Derived invariance of the numbers $h^{0,p}(X)$

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Derived invariance of the numbers h

(X )

Roland Abuaf

To cite this version:

Roland Abuaf. Derived invariance of the numbersh^0,p(X). 2019. �hal-02189788�

Derived invariance of the numbers h

(X )

Roland Abuaf ^∗

Abstract

LetX₁andX₂be derived equivalent smooth projective varieties over the eld of complex numbers. We prove that the numbersh^0,p(X₁)andh^0,p(X₂)are equal for any p.

∗rabuaf@gmail.com

Contents

1 Introduction 2

2 Cohomological rank functions, associated dual homological units 4

3 Derived invariance of the graded algebra H^•(X, ω_X) 7

1 Introduction

In his foundational work [Kon], M. Kontsevich proposed a formulation of Mirror Symmetry for a pair of (algebraic/symplectic) Calabi-Yau varieties (X, Y) as an equivalence of categories between the derived category of coherent sheaves on X and a possibly enlarged version of the derived Fukaya category of Lagrangian submanifolds ofY. He conjectured that this formulation of Mirror Symmetry should imply the numerical Hodge theoretic version of Mirror Symmetry in the Kähler case : if X is projective and Y is Kähler, then the Hodge diamond of Y is obtained from the Hodge diamond of X by a ^π₂-rotation. Kontsevich also expects that two derived equivalent projective varieties should have the same mirror on the symplectic side. In particular, a sub-conjecture can be derived form this expectation:

Conjecture 1.0.1 ([Kon]) Let X1 and X2 be smooth projective varieties (over C). Assume that D^b(X1)'D^b(X2), then h^p,q(X1) =h^p,q(X2) for all p, q.

This conjecture has attracted some attention in the past years. First note that it follows easily from the Hochschild-Kostant-Rosenberg isomorphism for Hochschild homology if X and Y have dimension one or two. Indeed, the Hochschild-Kostant-Rosenberg isomorphism implies a graded decomposition of Hochschild homology as:

HHr(Xi) = M

q−p=r

H^p(Xi,Ω^q_X

i),

for all r ∈ [−dim(Xi),dim(Xi)] and i = 1,2. Since derived equivalences between two smooth projective varieties can be lifted to a graded vector spaces isomorphism between their Hochschild homology spaces [Orl03], we immediately deduce thatX₁andX₂have the same Hodge numbers.

IfX1 and X2 have dimension three and are Calabi-Yau varieties, the same reasoning combined with Serre duality also shows thatX₁ andX₂have the same Hodge numbers. IfK_X1 is ample (or anti-ample), a famous result of Bondal and Orlov [BO01] implies that any derived equivalence betweenX1andX2comes from an algebraic isomorphism betweenX1andX2, so that conjecture 1.0.1 is also true in this case. If X₁ and X₂ are of general type, a result of Kawamata [Kaw02]

shows that derived equivalence betweenX₁andX₂ impliesK-equivalence, which in turn implies the equality of Hodge numbers byp-adic/Kontsevich's motivic integration [Bat98, DL99].

In higher dimensions and outside of the Fano and general-type cases (and also Calabi-Yau case in dimension three), this conjecture becomes more intriguing. The main diculty being that the lift to Hochschild homology of the equivalence betweenX1 andX2does not necessarily maps the(p, q)-part of HHr(X1) to the (p, q)-part ofHHr(X2) (put dierently, the lift to Hochschild homology of the derived equivalence does not necessarily respect the grading induced by the Hochschild-Kostant-Rosenberg isomorphism). This can already be checked for the Fourier Mukai equivalence between an abeliann-foldA and its dual abelian varietyAˆ : it maps Hⁿ(A, ωA)to H⁰( ˆA,OAˆ).

In [Orl05], Orlov conjectures a stronger variant of conjecture 1.0.1 : two derived equivalent varieties (over any eld) have isomorphic rational Chow motives. He proves his conjecture in the

case where the kernel giving the derived equivalence is supported on a scheme of codimension equal to dimX(= dimY) in the product X×Y (Orlov notes however notes that the Poincaré line bundle does not induce an isomorphism of rational motives in the case of an abelian variety and its dual, even though these rational motives are indeed isomorphic).

The rst general result toward conjecture 1.0.1 was obtained by Poppa and Schnell in [PS11]

: using a result of Rouquier [Rou11], they prove that derived equivalent smooth projective varieties over the eld of complex numbers have isogeneous Jacobian varieties. They deduce that if X₁ and X₂ are derived equivalent then h¹(X₁,OX1) =h¹(X₂,OX2). In dimension three, this argument in conjunction with the Hochschild-Kostant-Rosenberg isomorphism and the derived invariance of Hochshild homology is sucient to prove that h^p,q(X₁) = h^p,q(X₂) for all p, q. Various relations between derived invariance of Hodge numbers and derived invariance of non- vanishing loci have been explored in [Pop13, Lom14, LP15] following [PS11]. In [CP18], it is proved that derived equivalence betweenX₁andX₂having maximal Albanese dimension implies the equalityh^0,p(X₁) =h^0,p(X₂), for allp. Some results similar to that of Popa and Schnell have been obtained in positive characteristic, let us mention [HACMV18] for instance.

In [Abu17], the notion of homological unit was introduced for a large classe of triangulated category as a replacement of the algebraH^•(X,OX)when the category under study is not nec- essarily the derived category of an algebraic variety. We proved that the homological unit is invariant under derived equivalences in some cases, most notably when the derived equivalent varieties have dimension less or equal to four. Using this result, we deduced that derived equiv- alent varieties of dimension4 have the same Hodge numbers, except perhaps for their h^1,1 and h^2,2.

The homological unit is a natural sub-algebra of the Hochschild cohomology of a triangulated category. In the present paper, we introduce a dual version of the homological unit, which we call the dual homological unit. This is a natural graded sub-vector space of the Hochschild homology of a triangulated category. If X is a smooth projective variety, it is easily seen that the dual homological unit of D^b(X) is the graded vector space H^•(X, ω_X). Our main result is the following:

Theorem 1.0.2 LetX1 andX2be smooth projective varieties over the eld of complex numbers.

Assume that D^b(X1)'D^b(X2), then we have:

H^•(X₁, ω_X1)'H^•(X₂, ω_X2), as graded vector spaces.

As an obvious corollary, we get:

Corollary 1.0.3 LetX₁andX₂be smooth projective varieties over the eld of complex numbers.

Assume that D^b(X₁)'D^b(X₂), then we have:

h^0,p(X1) =h^0,p(X2), for all p.

The proof of Theorem 1.0.2 is based on a detailed study of dual homological units associated to Chow-theoretic rank functions (see section2) on a smooth projective variety. Using a recent result of Vishik [Vis], we prove that the dual homological unit associated to any non-trivial Chow-theoretic rank function on X embeds in H^•(X, ωX). Theorem 1.0.2 then easily follows from this . The result of Vishik uses in a crucial way the existence of embedded resolution of singularities for any algebraic variety (which is only known over a eld of characteristic zero). If

such resolutions were proved to exist for algebraic varieties over elds of positive characteristics, our results and their proofs could be extended in this setting with only minor modications. In prticular, it seems then very likely that our arguments can be mimicked for threefolds over elds of positive characteristics, since embedded resolution is known for surfaces over any eld.

2 Cohomological rank functions, associated dual homological units

In this section, we give some denitions related to[Abu17], adapted in the special case of the derived category of a smooth projective variety. We work over a eld (not necessarilyCfor now) denoted by k. If X is a projective variety over k, then D^b(X) denotes the bounded derived category of coherent sheaves on X.

Denition 2.0.1 Let X be a smooth projective variety over k. A rank function on D^b(X) is a function rk : D^b(X) −→ Z which is additive with respect to exact triangles and such that rk(F[1]) =−rk(F), for any F ∈D^b(X).

We say that the rank function is trivial if it is the zero function and we say that it is a Chow- theoretic rank function if there is an object v ∈ CH^•(X) such that for any E ∈D^b(X), we have:

rk(E) = Z

X

ch(E)∩v,

where ch : D^b(X) −→CH^•(X) is the Chern character to the Chow ring of X. In that case, we denote by rk_v the rank function associated to v∈CH^•(X).

Denition 2.0.2 (Dual homological unit) LetX be a smooth projective variety andrkbe a non-trivial rank function on D^b(X). A graded algebra I^•_X is called a dual homological unit for X with respect to rk, if I^•_X is maximal for the following property. For any object E ∈ D^b(X), there exists a pair of morphisms i_E : I^•_X −→ Hom^•_Db(X)(E, S_X(E)[−dimX]) and t_E : Hom^•_Db(X)(E, SX(E)[−dimX])−→I^•_X (whereSX( ) =⊗ω_X[dimX]is the Serre functor ofX) such that:

• the morphism i_E :I^•_X −→ Hom^•_Db(X)(E, S_X(E)[−dimX]) is a graded vector spaces mor- phism which is functorial in the following sense. Let E,F ∈ D^b(X) and let a ∈ I^k_X for somek. Then, for any morphism ψ:E −→F, there is a commutative diagram:

E ^iE(a)//

ψ

SX(E)[k−dimX]

ψ[k]

F ^iF(a)//SX(F)[k−dimX]

• the morphism t_E : Hom^•_Db(X)(E, SX(E)[−dimX])−→ I^•_X is a graded vector spaces mor- phism such that for anyE ∈D^b(X) and any a∈I_X, we have t_E(i_E(a)) = rank(E).a. With hypotheses as above, an objectE ∈D^b(X) is said to be unitary, if

Hom^•_Db(X)(E, SX(E)[−dimX])'I^•_X

as graded vector spaces, where I^•_X is a dual homological unit for D^b(X) with respect to rk. In case the rank function is cohomological and associated to a classv∈CH^•(X), a dual homological unit with respect to v will be denoted by I^•_X,v.

Remark 2.0.3 1. By maximal with respect to inclusion, we mean that for any algebra B^• satisfying the conditions in the above denition, any graded vector spaces injective morphismr^•:I^•_X ,−→B^• is necessarily an isomorphism.

2. LetX be a smooth projective variety and let us consider the rank of anOX-module as a rank function onD^b(X). This is the Chow-theoretic rank function associated to the class of any point[k(x)]∈CH^dimX(X). In such a case, we have I^•_X =H^•(X, ωX), as follows from the maximality condition imposed in the denition of dual homological unit. Furthermore, for any E ∈ D^b(X), the morphism i_E is the tensor product (over OX) with the identity map of E and the morphism t_E is the trace mapHom^•_Db(X)(E,E ⊗ωX)−→H^•(X, ωX). 3. In the above denition, the existence of the morphism i_E for all E ∈ D^b(X) and its

functorial properties is equivalent to the existence of a morphism of graded vector spaces:

I^•_X −→HH•(X)[−dimX],

where HH•(X) is the Hochschild homology of X. If the rank function on D^b(X) is non- trivial, the splitting property of t^• implies that the map T^•_X −→ HH•(X)[−dimX] is injective.

4. On the other hand, the denition and the (splitting) properties of the morphisms t_E, for E ∈ D^b(X) with non-zero rank do not seem to be easily written using only the notion of graded morphisms between HH•(X) and T^•_X. It appears that there is no obvious way to write that t_E splits i_E whenever the rank of E is not zero only in terms of Hochschild homology.

5. If D^b(X) contains a unitary object whose rank is not zero, then the dual homological unit (with respect to the given rank function) ofD^b(X) is necessarily unique (though the embedding of the homological unit in HH•(X)[−dimX] is certainly not unique). This follows from the maximality condition imposed in denition 2.0.2.

We now state and prove the main result of tis section:

Theorem 2.0.4 Let X be smooth projective variety over C and let v ∈ CH^•(X) such that the associated rank functionrk_v is non trivial. LetI^•_X,v be a dual homological unit associated to rk_v. We have an graded vector spaces injective morphism I^•_X,v ,−→H^•(X, ωX).

Before proving Theorem 2.0.4, we shall recall a result due to Vishik [Vis] which will be central in our argument.

Theorem 2.0.5 (Theorem 6.1 in [Vis]) Let X be a smooth projective variety over C and v ∈ CH^•(X). There exists a blow-up π : ˜X −→ X, with X˜ smooth such that π^∗(v) is in the Z-algebra generated by Chern classes of divisors in CH^•( ˜X).

With this result in hand, we can turn to the proof of Theorem 2.0.4:

Proof of Theorem 2.0.4 : I

The rank function associated to v is non-trivial, hence there exists E ∈ D^b(X) such that R

Xch(E) ∩v 6= 0. By the above-quoted result, there exists a blow-up π : ˜X −→ X with X˜ smooth such that π^∗(ch(E)) is in the Z-algebra generated by Chern classes of divisors in CH^•( ˜X). Let us deduce that there exists L ∈ Pic( ˜X) such that rkv(Rπ∗(L)) 6= 0. Indeed, as- sume that rk_v(Rπ∗(L)) = 0 for all L ∈ Pic( ˜X). For any n ∈ N, for any P ∈ Z[X₁, . . . , X_n] and L1, . . . , Ln∈Pic( ˜X), we denote by P(L1, . . . , Ln) the corresponding class in K(X) (where

multiplication is the tensor product). Since rk_v(Rπ∗(L)) = 0for all L ∈ Pic( ˜X), we have, for any P ∈Z[X₁, . . . , X_n]and anyL₁, . . . , L_n∈Pic( ˜X):

0 = rkv(π!(P(L1, . . . , Ln)))

= Z

X

ch(π_!(P(L₁, . . . , L_n)))∩v

= Z

X

Td(X)⁻¹∩Td(X)∩ch(π_!(P(L₁, . . . , L_n)))∩v

= Z

X

Td(X)⁻¹∩v∩π∗(ch(P(L₁, . . . , L_n))∩Td( ˜X)), by the Grothendieck-Riemann-Roch Theorem,

= Z

X˜

π^∗(Td(X)⁻¹)∩π^∗(v)∩ch(P(L₁, . . . , L_n))∩Td( ˜X), by the projection formula,

= Z

X˜

π^∗(Td(X)⁻¹)∩π^∗(v)∩P(ch(L₁), . . . ,ch(L_n))∩Td( ˜X), by the properties of the Chern character. On the other hand, we have:

rkv(E) = Z

X

ch(E)∩v

= Z

X

Td(X)⁻¹∩Td(X)∩ch(E)∩v

= Z

X

Td(X)⁻¹∩v∩ch(E)∩ch(OX)∩Td(X)

= Z

X

Td(X)⁻¹∩v∩ch(E)∩ch(Rπ∗OX˜)∩Td(X), since Rπ∗OX˜ =OX,

= Z

X

Td(X)⁻¹∩v∩ch(E)∩π∗(ch(OX˜))∩Td( ˜X)), by the Grothendieck-Riemann-Roch Theorem,

= Z

X˜

π^∗(Td(X)⁻¹)∩π^∗(v)∩π^∗(ch(E))∩ch(OX˜)∩Td( ˜X), by the projection formula,

= Z

X˜

π^∗(Td(X)⁻¹)∩π^∗(v)∩π^∗(ch(E))∩Td( ˜X).

We know that π^∗(ch(E))is in the Z-algebra generated by Chern classes of divisors inCH^•(X). Since theQ-algebra generated by classes of divisors is equal to theQ-algebra generated by Chern characters of line bundles in CH^•(X)_Q, we deduce that that there exists integers m, n ∈ N, a polynomialP ∈Z[X₁, . . . , X_n]and line bundlesL₁, . . . , L_n∈Pic( ˜X) such that:

m.π^∗(ch(E)) =P(ch(L₁), . . . ,ch(L_n)).

The vanishing ofR

X˜π^∗(Td(X)⁻¹)∩π^∗(v)∩P(ch(L1), . . . ,ch(Ln))∩Td( ˜X)and the non-vanishing of R

X˜π^∗(Td(X)⁻¹)∩π^∗(v)∩π^∗(ch(E))∩Td( ˜X) is a contradiction. As a partial conclusion:

There exists a smooth blow-up π: ˜X −→X and L∈Pic( ˜X), such that rk_v(Rπ∗L)6= 0

LetL∈Pic( ˜X) such a line bundle. By denition of the dual homological unit, we have:

• a injective morphism of graded vector spaces i^• :I^•_X,v ,−→HH•(X)[−dimX],

• for any a∈I^•_X,v,t^•_Rπ

∗(L)(i^•_Rπ

∗(L)(a)) = rkv(Rπ∗(L)).a, withrkv(Rπ∗(L))6= 0.

Since Rπ∗OX˜ = OX, the functor Lπ^∗ : D^b(X) −→ D^b( ˜X) is fully faithful with left and right adjoint. The category D^b(X) is then a semi-orthogonal component of D^b( ˜X). A Theorem of Kuznetsov [Kuz09] ensures that we have graded direct sum decomposition:

HH•( ˜X) = HH•(X)⊕HH•(A),

where A is the right orthogonal complement of D^b(X) in D^b( ˜X). As a consequence, for any k∈Nand anya∈I^k_X,v, we have Lπ^∗(i^k(a))⊕0∈HH•( ˜X)and there is a commutative diagram:

Lπ^∗Rπ∗L ^//

Lπ^∗(i_Rπ∗(L)(a))

L ^//

Lπ^∗(i_Rπ∗(L)(a))⊕0

CL

0

Lπ^∗(Rπ∗L⊗ω_X)[k] ^//L⊗ω_˜X[k] ^//S_A(CL)[k−dimX]

whereS_A is the Serre functor ofA andCLis the cone of the adjunction morphism Lπ^∗Rπ∗L−→

L. The map Lπ^∗(i_Rπ∗(L)(a))⊕0 being non-zero as long as a 6= 0, we deduce that the graded vector space morphism:

Lπ^∗(i^•_Rπ∗(L)( ))⊕0 :I^•_X,v −→Hom^•_Db( ˜X)(L, L⊗ωX˜) =H^•( ˜X, ωX˜)

is injective. We conclude the proof of Theorem 2.0.4 with the Grauert-Riemenschneider Theorem

:H^•( ˜X, ω_X˜) =H^•(X, ω_X). J

Remark 2.0.6 The proof of Theorem 6.1 in [Vis] relies on the existence of embedded resolution of singularities for any algebraic variety embedded in a smooth algebraic variety. This is why this result is stated over C. Since embedded resolutions exist for surfaces in positive characteristic, it is likely that Theorem 6.1 in [Vis] is also true for threefolds dened over a eld of positive characteristic. The proof of Theorem 2.0.4 could then certainly be adapted so that it holds for threefolds dened over any eld.

3 Derived invariance of the graded algebra H

(X, ω

)

In this section, we prove our main result, namely Theorem 1.0.2. LetX1 andX2 be two smooth projective varieties overCsuch thatD^b(X₁)'D^b(X₂). By a result of Orlov [Orl03], there exists E ∈D^b(X₁×X₂) such that the equivalence between D^b(X₁) andD^b(X₂) is of the form:

Φ_E : D^b(X2)−→D^b(X1) G 7−→Rp∗((Lq^∗G)⊗^LE),

where p and q are the natural projections from X₁ ×X₂ to X₁ and X₂. In the following, we will also denote by Φ_E the corresponding transformation fromCH^•(X2)to CH^•(X1). This map is not graded, but it is an isomorphism of vector spaces since Φ_E^∨⊗q^∗ωX2 ◦Φ_E = Φ_O∆ and Φ_E^∨⊗q^∗ωX2 ◦Φ_E = Φ_O∆. Let us nally recall that for anyF,G ∈D^b(X₂), we have:

Z

X1

Φ_v(E)(ch(F))∩(Φ_v(E)(ch(G)))^∨∩exp

c1(X1) 2

= Z

X2

ch(F)∩(ch(G))^∨∩exp

c1(X2) 2

, where v(E) = ch(E)∩p

Td(X1×X2). This means that Φ_v(E) is an isometry with respect to the Mukai Pairing, see [Cal05] for a proof of this result.

We start with a result relating the natural rank function on D^b(X₂) to rank functions on X1:

Proposition 3.0.1 Let X₁ and X₂ be smooth projective varieties over C with an equivalence Φ_E : D^b(X₁) ' D^b(X₂). Let rk₂ : D^b(X₁) −→ Z be the rank function dened by rk₂(F) = rk_OX

2(Φ_E(F)), where rk_OX

2 is the rank of an object inD^b(X2) as aOX2-module. Then rk2 is a non-trivial Chow-theoretic rank function on D^b(X1).

Proof :

I We use the same notations as above and introduceα₂ ∈CHⁿ(X₂)the class of a point (where n= dimX2). For allF ∈D^b(X1), we have:

rk₂(Φ_E(F)) = Z

X2

α₂∩ch(Φ_E(F))

= Z

X2

α2∩ch(Φ_E(F))∩p

Td(X2), since p

Td(X2) = 1 +. . .,

= Z

X2

α2∩Φ_v(E)(v(F)), by the Grothendieck-Riemann-Roch Theorem,

= (−1)ⁿ Z

X2

(α₂)^∨∩Φ_v(E)(v(F)), since α₂ is concentrated in degreen, (−1)ⁿ

Z

X2

(α2)^∨∩Φ_v(E)(v(F))∩exp

c₁(X₂) 2

, sinceexp

c₁(X₂) 2

= 1 +. . .,

= (−1)ⁿ Z

X2

Φ_v(E)(Φ⁻¹_v(E)(α2)) ∨

∩Φ_v(E)(v(F))∩exp

c1(X2) 2

= (−1)ⁿ Z

X1

Φ⁻¹_v(E)(α2) ∨

∩v(F)∩exp

c1(X1) 2

, since Φ_v(E) is isometric with respect to the Mukai pairing,

= Z

X1

ch(F)∩β₁, withβ₁ = (−1)ⁿ

Φ⁻¹_v(E)(α₂)∨

∩exp

c₁(X₁) 2

∩p

Td(X₁).

This proves that rk2 is indeed a Chow-theoretic rank function on D^b(X1). It is obviously non- trivial sinceΦ_E induces an equivalence betweenD^b(X₁)'D^b(X₂) andD^b(X₂)contains object

which rank are not zero asOX2-modules. J

Using this proposition and Theorem 2.0.4, we an now prove:

Theorem 3.0.2 Let X1 and X2 be smooth projective varieties over C such that D^b(X1) ' D^b(X₂). Then there is a graded isomorphism of vector spaces H^•(X₁, ω_X1) 'H^•(X₂, ω_X2). In particular, we have h^0,p(X1) =h^0,p(X2) for all p.

Proof :

I Let E ∈ D^b(X1×X2) be the kernel giving the equivalence and let rk2 : D^b(X1) −→ Z be the rank function dened by rk₂(F) = rk_OX

2(Φ_E(F)), whererk_OX

2 is the rank of an object in D^b(X₂)as anOX2-module. Proposition 3.0.1 implies thatrk₂is a non trivial Chow-theoretic rank function on D^b(X1). Furthermore, since rk2 is the natural rank function on D^b(X2)'D^b(X1), the homological unit associated to rk2 is H^•(X2, ω_X2) (see remark 2.0.3). Theorem 2.0.4 then implies that we have a graded vector spaces embedding:

H^•(X2, ω_X2),−→H^•(X1, ω_X1).

The role ofX1 and X2 being symmetric, we also have a graded embedding:

H^•(X1, ωX1),−→H^•(X2, ωX2).

This nally implies the existence of a graded vector space isomorphism H^•(X₁, ω_X1) '

H^•(X₂, ω_X2). J

As mentioned in the introduction the equality of Hodge numbers for derived equivalent smooth projective varieties is related to the existence of isomorphisms between the corresponding non-vanishing loci. Namely we have:

Corollary 3.0.3 Let X1 and X2 be two smooth projective varieties over C. For i = 1,2 and for any k ∈ [0, . . . ,dimXi], dene V^k(Xi, ω_Xi) := {L ∈ Pic⁰(Xi), h^k(Xi, L⊗ω_Xi) 6= 0}. If D^b(X₁)'D^b(X₂), then for all k∈[0, . . . ,dimX₁], we have

V^k(X₁, ω_X1)₀ 'V^k(X₂, ω_X2)₀,

whereV^k(Xi, ωXi)0 is the union of the irreducible components of V^k(Xi, ωXi)passing through 0. Proof :

I Sinceh^0,k(X₁) =h^0,k(X₂)for allkby Theorem 3.0.2, this corollary is a direct application of the main result in [LP15].

J Question 3.0.4 Theorem 3.0.2 in the present paper can be restated as follows : let X₁ and X₂ be smooth projective varieties over C such that D^b(X₁) ' D^b(X₂), then there exists an isomorphism graded of vector spacesH^•(X1,OX1)'H^•(X2,OX2). In [Abu17], we proved that if X₁ and X₂ are derived equivalent smooth projective varieties of dimension less or equal to 4, then there is a graded algebras isomorphism H^•(X₁,OX1)'H^•(X₂,OX2). It would certainly be desirable to know if such an algebra isomorphism exists in dimension bigger than 4.

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