , d) is a minimal Sullivan model of a simply connected finite type CWcomplex X , then eK(X ) =
eK( V , d), where eK(X ) denotes the classical Toomer invariant introduced in [
23
]. If X is taken in the range
of Anick, applying a similar argument yields in odd characteristic the coincidence eK(X ) = eK( V , d).
Now, consider an arbitrary field K and denote by (T (W ), d) the free model of X introduced in [
14
]. In a
similar way, Halperin and Lemaire defined the invariant eK(T (W ), d) with respect to the projection
and showed that eK(X ) = eK(T (W ), d).
Recall finally that an alternative version of the Toomer invariant is given in terms of the Milnor–Moore
spectral sequence
pn : T (W ) → T (W )/T ≥n+1W
Ext Hp,∗q( (X),K)(K, K) ⇒ H p+q (X, K)
eK(X ) = sup{ p ∈ N  E ∞p,∗ = 0}
Hom(A,d)(( P, d), ( A, d)) −→ ( A, d)
ev(A,d) : E x t(A,d)(K, ( A, d)) −→ H ∗( A, d).
is given by f → f (z), where z ∈ P is a cocycle representing 1K. Passing to the cohomology, we obtain the
evaluation map of ( A, d):
Note that the definition of ev(A,d) is independent on the choice of ( P, d) and z. Moreover, it is natural with
respect to ( A, d). As a particular case, evC∗(X,K) is called the evaluation map of X over K
On the other hand, the authors of [
7
] introduced the concept of a Gorenstein space over K. It is a space X
such that dim E x tC∗(X,K)(K, C ∗(X, K)) = 1. In addition, if dim H ∗(X, K) < ∞, then X satisfies the Poincaré
duality property over K and its fundamental class is closely related to the evaluation map (See, [
11,16,21
] for
more details).
(2.3.1)
(2.3.2)
(2.4.1)
Remark 2.2 In [
4
], it is shown that for any minimal Sullivan model ( V , d) of X , the Milnor–Moore spectral
sequence (2.4.1) and the following one:
H p,q ( V , d2) ⇒ H p+q ( V , d),
(2.4.2)
are isomorphic from their second terms. Here d2 designates the quadratic part of the differential d.
Therefore, whenever X satisfies the Poincaré duality property and denoting by ω its fundamental class,
eK(X ) = eK( V , d) = sup{ p/ω can be represented by a cocycle in
≥ p V }
[4, Lemma 10.1]. Similarly, when K is any field and (T (W ), d) is a minimal free model of X over K, by [2,
Lemma 2.1], we have
eK(T (W ), d) = sup{ p/ω can be represented bya cocycle in T ≥ p W }.
3 Main results
In this section, we first introduce the spectral sequence (1.0.2) and then use it, in conjunction with (1.0.1), to
give the proofs of our main results.
3.1 Eilenberg–Moore spectral sequence of a free cdga
Let ( V , d) be a free cdga over a field K whose char(K) = 2 and assume that d =
and k ≥ 2. The map d is an algebra derivation defined on V ; that is,
i≥k di , with di (V ) ⊆
i V
d(x y) = d(x )y + (−1)xx d(y),
∀x , y ∈ V .
So, by extension, we have d( i V ) ⊆ ≥i+k−1V , ∀i ≥ 1. Hence each di is also a derivation defined on V
and particularly, for any x ∈ V , dk2(x ) ∈ 2k−1V is by wordlength reason the (2k − 1)−th homogeneous part
of d2(x ). Whence dk2 = 0 and then ( V , dk ) is also a free cdga.
As char(K) may be nonzero, in the sequel, we will use the divided power algebra (s V ) (see for instance
[
12
]). If {vi }i∈I is a well ordered basis of V and V<i denotes the subspace generated by {v j , j < i }, the
differential D on the product algebra V ⊗ (s V ) restricts to dV on V and on s V it is given by:
D(svi ) = vi + φ, φ ∈
V<i ⊗
(s V<i ),
∀i ∈ I.
[21, Remark 1.2]. So, by extension, we have
D(γ p(sv)) = D(sv)γ p−1(sv); ∀ p ≥ 1,
∀sv ∈ (s V )even.
( V ⊗
(s V ), d) is a differential graded algebra and also an ( V , d)semifree module. Therefore, the
projection ( V ⊗ (s V ), D) → K is a semifree resolution of K called an acyclic closure of ( V , d) (cf.
section 2 in [
12
]). When K = Q, (s V ) is replaced by the free cdga (s V ).
Consider now on A = Hom( V,d)(( V ⊗ (s V ), D), ( V , d)) the filtration
F
p
= { f ∈ Hom V ( V ⊗
(s V ), V )  f ( (s V )) ⊆
≥ p V },
∀ p ≥ 0
and the differential defined by
(3.1.1)
(3.1.2)
(3.1.3)
D( f ) = d ◦ f + (−1) f +1 f ◦ D,
∀ f ∈ A.
Lemma 3.1 The filtration (3.1.3) verifies the following:
(i) (F p) p≥0 is decreasing,
(ii) F 0(A) = A,
(iii) D(F p(A)) ⊆ F p(A).
Proof Properties (i) and (ii) are immediate. Moreover, the property (iii) follows from the definition of D on
A, specially, the relation D(γ p(sv)) = (v + φ)γ p−1(sv) on (s V )even [cf. (3.1.1) and (3.1.2) above].
The general term of the spectral sequence induced by the filtration (3.1.3) is given by:
p
Er =
{ f ∈ F p, D( f ) ∈ F p+r }
{ f ∈ F p+1, D( f ) ∈ F p+r } + F p ∩ D(F p−r+1) .
Moreover, Ar,s = (Homr V ( V ⊗ (s V ), V ))r+s = { f ∈ A, f ( (s V )) ⊆
(Fr /Fr+1)r+s . A straightforward calculation permits to prove that
r V }r+s is isomorphic to
Ek
p,q ∼
=
Ker[A p,q →Dk A p+k−1,q−k+2]
Im[A p−k+1,q+k−2 Dk
→ A
p,q
]
,
where Dk stands for the differential of Hom( V,dk )(( V ⊗
isomorphism of graded modules Ek∗,∗ ∼= ⊕ p,q≥0E x t(p,qV,dk )(K, ( V , dk )). This yields the spectral sequence
(s V ), Dk ), ( V , dk )). Hence, we obtain the
which we call the Eilenberg–Moore spectral sequence of ( V , d).
Remark 3.2 1. When dim(V ) < ∞, the filtration (3.1.3) is clearly bounded in the sense of Theorem 2.6 in
[
20
]; that is, for each dimension n, there exists s = s(n) and t = t (n) such that
{0} = F s (An) ⊆ F s−1(An) ⊆ · · · ⊆ F t+1(An) ⊆ F t (An) = An.
So the spectral sequence (3.1.4) is convergent. Furthermore, if V is of finite type, the convergence is a
consequence of Theorem 1.2 (1).
2. Consider on ( V , d) the filtration defined by
(3.1.4)
(3.1.5)
(3.1.6)
An easy calculation shows that it induces the following spectral sequence:
F p =
≥ p V =
∞
i= p
i V .
H p,q ( V , dk ) ⇒ H p+q ( V , d)
whose convergence is guaranteed if ( V , d) is a minimal Sullivan algebra (cf. §9. in [
4
]). Moreover, the
chain map (2.3.1) is filtrationpreserving, so that, the evaluation map ev( V,d) : E x t( V,d)(K, ( V , d)) −→
H ∗( V , d) is a morphism of the spectral sequences (3.1.4) and (3.1.6). Notice that if k = 2, we find the
spectral sequence (2.4.2).
3.2 Proofs of the main results
Let us note at the outset that the proofs of our main results are consequences of the following Proposition:
Proposition 3.3 Let X be a simplyconnected CWcomplex of finite type (or else in the range of Anick) and
denote by ( V , d) its minimal Sullivan model. Then, the cohomological Eilenberg–Moore spectral sequence
of C∗( (X ), K) is convergent and it is isomorphic to the one of ( V , d).
Before giving the proof of this proposition, we recall beforehand the construction of the Eilenberg–Moore
spectral sequence associated with C∗( (X ), K) as well as that of the one introduced by Bisiaux in [
2
]. In fact,
we will show that the latter is isomorphic to each of the two spectral sequences of Eilenberg–Moore, namely
(1.0.1) and (1.0.2).
Notice first that for any field K and any simply connected CWcomplex X of finite type, the Adams–Hilton
model A = (T (W ), d) → C∗( (X ), K) is a finite type free model of C∗( (X ), K) [
14
]. Now, consider on
A ⊗ B( A), the filtration defined by
IF q =
A ⊗ T n(s A¯), ∀q ≥ 0
n≤q
and endow Hom A( A ⊗ B( A), A) with the following one:
IF
q = { f  f (IF k ) = 0, ∀k < q}, ∀q ≥ 0.
Clearly IF 0 = Hom A( A ⊗ B( A), A), (IF q )q≥0 is decreasing and it is stable with respect to the differential
of Hom A( A ⊗ B( A), A). So it induces a cohomological spectral sequence which is, by Remark 2.1, the
Eilenberg–Moore spectral sequence (1.0.1) (see for instance [
9
], §20(d)).
On the other hand, let (B, d) = (K ⊕ B≥2, d) be a dga quasiisomorphic to C ∗(X, K) (K being any field)
and denote by (T (Z ), d) → (B, d) its free minimal model [
14
]. An acyclic closure of (T (Z ), d) has the form
(T (Z ) ⊗ (s Z ⊕ K), D) [
2
]. Therefore, by taking on (HomT (Z)(T (Z ) ⊗ (s Z ⊕ K), T (Z )), D) the filtration
q
IF (T (Z),d) = { f  f (T (Z ) ⊗ (s Z ⊕ K)) ⊆ T ≥q Z }, ∀q ≥ 0,
we obtain the following convergent spectral sequence introduced by Bisiaux [
2
]:
E x t(pT,q(Z),d2)(K, (T (Z ), d2)) ⇒ E x t(pT+(qZ),d)(K, (T (Z ), d)).
Proof of Proposition 3.3 Let us denote by ( A) = (T (s−1 A∨), d) the dual of B( A). Thus ( A ⊗ B( A))∨ =
( A) ⊗ A∨ is a left ( A)module and then the filtration
F q = ⊕n≤q ( A) ⊗ An∨,
∀q ≥ 0
exhibits it as an ( A)semifree resolution of K [
7
]. Again, by Remark 2.1, we can assume that it is an
Eilenberg–Moore semifree resolution of K. Notice that ( A), denoted thereafter by (T (W ), d), is a
finitetype free model of C ∗(X, K) [14, Proposition 1.6].
The rest of the proof falls into two steps:
Step 1. (In this step, we assume K an arbitrary field). Using [7, Remark 1.3], we will replace C∗( (X ), K)
and C ∗(X, K), respectively, by A and ( A). Thus, referring to [7, Theorem 2.1], the isomorphism:
(3.2.1)
(3.2.2)
(3.2.3)
E x tC∗( (X),K)(K, C∗( (X ), K)) ∼= E x tC∗(X,K)(K, C ∗(X, K))
(3.2.4)
is deduced from the following isomorphisms of complexes:
∼ϕA
(Hom A( A ⊗ B( A), A), D) −→ (End A⊗B(A)( A ⊗ B( A)), [d, ])
−∼=→∨ (End (A)⊗A∨ ( ( A) ⊗ A∨), [d∨, ])
∼←ϕ−(A) (Hom (A)( ( A) ⊗ A∨, ( A)), D).
Here, ϕA( f ) = ( f ⊗ i dB(A)) ◦ (i dA ⊗ B(A)) and its inverse map is the projection ψA⊗B A(g) =
(I dA ⊗ εB(A)) ◦ g. B(A) and εB(A) are, respectively, the diagonal and the counity of B A (see for
instance [18, Proposition 1.5.14]).
We endow respectively Hom A( A ⊗ B( A), A) and Hom (A)( ( A) ⊗ A∨, ( A)) with the filtrations
IF q and IF q(T (W ),d) = { f  f ( ( A) ⊗ A∨) ⊆ ( A)≥q }. If f ∈ IF q , then
ϕA( f )( A ⊗ B( A)<q ) = 0 and ϕA( f )( A ⊗ B( A)≥q ) ⊆ A ⊗ B( A)≥q .
So
∨ ◦ ϕA( f )( ( A)<q ⊗ A∨) = 0 and ∨ ◦ϕA( f )( ( A)≥q ⊗ A∨) ⊆
( A)≥q ⊗ A∨.
Hence ∨ ◦ ϕA( f )( ( A) ⊗ A∨) ⊆ ( A)≥q ⊗ A∨.
Applying ϕ−1A, we conclude that ϕ−1A ◦ ∨ ◦ ϕA( f )( ( A)≥q ⊗ A∨) ⊆
( A)≥q . Therefore,
q q
ϕ−1A ◦ ∨ ◦ ϕA(IF ) ⊆ IF (T (W ),d)
and then the composition isomorphism is one of filtered complexes. Consequently, the spectral
sequence (1.0.1) is isomorphic to the one (3.2.3) and then it is convergent.
Step 2. (Here we assume K = Q and notice that the same proof remains valid in odd characteristic with X in
the range in Anick (cf. Sect. 2.1).
Let ( V , d) be a minimal Sullivan model of X and consider its minimal free model given by a
quasiisomorphism ϕ : (T (Z ), d) → ( V , d). Since (T (Z ), d) is also a free model of C ∗(X, Q),
the spectral sequences of the form (3.2.3) induced by both (T (W ), d) and (T (Z ), d) are isomorphic.
Now, by degree reason, ϕ preserves filtrations and then by [14, Proposition 3.6] it induces a
quasiisomorphism E2(ϕ) : (T (Z ), d2) −→ ( V , d2), where d2 stands for the quadratic part of d. Also, ϕ
induces on ( V ⊗ (s V ), D) the structure of a (T (Z ), d)module and then [9, Proposition 6.4.] the
following diagram
is completed by a quasiisomorphism
(T (Z ) ⊗ (s Z ⊕ Q), D) −→
: (T (Z ) ⊗ (s Z ⊕ Q), D) −→ ( V ⊗
(s V ), D)
( V ⊗
(s V ), D)
↓
Q
of (T (Z ), d)modules between acyclic closures of (T (Z ), d) and ( V , d).
We then provide HomT (Z)(T (Z ) ⊗ (s Z ⊕ Q), V ), D) with the filtration
IH
q = { f  f (T (Z ) ⊗ (s Z ⊕ Q)) ⊆ ( V )≥q }
(3.2.5)
and define the differential graded morphism of complexes
: HomT (Z)(T (Z ) ⊗ (s Z ⊕ Q), T (Z )) → HomT (Z)(T (Z ) ⊗ (s Z ⊕ Q),
V ))
(resp.,
: Hom V ( V ⊗
(s V ),
V )) → HomT (Z)(T (Z ) ⊗ (s Z ⊕ Q),
V ))
by putting ( f ) = ϕ ◦ f (resp., (g) = g ◦ ). These morphisms preserve filtrations (3.2.2)
and (3.2.5) (resp., (3.1.3) and (3.2.5)) and induce morphisms of spectral sequences. Finally, since
( V ⊗ (s V ), D) yields an acyclic closure ( V ⊗ (s V ), D2) of ( V , d2) ([10, Prop. 3.12] and
[21, Lemma 2.1]), the quasiisomorphism E2(ϕ) aforementioned implies that the later morphisms are
in fact isomorphisms of spectral sequences. Hence, the two spectral sequences (3.2.3) and (1.0.2) are
isomorphic.
Composing the obtained isomorphisms, we deduce that the two spectral sequences (1.0.1) and (1.0.2) are
isomorphic.
Proof of Theorem 1.1 The inequality depth(H∗( (X ), K)) ≤ r(X, K) comes immediately from the
convergence of the EilenbergMoore spectral sequence (1.0.1) established in the first step of the proof of
Proposition 3.3. Now, X being a Gorenstein space, we have dim E x t(T (W ),d)(K, (T (W ), d)) = 1. There exists then
an unique pair ( p, q) of integers and an unique [ f ] of bidegree ( p, q) [with respect to the filtration (3.2.2)]
generating E x t(T (W ),d)(K, (T (W ), d)). It follows from the isomorphism between (1.0.1) and (3.2.3) that
p = r(X, K). The hypothesis evK(T (W ), d) = 0 implies that ev([ f ]) = [ f (1)] = 0. But f (1) ∈ T ≥ p(W ),
so eK(T (W ), d) ≥ p by the characterization given in Remark 2.2. Consequently, r(X, K) ≤ eK(X ).
Proof of Theorem 1.2 It remains to prove the second assertion. By Proposition 3.3, we have r(X, K) =
r( V , d). Moreover, remark that the spectral sequence (1.0.2) relative to ( V , dk ) degenerates at its first
term. Hence, r( V , d) ≤ r( V , dk ) is then a consequence of the convergence of (1.0.2) (relative to ( V , d)).
Now suppose that dim V < ∞. Since d is decomposable, [21, Proposition 3.1] asserts that both ( V , dk ) and
( V , dk ) are Gorenstein algebras. Therefore, in (1.0.2), there exists a unique pair ( p, q) such that
E x t(∗,∗V,dk )(K, ( V , dk )) = E x t(p,qV,dk )(K, ( V , dk )) = E∞
p,q
= E x t(∗,∗V,d)(K, ( V , d))
and then the desired equality follows.
Remark 3.4 Following the same approach of [
7
], we state below for any minimal Sullivan algebra ( V , d)
some properties for r( V , d) similar to that listed in Remark 2.2.
V , d ) = sup{ p 
p .
can be represented by a cocycle in F }
We end this paper by asking the following question.
Question Recall first that a minimal Sullivan algebra ( V , d ) is said to be elliptic, if both V and H ( V , d )
are finite dimensional. The main result established by Lechuga and Murillo [
17
] states that if ( V , dk ) is also
elliptic, then e0( V , d ) = e0( V , dk,σ ), where dk,σ designates the pure differential associated to dk [
13
]. So
the following natural question arises:
Under what hypothesis, r(
V , d ) = r(
V , dσ )?
r(
By Theorem 1.2, since dσ,k = dk,σ , if the answer to this question is positive, we will have r(
V , dσ,k ) = r( V , dk,σ ) when dim(V ) < ∞.
V , d ) =
Acknowledgements I am indebted to J. C. Thomas for very useful conversations which enabled me to improve my results
significantly. I am also grateful to the reviewers whose comments, gave me the opportunity to improve the drafting of this work.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://
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