duality on the (co)chain type levels of...

Duality on the (co)chain type levels of

Katsuhiko KURIBAYASHI

Shinshu University

The University of Tokyo, EACAT4, December 6, 2011

§1 Overview of the levels of DG modules over a DG algebra

DGM−A : the category of differential graded right modules (DG

modules) over a DG algebra A / K a field.

Definitoin A DG module F : A-semifree if ∃ a filtration

F (0) ⊂ F (1) ⊂ · · · ⊂ F (k) ⊂ · · · ⊂∪k

F (k) = F

s.t. F (0) and F (k)/F (k − 1) are A-free on a basis of cycles.

For any M in DGM−A, ∃ ΓM ' //M : a semifree resolution of M.

§1 Overview of the levels of DG modules over a DG algebra

DGM−A : the category of differential graded right modules (DG

modules) over a DG algebra A / K a field.

Definitoin 1.1 A DG module F : A-semifree if ∃ a filtration

F (0) ⊂ F (1) ⊂ · · · ⊂ F (k) ⊂ · · · ⊂∪k

F (k) = F

s.t. F (0) and F (k)/F (k − 1) are A-free on a basis of cycles.

For any M in DGM−A, ∃ ΓM ' //M : a semifree resolution of M.

D(A) : the derived category of DG A-modules;

ObD(A) := Ob(DGM−A)

HomD(A)(X,Y ) := HomDGM−A(ΓX,ΓY )/chain homotopy '

D(A) : a triangulated cat. with the shift Σ; (ΣM)n = Mn+1.

The distinguished triangles comes from mapping cone construc-

tions in DGM−A,

//N //C(φ) //ΣM ; C(φ) = N ⊕ ΣM, dC(φ) =

(dN φ0 −dM

D(A) : the derived category of DG A-modules;

ObD(A) := Ob(DGM−A)

HomD(A)(X,Y ) := HomDGM−A(ΓX,ΓY )/chain homotopy '

D(A) : a triangulated cat. with the shift Σ; (ΣM)n = Mn+1.

The distinguished triangles comes from mapping cone construc-

tions in DGM−A,

//N //C(φ) //ΣM ; C(φ) = N ⊕ ΣM, dC(φ) =

(dN φ0 −dM

A : a DGA over a field KD(A) : the derived category of DGM’s over A

C ∈ Ob(D(A))

Definition (the level of M)

(Avramov, Buchweitz, Iyengar, Miller, 2006)

The 0th thickening thick0D(A)(C) := 0

thick1D(A)(C) : the smallest strict full subcategory which con-

tains C and is closed under taking finite coproducts, retracts

and all shifts.

Moreover for n > 1 define inductively the nth thickening

thicknD(A)(C)

by the smallest strict full subcategory of D(A) which is closed

under retracts and contains objects M admitting a distinguished

triangle M1 →M →M2 → ΣM1, where

M1 ∈ thickn−1D(A)(C) and M2 ∈ thick1

D(A)(C).

The C-level of M

levelCD(A)(M) := infn ∈ N ∪ 0 |M ∈ thicknD(A)(C).

Moreover for n > 1 define inductively the nth thickening

thicknD(A)(C)

by the smallest strict full subcategory of D(A) which is closed

under retracts and contains objects M admitting a distinguished

triangle M1 →M →M2 → ΣM1, where

M1 ∈ thickn−1D(A)(C) and M2 ∈ thick1

D(A)(C).

The C-level of M

levelCD(A)(M) := infn ∈ N ∪ 0 |M ∈ thicknD(A)(C).

levelCD(A)(M)

high level

... ... ...

· · ·

ΣN2C2=

= · · ·

level ≤ 4

ΣN1C1;

@@ · · ·

level ≤ 2

ΣN0C<

level = 1 · · ·

AA · · ·

low level6

A triangular inequality on the level:

Proposition 1.2. For any M, C and C′ in D(A),

levelCD(A)M ≤ levelCD(A)C′ · levelC

′D(A)M

§2 The cochain type levels

T OPB : the category of spaces over a space B:

Objects α : Y → B. Morphisms Yφ

α AAA

AAAA Y ′

α′||||

C∗(X;K) : the singular cochain complex of a space X withcoefficients in a field K.For α : s(α) → B ∈ Ob(T OPB),C∗(s(α);K) is a DGM over the DGA C∗(B;K).

C∗(s( );K) : T OPB //D(C∗(B;K))

levelD(C∗(B;K))(s(α)) := levelC∗(B;K)

D(C∗(B;K))(C∗(s(α);K)).

α AAA

AAAA Y ′

α′||||

C∗(s( );K) : T OPB //D(C∗(B;K))

D(C∗(B;K))(C∗(s(α);K)).

α AAA

AAAA Y ′

α′||||

C∗(s( );K) : T OPB //D(C∗(B;K))

D(C∗(B;K))(C∗(s(α);K)).

Proposition 2.1 [K, 2008, 2010] Suppose that there exists a se-

quence of fibrations

Sm1 → Y1π1−→ B, Sm2 → Y2

π2−→ Y1, .....,

Smc → Ycπc−→ Yc−1

in which B is simply-connected and mj ≥ 2 for any j. We regard

Yc as a space over B via the composite π1 · · · πc. Then

levelD(C∗(B;K))(Yc) ≤ 2c (levelD(C∗(B;Q))(Yc) ≤ c+ 1 if mj is odd).

The cochain type level : debut in ECAT2, 2008

”The level is related to the Lusternik-Schnirelmann category.”

The cochain type level : debut in ECAT2, 2008

”The level is related to the Lusternik-Schnirelmann category.”

§3 The chain type levels and the L.-S. category

For any object f : s(f) → B in T OPB,

holonomy act.

Ff // s(f)f

C∗(F(−);K) : T OPB → D(C∗(ΩB;K))

levelD(C∗(ΩB;K))(Ff) := levelC∗(ΩB;K)D(C∗(ΩB;K))(C∗(Ff ;K))

§3 The chain type levels and the L.-S. category

For any object f : s(f) → B in T OPB,

holonomy act.

Ff // s(f)f

C∗(F(−);K) : T OPB → D(C∗(ΩB;K))

levelD(C∗(ΩB;K))(Ff) := levelC∗(ΩB;K)D(C∗(ΩB;K))(C∗(Ff ;K))

B(K, A,A) → K → 0 : the bar resolution of K as a right A-module.

Define a sub A-module EnA of B(K, A,A) by EnA = T (ΣA)≤n⊗A.

Definition 3.1 [Kahl, 2003] The E-category for M in DGM-A.

EcatAM := infn | ∃M → EnA in DGM-A

Theorem 3.2 [Kahl] For a map f : X → Y from a connected

space to a simply-connected space,

EcatC∗(ΩY )C∗(Ff)≤catf.

B(K, A,A) → K → 0 : the bar resolution of K as a right A-module.

Define a sub A-module EnA of B(K, A,A) by EnA = T (ΣA)≤n⊗A.

Definition 3.1 [Kahl, 2003] The E-category for M in DGM-A.

EcatAM := infn | ∃M → EnA in DGM-A

Theorem 3.2 [Kahl] For a map f : X → Y from a connected

space to a simply-connected space,

EcatC∗(ΩY )C∗(Ff)≤catf.

Theorem 3.3. [K, 2010] Let f : X → Y be a map from a con-

nected space to a simply-connected space. Then one has

EcatC∗(ΩY )C∗(Ff)≤ levelD(C∗(ΩY ))(Ff) − 1≤dimH∗(X) − 1.

• If X and Y : 1-connected, EcatC∗(ΩY )C∗(Ff) = Mcatf in the

sense of Halperin and Lemaire [Kahl].

•Mcat(idX) = catX for a rational 1-conn. space X [Hess, 1991].

Corollary 3.4. Let X be a simply-connected rational space. Then

catX ≤ levelD(C∗(ΩX;Q))Q − 1.

Example 3.5. levelC∗(ΩX;Q)Q = levelC∗(ΩX;Q)(FidX).

X : a simply-connected rational H-space with dimH∗(X;Q) <∞.

H∗(X;Q) = ∧(x1, ..., xl): primitively generated.

H∗(ΩX;Q) ∼= Q[y1, ..., yl] as an algebra, where deg yi = degxi − 1.

l = c(X) ≤ catX ≤ levelD(C∗(ΩX;Q))Q − 1 ≤ pdH∗(ΩX)Q = l.

We have catX + 1 = levelD(C∗(ΩY ;Q))Q = l+ 1.

Example 3.5. levelC∗(ΩX;Q)Q = levelC∗(ΩX;Q)(FidX).

X : a simply-connected rational H-space with dimH∗(X;Q) <∞.

H∗(X;Q) = ∧(x1, ..., xl): primitively generated.

H∗(ΩX;Q) ∼= Q[y1, ..., yl] as an algebra, where deg yi = degxi − 1.

l = c(X) ≤ catX ≤ levelD(C∗(ΩX;Q))Q − 1 ≤ pdH∗(ΩX)Q = l.

We have catX + 1 = levelD(C∗(ΩY ;Q))Q = l+ 1.

For an 1-conn. space B,

D(C∗(ΩB;K)) T OP/BC∗(F( ))

ooC∗(s( ))

//D(C∗(B;K))

the chain type level the cochain type level

Koszul duality: For a nice DGA A,

D(ExtA(K,K)-mod)h' //

Adams’ cobar construction:

ExtC∗(B)(K,K) ∼= H∗(ΩB;K) as an algebra.

ooC∗(s( ))

//D(C∗(B;K))

ooC∗(s( ))

//D(C∗(B;K))

The cochain type level : ObD(C∗(X;K)) → N ∪ 0,∞≤ #fibrations which construct a given space

The chain type level : ObD(C∗(ΩX;K)) → N ∪ 0,∞≥ The L.-S. category

Duality ??

The cochain type level : ObD(C∗(X;K)) → N ∪ 0,∞≤ #fibrations which construct a given space

The chain type level : ObD(C∗(ΩX;K)) → N ∪ 0,∞≥ The L.-S. category

Duality ??

§4 Duality on the (co)chain type levels

Theorem 4.1. [K, 2010] Let B be a simply-connected space

and f : X → B an object in T OPB. Then one has (in)equalities

(1) dimH∗(X;K) ≥ level C∗(ΩB)D(C∗(ΩB))(C∗(Ff)) = level K

D(C∗(B))(C∗(X))

(2) dimH∗(Ff ;K) ≥ level C∗(B)

D(C∗(B))(C∗(X)) = level K

D(C∗(ΩB))(C∗(Ff)).

On the equalities on Theorem 4.1:

Theorem 4.2. [K, 2010] One has commutative diagrams

T OPBC∗(F(−))

C∗(s(−))

D(C∗(ΩB))RC∗(ΩB)

//D(B(C∗(ΩB))tD

//D(B(C∗(ΩB)∨))ψ∗' //

D(C∗(B)),−⊗L

C∗(B)B(C∗(ΩB))∨oo

T OPBC∗(F(−))

C∗(s(−))

D(C∗(ΩB)) φ∗

**VVVVVVVVVVVVVVVVVVVVVV D(B(C∗(B))∨) D(B(C∗(B)))tD

oo D(C∗(B)),RC∗(B)

D(ΩC∗(B))

−⊗LΩC∗(B)

C∗(ΩB)'

jjVVVVVVVVVVVVVVVVVVVVVV

in which all the functors between derived categories are exact.

On the equalities on Theorem 4.1:

Theorem 4.2. [K, 2010] One has commutative diagrams

T OPBC∗(F(−))

C∗(s(−))

//D(B(C∗(ΩB))tD

//D(B(C∗(ΩB)∨))ψ∗' //

D(C∗(B)),−⊗L

T OPBC∗(F(−))

C∗(s(−))

D(C∗(ΩB)) φ∗

**VVVVVVVVVVVVVVVVVVVVVV D(B(C∗(B))∨) D(B(C∗(B)))tD

oo D(C∗(B)),RC∗(B)

D(ΩC∗(B))

−⊗LΩC∗(B)

C∗(ΩB)'

jjVVVVVVVVVVVVVVVVVVVVVV

in which all the functors between derived categories are exact.

• level C∗(ΩB)D(C∗(ΩB))(C∗(Ff)) = level K

D(C∗(B))(C∗(X))

T OPBC∗(F(−))

C∗(s(−))

//D(B(C∗(ΩB))tD

//D(B(C∗(ΩB)∨))ψ∗' / /

D(C∗(B)), exact−⊗L

ψ∗ tD RC∗(ΩB)(C∗(ΩB)) = KC∗(B)

level C∗(ΩB)D(C∗(ΩB))(C∗(Ff)) ≥ level K

D(C∗(B))(C∗(s(f))

D(C∗(B))(C∗(X))

T OPBC∗(F(−))

C∗(s(−))

//D(B(C∗(ΩB))tD

//D(B(C∗(ΩB)∨))ψ∗' / /

D(C∗(B))(C∗(s(f))

D(C∗(B))(C∗(X))

T OPBC∗(F(−))

C∗(s(−))

//D(B(C∗(ΩB))tD

//D(B(C∗(ΩB)∨))ψ∗' / /

D(C∗(B))(C∗(s(f))

The second diagram in Theorem 4.2:

T OPBC∗(F(−))

C∗(s(−))

D(C∗(ΩB)) φ∗

++WWWWWWWWWWWWWWWWWWWW D(B(C∗(B))∨) D(B(C∗(B)))tD

oo D(C∗(B)), exactRC∗(B)

D(ΩC∗(B))Θ'OO

−⊗LΩC∗(B)

C∗(ΩB)'kkWWWWWWWWWWWWWWWWWWWW

tD RC∗(B)(KC∗(B)) = Θ φ∗(C∗(ΩB)).

level C∗(ΩB)D(C∗(ΩB))(C∗(Ff)) ≤ level K

D(C∗(B))(C∗(s(f)))

The second diagram in Theorem 4.2:

T OPBC∗(F(−))

C∗(s(−))

D(C∗(ΩB)) φ∗

++WWWWWWWWWWWWWWWWWWWW D(B(C∗(B))∨) D(B(C∗(B)))tD

oo D(C∗(B)), exactRC∗(B)

D(ΩC∗(B))Θ'OO

−⊗LΩC∗(B)

C∗(ΩB)'kkWWWWWWWWWWWWWWWWWWWW

tD RC∗(B)(KC∗(B)) = Θ φ∗(C∗(ΩB)).

level C∗(ΩB)D(C∗(ΩB))(C∗(Ff)) = level K

D(C∗(B))(C∗(s(f)))

Corollary 4.3. Let f : X → B be a map with B simply-connected.

(1) level C∗(ΩB)D(C∗(ΩB))(C∗(Ff)) is finite if and only if so is dimH∗(X;K).

(2) level C∗(B)

D(C∗(B))(C∗(X)) is finite if and only if so is dimH∗(Ff ;K).

If levelKD(A)(M) <∞, then dimH∗(M) <∞.

dimH∗(X;K) ≥ level C∗(ΩB)D(C∗(ΩB))(C∗(Ff)) = level K

D(C∗(B))(C∗(X))

dimH∗(Ff ;K) ≥ level C∗(B)

D(C∗(ΩB))(C∗(Ff)).

Corollary 4.3. Let f : X → B be a map with B simply-connected.

(1) level C∗(ΩB)D(C∗(ΩB))(C∗(Ff)) is finite if and only if so is dimH∗(X;K).

(2) level C∗(B)

D(C∗(B))(C∗(X)) is finite if and only if so is dimH∗(Ff ;K).

If levelKD(A)(M) <∞, then dimH∗(M) <∞.

dimH∗(X;K) ≥ level C∗(ΩB)D(C∗(ΩB))(C∗(Ff)) = level K

D(C∗(B))(C∗(X))

dimH∗(Ff ;K) ≥ level C∗(B)

D(C∗(ΩB))(C∗(Ff)).

§5 A computational example of the cochain type level.

Example 5.1. Let BG be the classifying space of a connected

Lie group G. Consider the sequence of (homotopy) fibrations

BG∆→ (BG)×2 → · · · 1×∆→ (BG)×n.

Suppose that H∗(BG;K) is a polynomial algebra.

n ≤ levelD(C∗((BG)×n;K))(BG) ≤ (n− 1)dimQH∗(BG;K) + 1.

In particular, levelD(C∗((BS1)×n;K))(BS1) = n.

Let A be a DGA. A map f : M → N in D(A) : a ghost if H(f) = 0.For M ∈ D(A), the ghost length of M [Hovey and Lockridge]:

gh.len.M = supn |M f1→ Y1

f2→ · · · fn→ Yn non-trivial in D(A), fi :ghost

Lemma 5.2. [Schmidt 2008] For any M ∈ D(A), one has

gh.len.M + 1 ≤ levelAD(A)(M).

Each integration along the fibre (1 × ∆)! is a ghost.

C∗(BG)∆!→ C∗((BG)×2) → · · · (1×∆)!→ C∗((BG)×n).

G→ BGl−1 1×∆→ BGl

The composite (1×∆)!· · ·∆! : C∗(BG) → C∗(BG×n) is non-trivialin D(C∗(BG×n)).

C∗(BG)∆!→ C∗((BG)×2) → · · · (1×∆)!→ C∗((BG)×n).

For the free loop space LBG, the homotopy pull-back

Gn−1 //LBG×BG · · · ×BG LBG ∆ //

evaluation

ev. at n points

Gn−1 //BG∆(n):=(1×∆)···∆

//BG×n

The integration along the fibre (∆)!, which is an ”extension”

of (∆(n))! = (1×∆)! · · · ∆! in D(C∗(BG×n)), is non-trivial (the

Eilenberg-Moore spectral sequence argument).

(∆(n))! 6= 0 in D(C∗(BG×n)). We have

n− 1 + 1 ≤ gh.len.C∗(BG) + 1 ≤ levelD(C∗((BG)×n;K))(BG).

For the free loop space LBG, the homotopy pull-back

Gn−1 //LBG×BG · · · ×BG LBG ∆ //

evaluation

ev. at n points

Gn−1 //BG∆(n):=(1×∆)···∆

//BG×n

The integration along the fibre (∆)!, which is an ”extension”

of (∆(n))! = (1×∆)! · · · ∆! in D(C∗(BG×n)), is non-trivial (the

Eilenberg-Moore spectral sequence argument).

(∆(n))! 6= 0 in D(C∗(BG×n)). We have

n− 1 + 1 ≤ gh.len.C∗(BG) + 1 ≤ levelD(C∗((BG)×n;K))(BG).

Prospect:

The (co)chain type levels give ”estimates” for the length of loop

(co)products in string topology on Gorenstein spaces containing BG

and manifolds.

• In rational case, L.-S. category 6= the chain type level in

general.

X : an infinite wedge of spheres of the form∨α S

catXQ = catX = 1.

By applying Corollary,

level C∗(ΩX)D(C∗(ΩX))Q = ∞.

In fact, H∗(X;Q) is of infinite dimension.

On coderived categories:

(A, dA, εA) : an augmented DG algebra over K.(C, dC, εC) : a cocomplete, coaugmented DG coalgebra over K.

τ : C → A : a twisted cochain, a K-linear map of degree +1 suchthat εA τ εC = 0 and

dA τ + τ dC + µA (τ ⊗ τ) ∆C = 0.

M : a right DG module over A.The twisted tensor product M ⊗τ C : the comodule M ⊗C over Cwith

d = dM ⊗ 1 + 1 ⊗ dC − (µM ⊗ 1)(1 ⊗ τ ⊗ 1)(1 ⊗ ∆C).

For a DG comodule N over A, we define the DG module M⊗τ Asimilarly.

On coderived categories:

(A, dA, εA) : an augmented DG algebra over K.(C, dC, εC) : a cocomplete, coaugmented DG coalgebra over K.

τ : C → A : a twisted cochain, a K-linear map of degree +1 suchthat εA τ εC = 0 and

dA τ + τ dC + µA (τ ⊗ τ) ∆C = 0.

M : a right DG module over A.The twisted tensor product M ⊗τ C : the comodule M ⊗C over Cwith

d = dM ⊗ 1 + 1 ⊗ dC − (µM ⊗ 1)(1 ⊗ τ ⊗ 1)(1 ⊗ ∆C).

For a DG comodule N over A, we define the DG module M⊗τ Asimilarly.

D(C) : the coderived category which is a triangulated category,namely the localization of the category ComcC of cocompletecomodules over C.

Theorem 4.2 [Lefevre-Hasegawa, 2003]Let τ : C → A be a twist-ing cochain. Then one has adjoint functors

D(C)L:=−⊗τA//

D(A)R:=−⊗τCoo

between triangulated categories.

Let A be a DG algebra. By using the natural twisting cochainτ : B(A) → A; [x] 7→ x, we have a pair of adjoint functors

D(B(A))LA:=−⊗τA//

D(A).RA:=−⊗τB(A)

D(C)L:=−⊗τA//

D(A)R:=−⊗τCoo

D(C)L:=−⊗τA//

D(A)R:=−⊗τCoo

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