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Duality on the (co)chain type levels of
maps
Katsuhiko KURIBAYASHI
Shinshu University
The University of Tokyo, EACAT4, December 6, 2011
1
§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.
FACT
For any M in DGM−A, ∃ ΓM ' //M : a semifree resolution of M.
2
§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.
FACT
For any M in DGM−A, ∃ ΓM ' //M : a semifree resolution of M.
2-a
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,
Mφ
//N //C(φ) //ΣM ; C(φ) = N ⊕ ΣM, dC(φ) =
(dN φ0 −dM
)
3
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,
Mφ
//N //C(φ) //ΣM ; C(φ) = N ⊕ ΣM, dC(φ) =
(dN φ0 −dM
)
3-a
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.
4
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).
5
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).
5-a
levelCD(A)(M)
high level
...
C3
... ... ...
· · ·
===
====
===
C2=
====
====
=
===
====
===
ΣN2C2=
====
====
= · · ·
level ≤ 4
;;;
;;;;
;;;;
@@
C1;
;;;;
;;;;
;;
@@
ΣN1C1;
;;;;
;;;;
;;
@@ · · ·
level ≤ 2
<<<
<<<<
<<<
AA
ΣN0C<
<<<<
<<<<
<
AA
<<<
<<<<
<<<
AA
level = 1 · · ·
AA
AA C
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
7
§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′||||
|||
B
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)).
8
§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′||||
|||
B
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)).
8-a
§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′||||
|||
B
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)).
8-b
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).
9
The cochain type level : debut in ECAT2, 2008
”The level is related to the Lusternik-Schnirelmann category.”
10
The cochain type level : debut in ECAT2, 2008
”The level is related to the Lusternik-Schnirelmann category.”
10-a
§3 The chain type levels and the L.-S. category
For any object f : s(f) → B in T OPB,
ΩB
holonomy act.
Ff // s(f)f
//B
C∗(F(−);K) : T OPB → D(C∗(ΩB;K))
levelD(C∗(ΩB;K))(Ff) := levelC∗(ΩB;K)D(C∗(ΩB;K))(C∗(Ff ;K))
11
§3 The chain type levels and the L.-S. category
For any object f : s(f) → B in T OPB,
ΩB
holonomy act.
Ff // s(f)f
//B
C∗(F(−);K) : T OPB → D(C∗(ΩB;K))
levelD(C∗(ΩB;K))(Ff) := levelC∗(ΩB;K)D(C∗(ΩB;K))(C∗(Ff ;K))
11-a
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.
12
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.
12-a
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.
FACT
• 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.
13
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.
FACT
• 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.
13-a
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.
FACT
• 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.
13-b
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.
14
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.
14-a
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' //
D(A)t
oo
Adams’ cobar construction:
ExtC∗(B)(K,K) ∼= H∗(ΩB;K) as an algebra.
15
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' //
D(A)t
oo
Adams’ cobar construction:
ExtC∗(B)(K,K) ∼= H∗(ΩB;K) as an algebra.
15-a
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' //
D(A)t
oo
Adams’ cobar construction:
ExtC∗(B)(K,K) ∼= H∗(ΩB;K) as an algebra.
15-b
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 ??
16
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 ??
16-a
§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)).
17
On the equalities on Theorem 4.1:
Theorem 4.2. [K, 2010] One has commutative diagrams
T OPBC∗(F(−))
rr
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(−))
rr
C∗(s(−))
))
D(C∗(ΩB)) φ∗
**VVVVVVVVVVVVVVVVVVVVVV D(B(C∗(B))∨) D(B(C∗(B)))tD
oo D(C∗(B)),RC∗(B)
oo
D(ΩC∗(B))
Θ'OO
−⊗LΩC∗(B)
C∗(ΩB)'
jjVVVVVVVVVVVVVVVVVVVVVV
in which all the functors between derived categories are exact.
18
On the equalities on Theorem 4.1:
Theorem 4.2. [K, 2010] One has commutative diagrams
T OPBC∗(F(−))
rr
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(−))
rr
C∗(s(−))
))
D(C∗(ΩB)) φ∗
**VVVVVVVVVVVVVVVVVVVVVV D(B(C∗(B))∨) D(B(C∗(B)))tD
oo D(C∗(B)),RC∗(B)
oo
D(ΩC∗(B))
Θ'OO
−⊗LΩC∗(B)
C∗(ΩB)'
jjVVVVVVVVVVVVVVVVVVVVVV
in which all the functors between derived categories are exact.
18-a
• level C∗(ΩB)D(C∗(ΩB))(C∗(Ff)) = level K
D(C∗(B))(C∗(X))
T OPBC∗(F(−))
rr
C∗(s(−))
**
D(C∗(ΩB))RC∗(ΩB)
//D(B(C∗(ΩB))tD
//D(B(C∗(ΩB)∨))ψ∗' / /
D(C∗(B)), exact−⊗L
C∗(B)B(C∗(ΩB))∨oo
ψ∗ tD RC∗(ΩB)(C∗(ΩB)) = KC∗(B)
level C∗(ΩB)D(C∗(ΩB))(C∗(Ff)) ≥ level K
D(C∗(B))(C∗(s(f))
19
• level C∗(ΩB)D(C∗(ΩB))(C∗(Ff)) = level K
D(C∗(B))(C∗(X))
T OPBC∗(F(−))
rr
C∗(s(−))
**
D(C∗(ΩB))RC∗(ΩB)
//D(B(C∗(ΩB))tD
//D(B(C∗(ΩB)∨))ψ∗' / /
D(C∗(B)), exact−⊗L
C∗(B)B(C∗(ΩB))∨oo
ψ∗ tD RC∗(ΩB)(C∗(ΩB)) = KC∗(B)
level C∗(ΩB)D(C∗(ΩB))(C∗(Ff)) ≥ level K
D(C∗(B))(C∗(s(f))
19-a
• level C∗(ΩB)D(C∗(ΩB))(C∗(Ff)) = level K
D(C∗(B))(C∗(X))
T OPBC∗(F(−))
rr
C∗(s(−))
**
D(C∗(ΩB))RC∗(ΩB)
//D(B(C∗(ΩB))tD
//D(B(C∗(ΩB)∨))ψ∗' / /
D(C∗(B)), exact−⊗L
C∗(B)B(C∗(ΩB))∨oo
ψ∗ tD RC∗(ΩB)(C∗(ΩB)) = KC∗(B)
level C∗(ΩB)D(C∗(ΩB))(C∗(Ff)) ≥ level K
D(C∗(B))(C∗(s(f))
19-b
The second diagram in Theorem 4.2:
T OPBC∗(F(−))
rr
C∗(s(−))
**
D(C∗(ΩB)) φ∗
++WWWWWWWWWWWWWWWWWWWW D(B(C∗(B))∨) D(B(C∗(B)))tD
oo D(C∗(B)), exactRC∗(B)
oo
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)))
20
The second diagram in Theorem 4.2:
T OPBC∗(F(−))
rr
C∗(s(−))
**
D(C∗(ΩB)) φ∗
++WWWWWWWWWWWWWWWWWWWW D(B(C∗(B))∨) D(B(C∗(B)))tD
oo D(C∗(B)), exactRC∗(B)
oo
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)))
20-a
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∗(B))(C∗(X)) = level K
D(C∗(ΩB))(C∗(Ff)).
21
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∗(B))(C∗(X)) = level K
D(C∗(ΩB))(C∗(Ff)).
21-a
§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.
22
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)).
23
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)).
23-a
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)).
23-b
For the free loop space LBG, the homotopy pull-back
Gn−1 //LBG×BG · · · ×BG LBG ∆ //
evaluation
LBG
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).
24
For the free loop space LBG, the homotopy pull-back
Gn−1 //LBG×BG · · · ×BG LBG ∆ //
evaluation
LBG
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).
24-a
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.
25
• In rational case, L.-S. category 6= the chain type level in
general.
X : an infinite wedge of spheres of the form∨α S
nα.
catXQ = catX = 1.
By applying Corollary,
level C∗(ΩX)D(C∗(ΩX))Q = ∞.
In fact, H∗(X;Q) is of infinite dimension.
26
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.
27
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.
27-a
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)
oo
28
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)
oo
28-a
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)
oo
28-b