DG-enhanced cyclotomic KLR algebras and categori cation of Verma modules

DG-enhanced cyclotomic KLR algebras and categorication of Verma modules

Pedro Vaz (Université catholique de Louvain)

M^p(V_β) = U_q(g) ⊗_Uq(p) V_β

Joint work with Grégoire Naisse and Ruslan Maksimau September 2018

A well known story-I

Pick your favorite qKM algebra g and let Λ be a dominant integral weight.

L(Λ) Integrable irreducible

(n. dim. if g of nite type)

We think of these as modules over k(q) and k(q, β₁, . . . , β_`).

A well known story-I

Pick your favorite qKM algebra g and let Λ be a dominant integral weight.

L(Λ) Integrable irreducible

∆(Λ) Verma

N(Λ)

M(β) Universal Verma β_i 7→ q^Λi

We think of these as modules over k(q) and k(q, β₁, . . . , β_`).

A well known story-I

Pick your favorite qKM algebra g and let Λ be a dominant integral weight.

L(Λ) Integrable irreducible

∆(Λ) Verma

N(Λ)

M(β) Universal Verma β_i 7→ q^Λi

We think of these as modules over k(q) and k(q, β₁, . . . , β_`).

A well known story-I

Pick your favorite qKM algebra g and let Λ be a dominant integral weight.

L(Λ) Integrable irreducible

∆(Λ) Verma

N(Λ)

M(β) Universal Verma

β_i 7→ q^Λi

We think of these as modules over k(q) and k(q, β₁, . . . , β_`).

A well known story-I

Pick your favorite qKM algebra g and let Λ be a dominant integral weight.

L(Λ) Integrable irreducible

∆(Λ) Verma

N(Λ)

M(β) Universal Verma β_i 7→ q^Λi

We think of these as modules over k(q) and k(q, β₁, . . . , β_`).

A well known story-I

Pick your favorite qKM algebra g and let Λ be a dominant integral weight.

L(Λ) Integrable irreducible

∆(Λ) Verma

N(Λ)

M(β) Universal Verma β_i 7→ q^Λi

We think of these as modules over k(q) and k(q, β₁, . . . , β_`).

A well known story-II

It is well known that apart form the g-action, half quantum groups are almost like universal Vermas in the sense that we have a quotient map U_q⁻(g) → L(Λ) :

U_q⁻(g)

L(Λ)

M(β)

·

A well known story-II

It is well known that apart form the g-action, half quantum groups are almost like universal Vermas in the sense that we have a quotient map U_q⁻(g) → L(Λ) :

U_q⁻(g)

L(Λ)

M(β)

·

A well known categoried story-I

KhovanovLauda and Rouquier categoried part of this picture :

• Let R be the KLR algebra for g and R^Λ its cyclotomic quotient w.r.t. Λ and put

U_q⁻(g) = R-mod_g L(Λ) = R^Λ-mod_g U_q⁻(g)

L(Λ)

U_q⁻(g)

K₀ L(Λ)

K₀

A well known categoried story-I

KhovanovLauda and Rouquier categoried part of this picture :

• Let R be the KLR algebra for g and R^Λ its cyclotomic quotient w.r.t.

Λ and put

U_q⁻(g) = R-mod_g L(Λ) = R^Λ-mod_g

U_q⁻(g)

L(Λ)

U_q⁻(g)

K₀ L(Λ)

K₀

A well known categoried story-I

KhovanovLauda and Rouquier categoried part of this picture :

• Let R be the KLR algebra for g and R^Λ its cyclotomic quotient w.r.t.

Λ and put

U_q⁻(g) = R-mod_g L(Λ) = R^Λ-mod_g U_q⁻(g)

L(Λ)

U_q⁻(g)

K₀ L(Λ)

K₀

A well known categoried story-I

KhovanovLauda and Rouquier categoried part of this picture :

• Let R be the KLR algebra for g and R^Λ its cyclotomic quotient w.r.t.

Λ and put

U_q⁻(g) = R-mod_g L(Λ) = R^Λ-mod_g U_q⁻(g)

L(Λ)

U_q⁻(g)

K₀ L(Λ)

K₀

Today's story

The plan is to complete the diagram :

U_q⁻(g)

L(Λ)

M(β)

·

Categorication

L(Λ) U˙_q⁻(g)

M(β)

Today's story

The plan is to complete the diagram : U_q⁻(g)

L(Λ)

M(β)

·

Categorication

L(Λ) U˙_q⁻(g)

M(β)

Today's story

The plan is to complete the diagram : U_q⁻(g)

L(Λ)

M(β)

·

Categorication

L(Λ) U˙_q⁻(g)

M(β)

Today's story

The plan is to complete the diagram : U_q⁻(g)

L(Λ)

M(β)

·

Categorication

L(Λ)

U˙_q⁻(g)

M(β)

Today's story

The plan is to complete the diagram : U_q⁻(g)

L(Λ)

M(β)

·

Categorication

L(Λ) U˙_q⁻(g)

M(β)

Today's story

The plan is to complete the diagram : U_q⁻(g)

L(Λ)

M(β)

·

Categorication

L(Λ) U˙_q⁻(g)

M(β)

(cyclotomic) KLR algebras - I

Categorications of U_q⁻(g) and of L(Λ) are given through KLR algebras.

Fix (g, I, Λ) and a ground ring k.

KLR algebras can be dened by isotopy classes of diagrams / relations.

• Generators :

i

. . . and

i j

· · · (for all i, j ∈ I).

• Relations (for example) :

i j

=

i j

+

i j

if _{i j}

(cyclotomic) KLR algebras - I

Categorications of U_q⁻(g) and of L(Λ) are given through KLR algebras.

Fix (g, I, Λ) and a ground ring k.

KLR algebras can be dened by isotopy classes of diagrams / relations.

• Generators :

i

. . . and

i j

· · · (for all i, j ∈ I).

• Relations (for example) :

i j

=

i j

+

i j

if _{i j}

(cyclotomic) KLR algebras - II

For ν = P

i∈I ν_i · i ∈ N[I] let R(ν) be the algebra consisting of ν_i strands labelled i. Dene

R = M

ν∈N[I]

R(ν)

Theorem (KhovanovLauda, Rouquier '08) K₀(R) ∼= U_q⁻(g).

cyclotomic KLR algebras

Let I^Λ be the 2-sided ideal generated by all pictures like

i Λ_i

j k

· · ·

and put R^Λ = R/(I^Λ).

(cyclotomic) KLR algebras - II

For ν = P

i∈I ν_i · i ∈ N[I] let R(ν) be the algebra consisting of ν_i strands labelled i. Dene

R = M

ν∈N[I]

R(ν)

Theorem (KhovanovLauda, Rouquier '08) K₀(R) ∼= U_q⁻(g).

cyclotomic KLR algebras

Let I^Λ be the 2-sided ideal generated by all pictures like

i Λ_i

j k

· · ·

and put R^Λ = R/(I^Λ).

(cyclotomic) KLR algebras - III

Categorical g-action We dene

F^Λ_i (ν) : R^Λ(ν)-mod_g → R^Λ(ν + i)-mod_g

as the functor of induction for the map that adds a strand labeled i at the right of a diagram from R^Λ(ν), and E^Λ_i (ν) be its right adjoint (with an appropriated shift).

These functors have very nice properties, for example they are biadjoint and the composites E^Λ_i F^Λ_i (ν) and F^Λ_i E^Λ_i (ν) satisfy a direct sum

decomposition lifting the commutator relation. E^Λ_i F^Λ_i (ν) ' F^Λ_i E^Λ_i (ν) ⊕ ^νi±1−2ν⊕ ⁱ⁻¹

`=0

Id_ν{2`} if ν_i±1 ≥ 2ν_i, F^Λ_i E^Λ₁ (ν) ' E^Λ_i F^Λ_i (ν) ⊕ ^2νi−ν⊕^i±1−1

`=0

Id_ν{2`} if ν_i±1 ≤ 2ν_i, F_k^ΛE^Λ_j (ν) ' E^Λ_k F^Λ_j (ν) for j 6= k.

(cyclotomic) KLR algebras - III

Categorical g-action We dene

F^Λ_i (ν) : R^Λ(ν)-mod_g → R^Λ(ν + i)-mod_g

as the functor of induction for the map that adds a strand labeled i at the right of a diagram from R^Λ(ν), and E^Λ_i (ν) be its right adjoint (with an appropriated shift).

These functors have very nice properties, for example they are biadjoint and the composites E^Λ_i F^Λ_i (ν) and F^Λ_i E^Λ_i (ν) satisfy a direct sum

decomposition lifting the commutator relation.

E^Λ_i F^Λ_i (ν) ' F^Λ_i E^Λ_i (ν) ⊕ ^νi±1−2ν⊕ ⁱ⁻¹

`=0

Id_ν{2`} if ν_i±1 ≥ 2ν_i, F^Λ_i E^Λ₁ (ν) ' E^Λ_i F^Λ_i (ν) ⊕ ^2νi−ν⊕^i±1−1

`=0

Id_ν{2`} if ν_i±1 ≤ 2ν_i, F_k^ΛE^Λ_j (ν) ' E^Λ_k F^Λ_j (ν) for j 6= k.

(cyclotomic) KLR algebras - III

Categorical g-action We dene

F^Λ_i (ν) : R^Λ(ν)-mod_g → R^Λ(ν + i)-mod_g

as the functor of induction for the map that adds a strand labeled i at the right of a diagram from R^Λ(ν), and E^Λ_i (ν) be its right adjoint (with an appropriated shift).

These functors have very nice properties, for example they are biadjoint and the composites E^Λ_i F^Λ_i (ν) and F^Λ_i E^Λ_i (ν) satisfy a direct sum

decomposition lifting the commutator relation.

E^Λ_i F^Λ_i (ν) ' F^Λ_i E^Λ_i (ν) ⊕ ^νi±1

−2ν_i−1

⊕

`=0

Id_ν{2`} if ν_i±1 ≥ 2ν_i, F^Λ_i E^Λ₁ (ν) ' E^Λ_i F^Λ_i (ν) ⊕ ^2νi−ν⊕^i±1−1

`=0

Id_ν{2`} if ν_i±1 ≤ 2ν_i, F_k^ΛE^Λ_j (ν) ' E^Λ_k F^Λ_j (ν) for j 6= k.

(cyclotomic) KLR algebras - III

Categorical g-action We dene

F^Λ_i (ν) : R^Λ(ν)-mod_g → R^Λ(ν + i)-mod_g

as the functor of induction for the map that adds a strand labeled i at the right of a diagram from R^Λ(ν), and E^Λ_i (ν) be its right adjoint (with an appropriated shift).

These functors have very nice properties, for example they are biadjoint and the composites E^Λ_i F^Λ_i (ν) and F^Λ_i E^Λ_i (ν) satisfy a direct sum

decomposition lifting the commutator relation.

The Categorication Theorem (KangKashiwara, Webster,...)

K₀(R^Λ) ∼= L(Λ) (as g-modules)

DG enhancements

We cannot categorify M(λ) (nor ∆(Λ)) from R !

Idea : enhance cyclotomic KLR algebras to the DG world R

cyclotomic quotient

R^Λ

DG enhancement

(R^Λ, 0) (R, 0)

cyclotomic quotient

(R_β, d_Λ)

quasi iso.

DG enhancements

We cannot categorify M(λ) (nor ∆(Λ)) from R !

Idea : enhance cyclotomic KLR algebras to the DG world R

cyclotomic quotient

R^Λ

DG enhancement

(R^Λ, 0) (R, 0)

cyclotomic quotient

(R_β, d_Λ)

quasi iso.

DG enhancements

We cannot categorify M(λ) (nor ∆(Λ)) from R !

Idea : enhance cyclotomic KLR algebras to the DG world R

cyclotomic quotient

R^Λ

DG enhancement

(R^Λ, 0) (R, 0)

cyclotomic quotient

(R_β, d_Λ)

quasi iso.

DG enhancements

We cannot categorify M(λ) (nor ∆(Λ)) from R !

Idea : enhance cyclotomic KLR algebras to the DG world R

cyclotomic quotient

R^Λ

DG enhancement

(R^Λ, 0)

(R, 0)

cyclotomic quotient

(R_β, d_Λ)

quasi iso.

DG enhancements

We cannot categorify M(λ) (nor ∆(Λ)) from R !

Idea : enhance cyclotomic KLR algebras to the DG world R

cyclotomic quotient

R^Λ

DG enhancement

(R^Λ, 0) (R, 0)

cyclotomic quotient

(R_β, d_Λ)

quasi iso.

DG enhancements

We cannot categorify M(λ) (nor ∆(Λ)) from R !

Idea : enhance cyclotomic KLR algebras to the DG world R

cyclotomic quotient

R^Λ

DG enhancement

(R^Λ, 0) (R, 0)

cyclotomic quotient

(R_β, d_Λ)

quasi iso.

Extended KLR algebras - I

For each i ∈ I add a new generator to R, a oating dot :

i

i j

. . . (it oats)

and impose two relations

i

i i

j

. . . = 0 and

i j

j i

. . . = −

i

j i j

. . .

This is a multigraded superalgebra where the KLR generators are even while oating dot are odd. Call this algebra R_β. This is a minimal presentation.

Extended KLR algebras - I

For each i ∈ I add a new generator to R, a oating dot :

i

i j

. . . (it oats)

and impose two relations

i i

i j

. . . = 0 and

i

j j i

. . . = −

i j

i j

. . .

This is a multigraded superalgebra where the KLR generators are even while oating dot are odd. Call this algebra R_β. This is a minimal presentation.

Extended KLR algebras - II

We have R_β = L

ν∈N[I] R_β(ν).

One can introduce more general oating dots, that can be placed in arbitray regions of the diagrams and get a presentation that is easier to handle in computations.

These generators satisfy some relations, for example,

j i

· · · =

j i

· · · if _{i j}

Extended KLR algebras - II

We have R_β = L

ν∈N[I] R_β(ν).

One can introduce more general oating dots, that can be placed in arbitray regions of the diagrams and get a presentation that is easier to handle in computations.

These generators satisfy some relations, for example,

j i

· · · =

j i

· · · if _{i j}

Dierentials : DG enhanced KLR algebras

For Λ an integral dominant weight for g put

d_Λ

i

i

j

. . .

!

=

i Λ_i

j

. . .

This denes a dierential on R_β. Proposition (NaisseV. '17

The DG-algebras (R_β, d_Λ) and (R^Λ, 0) are quasi-isomorphic.

• We say that (R_β, d_Λ) is a DG-enhancement of R^Λ.

Categorifying the half quantum group

Let R_β(ν) -mod_lf be the category of left bounded, locally nite

dimensional, left supermodules over R_β(ν), with degree zero morphisms, and R_β-pmod_lf the (full) subcategory of projectives.

We can place diagrams aside each other to dene an inclusion of algebras R_β(ν) ⊗ R_β−µ(ν⁰) → R_β(ν + ν⁰).

Theorem (NaisseV. '17 :

There is an isomorphism of Z[q^±1]-algebras

K₀(⊕_δ∈Z[I]R_β+δ-pmod_lf) ∼= ˙U_q⁻(g).

Categorifying the half quantum group

Let R_β(ν) -mod_lf be the category of left bounded, locally nite

dimensional, left supermodules over R_β(ν), with degree zero morphisms, and R_β-pmod_lf the (full) subcategory of projectives.

We can place diagrams aside each other to dene an inclusion of algebras R_β(ν) ⊗ R_β−µ(ν⁰) → R_β(ν + ν⁰).

Theorem (NaisseV. '17 :

There is an isomorphism of Z[q^±1]-algebras

K₀(⊕_δ∈Z[I]R_β+δ-pmod_lf) ∼= ˙U_q⁻(g).

Categorifying Verma modules

Categorication of the weight spaces of M(β) Put

M(β) = R_β -mod_lf = M

ν∈N[I]

R_β(ν) -mod_lf .

Dene the functor

F_i(ν) : R_β(ν) -mod_lf → R_β(ν + i) -mod_lf

as the functor of induction for the map that adds a strand colored i at the right of a diagram from R_β(ν), and denote E_i(ν) its right adjoint (with some shift).

Categorifying Verma modules

There are several relations between these functors lifting the g-relations.

For example,

Proposition (NaisseV. '17)

There is a short exact sequence of functors

0 → F_iE_i(ν) → E_iF_i(ν) → Q(ν)hqshiftⁱ, 1i ⊕ ΠQ(ν)h−qshiftⁱ,−1i → 0 for all i ∈ I, and isomorphisms

F_iE_j(ν) ' E_jF_i(ν) for i 6= j.

Categorifying Verma modules

Put

F_i = M

ν∈N[I]

F_i(ν) and E_i = M

ν∈N[I]

E_i(ν).

The Categorication Theorem (NaisseV. '17) :

• Functors (F_i, E_i) are exact and form an adjoint pair.

• These functors induce an action of U_q(g) on the (topological) Grothendieck group of M(β).

With this action we have an isomorphism

K₀(M(β)) ∼= M(β), of U_q(g)-representations.

Categorication of L(Λ)

There is a SES 0 →

F_iE_i(ν), d_Λ

→

E_iF_i(ν), d_Λ

→

Q(ν)hqshiftⁱ, 1i ⊕ ΠQ(ν)h−qshiftⁱ,−1i, d_Λ

→ 0

This induces a LES in homology which splits ! Depending on the sign of

ν_i±1 − 2ν_k, the homology of the last term is concentrated in degree 0 or 1. Corollary 1 :

E^Λ_i F^Λ_i (ν) ' F^Λ_i E^Λ_i (ν) ⊕ ^νi±1−2ν⊕ ⁱ⁻¹

`=0

Id_ν{2`} if ν_i±1 ≥ 2ν_i, F^Λ_i E^Λ₁ (ν) ' E^Λ_i F^Λ_i (ν) ⊕ ^2νi−ν⊕^i±1−1

`=0

Id_ν{2`} if ν_i±1 ≤ 2ν_i, F_k^ΛE^Λ_j (ν) ' E^Λ_k F^Λ_j (ν) for j 6= k.

Categorication of L(Λ)

There is a SES 0 →

F_iE_i(ν), d_Λ

→

E_iF_i(ν), d_Λ

→

Q(ν)hqshiftⁱ, 1i ⊕ ΠQ(ν)h−qshiftⁱ,−1i, d_Λ

→ 0 This induces a LES in homology which splits ! Depending on the sign of

ν_i±1 − 2ν_k, the homology of the last term is concentrated in degree 0 or 1.

Corollary 1 :

E^Λ_i F^Λ_i (ν) ' F^Λ_i E^Λ_i (ν) ⊕ ^νi±1−2ν⊕ ⁱ⁻¹

`=0

Id_ν{2`} if ν_i±1 ≥ 2ν_i, F^Λ_i E^Λ₁ (ν) ' E^Λ_i F^Λ_i (ν) ⊕ ^2νi−ν⊕^i±1−1

`=0

Id_ν{2`} if ν_i±1 ≤ 2ν_i, F_k^ΛE^Λ_j (ν) ' E^Λ_k F^Λ_j (ν) for j 6= k.

Categorication of L(Λ)

There is a SES 0 →

F_iE_i(ν), d_Λ

→

E_iF_i(ν), d_Λ

→

Q(ν)hqshiftⁱ, 1i ⊕ ΠQ(ν)h−qshiftⁱ,−1i, d_Λ

→ 0 This induces a LES in homology which splits ! Depending on the sign of

ν_i±1 − 2ν_k, the homology of the last term is concentrated in degree 0 or 1.

Corollary 2 :

We have an isomorphism

K₀ D^c(R_β, d_Λ) ∼= L(Λ), of U_q(g)-representations.

Parabolic Verma modules

A subset I_f ⊆ I denes a parabolic subalgebra p ⊆ g and the construction above allow to category parabolic Verma modules as induced modules from the Levi factor l of p by redening the dierential d_Λ.

Let N be an integral dominant weight for l and put

d_N

i

i j

. . .

!

=















0 if i ∈ I_f

i N_i

j

. . . if i ∈ I\I_f

• This results in a categorication of the parabolic Verma module M^p(V_N).

(Ane) Hecke algebras

By BrundanKleshchev and Rouquier we know that in type A,

• (cyclotomic) KLR algebras are essentially (cyclotomic) Hecke algebras.

Let k be an algebraic closed eld and x q ∈ k^∗ and d ∈ N and let H_d = H_d(q) be the

• degenerate ane Hecke algebra (over k) if q = 1,

{s₁, . . . , s_d−1, X₁, . . . , X_d}/relations, or the

• (non-degenerate) ane Hecke algebra (over k) if q 6= 1, {T₁, . . . , T_d−1, X₁^±1, . . . , X_d^±1}/relations.

(Ane) Hecke algebras

By BrundanKleshchev and Rouquier we know that in type A,

• (cyclotomic) KLR algebras are essentially (cyclotomic) Hecke algebras.

Let k be an algebraic closed eld and x q ∈ k^∗ and d ∈ N and let H_d = H_d(q) be the

• degenerate ane Hecke algebra (over k) if q = 1,

{s₁, . . . , s_d−1, X₁, . . . , X_d}/relations, or the

• (non-degenerate) ane Hecke algebra (over k) if q 6= 1, {T₁, . . . , T_d−1, X₁^±1, . . . , X_d^±1}/relations.

Extended Hecke algebras

Denition :

Dene the superalgebra H_d by adding an odd variable θ to H_d and imposing the relations

θ² = 0,

and (

θX_r = X_rθ, s_iθ = θs_i for i > 1

s₁θs₁θ + θs₁θs₁ = 0 if q = 1 or

( θX_r = X_rθ, T_iθ = θT_i for i > 1

T₁θT₁θ + θT₁θT₁ = (q − 1)θT₁θ if q 6= 1

Extended Hecke algebras

Denition :

Dene the superalgebra H_d by adding an odd variable θ to H_d and imposing the relations

θ² = 0,

and (

θX_r = X_rθ, s_iθ = θs_i for i > 1

s₁θs₁θ + θs₁θs₁ = 0 if q = 1

or

( θX_r = X_rθ, T_iθ = θT_i for i > 1

T₁θT₁θ + θT₁θT₁ = (q − 1)θT₁θ if q 6= 1

Extended Hecke algebras

Denition :

Dene the superalgebra H_d by adding an odd variable θ to H_d and imposing the relations

θ² = 0,

and (

θX_r = X_rθ, s_iθ = θs_i for i > 1

s₁θs₁θ + θs₁θs₁ = 0 if q = 1 or

(θX_r = X_rθ, T_iθ = θT_i for i > 1

T₁θT₁θ + θT₁θT₁ = (q − 1)θT₁θ if q 6= 1

DG-enhanced cyclotomic Hecke algebras

Introduce a dierential ∂_Λ on H_d :

• ∂_Λ acts as zero on H_d while

∂_Λ(θ) =







Π_i∈I(X₁ − i)^Λi if q = 1, Π_i∈I(X₁ − qⁱ)^Λi if q 6= 1,

+ Leibniz rule

Here, Λ is an integral dominant weight of Lie type A_∞ or A⁽¹⁾_n−1.

Proposition (MaksimauV. '18) :

The DG-algebras (H_d, ∂_Λ) and (H_d^Λ,0) are quasi-isomorphic.

Cyclotomic Hecke algebra : H_d^Λ =











H_d

Π_i∈I(X₁ − i)^Λi if q = 1, H_d

Π_i∈I(X₁ − qⁱ)^Λi if q 6= 1.

DG-enhanced cyclotomic Hecke algebras

Introduce a dierential ∂_Λ on H_d :

• ∂_Λ acts as zero on H_d while

∂_Λ(θ) =







Π_i∈I(X₁ − i)^Λi if q = 1, Π_i∈I(X₁ − qⁱ)^Λi if q 6= 1,

+ Leibniz rule

Here, Λ is an integral dominant weight of Lie type A_∞ or A⁽¹⁾_n−1.

Proposition (MaksimauV. '18) :

The DG-algebras (H_d, ∂_Λ) and (H_d^Λ, 0) are quasi-isomorphic.

Cyclotomic Hecke algebra : H_d^Λ =











H_d

Π_i∈I(X₁ − i)^Λi if q = 1, H_d

Π_i∈I(X₁ − qⁱ)^Λi if q 6= 1.

The DG-enhanced BKR isomorphism

DG-enhanced cyclotomic KLR algebras are

DG-enhanced cyclotomic Hecke algebras :

• After a suitable completion, H_d gets a block decomposition where blocks are labelled by elements of N[I].

• The dierentials ∂_Λ give rise to dierentials on blocks.

WIP (MaksimauV.)

The DG-algebras (Rb_ν, d_Λ) and (Hb_d1_ν, ∂_Λ,ν) are isomorphic.

The DG-enhanced BKR isomorphism

DG-enhanced cyclotomic KLR algebras are

DG-enhanced cyclotomic Hecke algebras :

• After a suitable completion, H_d gets a block decomposition where blocks are labelled by elements of N[I].

• The dierentials ∂_Λ give rise to dierentials on blocks.

WIP (MaksimauV.)

The DG-algebras (Rb_ν, d_Λ) and (Hb_d1_ν, ∂_Λ,ν) are isomorphic.

Thanks for your attention !