7_wang.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 5, No. 4, 2012, 511-539

ISSN 1307-5543 – www.ejpam.com

Koszul Duality for Multigraded Algebras

F. T. Hawwa 1, J. William Hoffman1, Haohao Wang 2,∗

1 Department of Mathematics, Louisiana State University, Baton Rouge, LA, USA
2 Department of Mathematics, Southeast Missouri State University, Cape Girardeau, MO, USA

Abstract. Classical Koszul duality sets up an adjoint pair of functors, establishing an equivalence

F : Db(A)⇆ Db(A!) : G, where A is a quadratic algebra, A! is the quadratic dual, and Db refers to the

bounded derived category of complexes of graded modules over the graded algebra (i.e., A or A!). This

duality can be extended in many ways. We consider here two extensions: first we wish to allow a

Λ-graded algebra, where Λ is any abelian group (not just Z). Second, we will allow filtered algebras.

In fact we are considering filtered quadratic algebras with an (internal) Λ-grading.

2010 Mathematics Subject Classifications: 14F05, 16E05, 13D25, 13D02, 18E30.

Key Words and Phrases: Multigraded module, Functors, Koszul duality, BGG correspondence, Derived

category

1. Introduction

Koszul duality originated from Bernstein-Gelfand-Gelfand [2] in the mid 1970s. It led to

the observation that for certain pairs of associative algebras A and A!, there is a relationship

between the categories of A-modules and A!-modules. An example of such a pair is

S = k[x1, x2, . . . , xn], the polynomial algebra over a field k, and E =
∧

k(e1, e2, . . . , en), the

exterior algebra over k.

Bernstein-Gelfand-Gelfand constructed an adjoint pair of functors between the categories

of bounded chain complexes of S-modules and bounded chain complexes of E-modules which

induce an equivalence of the corresponding derived categories. This means that problems of

homological algebra for S-modules can be translated into problems of homological algebra

for E-modules and vice-versa. A key fact underlying this is that the Koszul complex of S and

E which is given by

· · · → Si ⊗ En→ Si+1 ⊗ En−1→ · · ·

∗Corresponding author.

Email addresses: fhawwa�math.lsu.edu (F. Hawwa), hoffman�math.lsu.edu (J. Hoffman),

hwang�semo.edu (H. Wang)

http://www.ejpam.com 511 c© 2012 EJPAM All rights reserved.



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 512

with differential

d( f ⊗ ei1
∧ ei2

∧ · · · ∧ eim
) =

m∑

j=1

(−1)mx i j
f (ei1

∧ · · · ∧cei j
∧ · · · ∧ eim

)

is acyclic in all degrees greater than zero. Given this fact we say that S and E are Koszul

algebras. This theory has been generalized by a number of people to algebras defined by

homogeneous relations. A general reference is [5] for background on quadratic algebras and

Koszul duality. Consider the free k-algebra k < x1, x2, . . . , xn > and define

A = k < x1, x2, . . . , xn > /R, where R is the ideal generated by homogeneous quadratic rela-

tions of the form Σci j x i x j = 0. In our example above, the algebra S is defined by the quadratic

relation x i x j − x j x i = 0 and the algebra E is defined by x i x j + x j x i = 0 and x2
i = 0.

More generally S and E can be replaced by a pair of dual quadratic algebras A and A!.

Note that this relation is symmetric, (A!)! = A. Here A and A! are quadratic algebras meaning

that each is the quotient of a free k-algebra by homogeneous quadratic relations. Given a pair

of quadratic algebras we may form the Koszul complex given by

. . .→ A⊗ (A!
2)
∗→ A⊗ (A!

1)
∗→ A

and we say that A is a Koszul algebra if the Koszul complex is exact in nonzero degrees. By

symmetry, if A is Koszul then A! is also Koszul.

Koszul duality is a relation between the complexes of A-modules and the complexes of

A!-modules and this establishes an equivalence of categories

F : Db(A)⇆ Db(A!) : G

where Db refers to the bounded derived category of complexes of graded modules over the

graded algebra A (or A!). For M ∈ C b(A) where C b(A) is the category of bounded chain

complexes of graded left A-modules, the functor F is given by

(F M)pq =
⊕

p=i+ j

q=l− j

A!
l ⊗M i

j .

For N ∈ C b(A!), the functor G is explicitly described as

(GN)pq =
⊕

p=i+ j

q=l− j

Homk(A−l , N i
j ).

Another example concerns a nondegenerate quadratic form, Q, in variables x0, x1, . . . , xn.

Let A= k[x0, x1, . . . , xn]/Q = Sym(V )/Q be the homogeneous coordinate ring of the quadric

Q = 0 in projective space Pn
k
. The dual A! is the graded Clifford algebra attached to Q. This is

generated by elements ξ ∈ V ∗ of tensor degree 1 and an element h of tensor degree 2 with

the relations

ξη+ηξ = 2Q(ξ,η)h, ξ,η ∈ V ∗.



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 513

The ring A is proven to be a Koszul ring in [4].

Not every quadratic algebra is Koszul. A counterexample is

A=
∧

k

(x , y, z, w)/(x y + zw).

Using the software package Singular to calculate the minimal free resolution of A, we observed

that quadratic entries appear within the matrices, violating the property that an algebra is

Koszul if and only if k has a linear free A-module resolution.

If we now allow the relations, R, of an algebra A = k < x1, x2, . . . , xn > /R to be nonho-

mogeneous, A will no longer be a graded algebra, but rather a filtered algebra. The dual of A

will no longer be the quadratic algebra A!, but rather the curved differential graded algebra

(CDGA) (A!, d , c). That is, we have a differential d : (A!)n → (A!)n+1 with the usual property

that d(x y) = d x(y) ± (x)d y but with d2(x) = [c, x] = cx ± xc, where c ∈ (A!
2)
∗ is the

curvature. In the case where c = 0 we refer to a CDGA as just a DGA (differential graded

algebra). Two canonical examples of DGAs are the Koszul complex and the deRham Complex.

Let us consider an example that illustrates dualizing a nonhomogeneous quadratic algebra.

Consider the algebra,

U = k < x , y > /P, P = (x2− y, x y − y x).

The associated graded algebra to U will be A = k < x , y > /(x2, x y − y x) which is dual to

A! = k < ξ,η > /(η2,ξη+ ηξ) which can be seen to be a curved differential graded algebra

(A!, d , c) with c = 0, dξ= 0 and dη= −ξ2.

A canonical example of a nonhomogeneous quadratic algebra which is filtered by tensor

degree is U = Ug, the universal enveloping algebra of a Lie algebra g. The dual of Ug is the

Chevellay-Eilenberg complex which is a CDGA (more specifically it is a DGA since c = 0).

Given a filtered algebra, we can add a Λ-grading to it. A Λ-graded filtered algebra, U , is

an algebra with a filtration, Fi , and also a grading

U =
⊕

λ∈Λ

Uλ for some abelian group Λ.

If we allow V to be Λ-graded, and P is Λ-homogeneous, then we know that

U = T (V )/P

will be filtered by tensor degree and Λ-graded. Two examples of Λ-graded algebras are the

universal enveloping algebra of a semisimple Lie algebra, and the coordinate ring of a projec-

tive toric variety.

There is no obvious way to extend Koszul duality to a Λ-graded situation when Λ 6= Z.
Consider the functor F given by

(F M)pq =
⊕

p=i+ j

q=l− j

A!
l ⊗M i

j .



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 514

This functor is well defined for the case Λ = Z but if A is Λ-graded then we would have i ∈ Z
but j ∈ Λ. Now the addition i + j no longer makes sense since i is a scalar and j is a vector.

The key step to our extension of Koszul duality is considering A!
l

not as a quadratic algebra

but as a CDGA, meaning

A!
l
= (A!)lλ l ∈ Z,λ ∈ Λ,

with the multiplication

(A!)lλ × (A
!)mµ −→ (A

!)l+m
λ+µ

.

The benefit of switching over from a quadratic algebra point of view to a CDGA point of view

is that now we are allowed two new levels of freedom. First we may insert a Λ-grading and

second we can now let A be a nonhomogeneous filtered algebra.

We define ComΛ(A
!, d , c) to be the category of curved differential graded modules over

the CDGA (A!, d , c). We define ComΛ(U) to be the category of complexes of Λ-graded left

U-modules. The version of Koszul duality that we are most interested in is that introduced by

Floystad [3] concerning the pair of adjoint functors

F : ComΛ(A
!, d , c)⇆ ComΛ(U) : G.

Here U is a Λ-graded filtered quadratic algebra and (A!, d , c) is a curved differential graded

algebra (cdga) which is dual to U . We have generalized the functors F and G for this Λ-graded

situation, i.e.,

F(N)
p

λ
=
⊕

µ+ν=λ

Uµ ⊗k N p
ν

G(M)
p

λ
=
∏

r≥0

∏

µ

Homk((A
!)rµ, M

p+r

λ+µ
).

Notice that now integers are added to integers and elements of Λ are added to other elements

of Λ.

Let KΛ(A
!, d , c) be the category ComΛ(A

!, d , c) with morphisms being chain homotopy

equivalence classes of maps. The null system, N , of KΛ(A
!, d , c) is defined to be all of the

complexes, X , such that F(X ) is acyclic. A similar definition is given for the null system of

KΛ(U). We define DΛ(A
!, d , c) to be the category KΛ(A

!, d , c)/N where N is the null system of

ComΛ(A
!, d , c). We also define DΛ(U) to be the category KΛ(U)/N where N is the null system

of ComΛ(U).

The main result of this paper is that the functors F and G given in Proposition 4 induce an

equivalence of categories between the quotient categories DΛ(A
!, d , c) and DΛ(U).

2. Graded and Filtered Algebras

Floystad’s Koszul duality concerns a pair of adjoint functors

F : ComΛ(A
!, d , c)⇆ ComΛ(U) : G



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 515

which induce an equivalence of homotopy categories. Here U is a Λ-graded filtered quadratic

algebra, and (A!, d , c) is a curved differential graded algebra (CDGA) dual to U . Here ComΛ(U)

is the category of complexes of Λ-graded left U-modules (of finite type). We let ComΛ(A
!, d , c)

be the category of curved differential graded left modules over the CDGA (A!, d , c). When

c = 0 these are just the usual differential graded algebras. It is important to note that this

form of duality is not symmetrical.

We will explain how classical Koszul duality relates to Floystad’s Koszul duality in the

special case U = A is a Z-graded Koszul quadratic algebra. So first let us recall the definitions

of some of the key terms we will be using.

Definition 1. Let k be a field. A Λ-graded associative k-algebra A with unit is an algebra together

with a decomposition into k-subspaces,

A=
⊕

λ∈Λ

Aλ

which obeys the multiplication law

Aλ ·Aµ ⊂ Aλ+µ.

Definition 2. A Λ-graded filtered algebra U over a field k is a Λ-graded algebra which has an

increasing sequence 0⊂ F0 ⊂ F1 ⊂ · · · Fi ⊂ · · · ⊂ U of k-subspaces of U such that

U =
⋃

i∈N

Fi

and the following property of the algebra multiplication holds:

∀m, n ∈ N, Fm · Fn ⊂ Fn+m.

In addition, the grading must be compatible with the filtration, meaning

Fi =
⊕

λ∈Λ

(Fi ∩ Uλ) given the grading U =
⊕

λ∈Λ

Uλ.

Definition 3. If U is a Λ-graded ring, then a Λ-graded module M is a left U-module with a

decomposition

M =
⊕

λ∈Λ

Mλ

into k-subspaces such that Uλ ·Mµ ⊂ Mλ+µ.

Remark 1. We allow the case where there is no Λ-grading in which case the algebra (resp.

module) U is just considered a filtered algebra (resp. module). We also will allow the case where

there is neither a filtration nor a grading in which case U is just an algebra.

Definition 4. Let ComΛ(U) be the category of chain complexes of Λ-graded left U-modules M p

(of finite type). The differentials will be U-linear maps which preserve the Λ degree, i.e.,

M
p

λ

d
−→ M

p+1

λ
.

If there is no Λ-grading then we use Com(U) to denote the category of chain complexes of left

U-modules.



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 516

A general construction that we will be utilizing is that of a quadratic filtered algebra. Let k

be a field, Λ an abelian group, and V a Λ-graded, finite dimensional vector space over k, i.e.,

V =
⊕

λ∈Λ

Vλ.

We can form the tensor algebra

T (V ) = V ⊕ (V ⊗ V )⊕ (V ⊗ V ⊗ V )⊕ · · ·

which will also be Λ-graded. For each v1 ⊗ . . . ⊗ vn ∈ T (V ), each vi has a Λ grading, λi,

and the total degree will be the sum of all the λi. Let P be a Λ-graded sub-vector space of

k⊕ V ⊕ (V ⊗ V ) such that P ∩ (k⊕ V ) = 0. Let p0(P), p1(P), and p2(P) be the projections of

P onto k, V , and (V ⊗ V ) respectively. Let R= p2(P), now we may then define

U = T (V )/ < P >

to be filtered by tensor powers and Λ-graded. In the case where P = R, U = A is said to be a

quadratic algebra defined by R and this quotient induces an epimorphism Φ : A→ grU . Here

grU is the associated graded algebra of U defined by

U =
⊕

i∈Z

Fi+1/Fi .

If Φ is an isomorphism then we say that U is of Poincaré-Birkho ff-Witt (PBW) type.

Definition 5. Given a quadratic algebra A defined by R ⊂ V ⊗ V , we may dualize this inclusion

and get an exact sequence

0→ R⊥→ V ∗ ⊗ V ∗→ R∗→ 0.

The algebra A! = T (V ∗)/R⊥ is called the quadratic dual algebra of A.

Now, assuming that (k ⊕ V ) ∩ P = 0, the map P → p2(P) = R is a bijection, so we can

define maps α : R→ V and β : R→ k as

α : R
p−1

2−−−→ P
p1−−−→ V, β : R

p−1
2−−−→ P

p0−−−→ k.

Then

P =
�

x +α(x)+ β(x)|x ∈ R
	

.

Now let A! be the dual algebra of A. Dualizing the maps α and β we have

A!
1 = V ∗

α∗

−−−→ A!
2 = R∗, k

β∗

−−−→ A!
2 = R∗.

Example 1. First let us examine the classical case. Consider the tensor algebra, T (V ), of the

Λ-graded vector space V , where Λ = Z. Now assign deg(v) = 1 for v ∈ V . It is easy to see that

the Λ-degree will equal the tensor degree meaning that the filtration and grading of T (V ) will be

one in the same. Given R⊂ V ⊗ V ,

U = A= T (V )/ < R> .



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 517

Example 2. Now let us examine a special case of a Λ-graded filtered algebra. Let g be a semisim-

ple Lie algebra, where Λ is the lattice of weights of g. It is known that we have a decomposition

g =
⊕

λ∈Λ

gλ, [gλ,gµ]⊂ gλ+µ, gλ = {x ∈ g | [h, x] = λ(h)x ∀h ∈ h}

where h is a Cartan subalgebra. If we consider the relation x⊗ y− [x , y] for x ∈ gλ and y ∈ gµ,

we can see that deg(x ⊗ y) = deg(([x , y]) = λ+ µ. By definition, the universal enveloping

algebra of g is

U = Ug = Tg/J

where Tg is the tensor algebra on g and J =< x ⊗ y − [x , y] >. We can see that since J is

generated by homogeneous relations, U is graded by Λ and is filtered by the tensor degree.

3. Curved Differential Graded Algebras

A Λ-graded curved differential graded algebra (CDGA) (B, d , c) over k is a cohomologically

graded k-algebra such that

B =
⊕

p∈Z

Bp, Bp =
⊕

λ∈Λ

B
p

λ
, Bp · Bq ⊂ Bp+q, B

p

λ
· Bq
µ = B

p+q

λ+µ
,

where the differential d is a k-linear map such that d : Bp → Bp+1 with

d2(b) = [c, b] = cb+ (−1)deg(b)deg(c)bc, d(b1b2) = d(b1)b2+ (−1)|b1|b1d(b2).

Note that when Λ = 0 we have the notion of a curved differential graded algebra, if c = 0 we

have the notion of a Λ-graded differential graded algebra, and if both are zero then we simply

have a differential graded algebra.

Definition 6. A Λ-graded left curved differential graded module (CDGM) (N , d , c) over a CDGA

(B, d , c) is a graded left B-module N with a k-linear map dN such that

N =
⊕

i∈Z

N i, N i =
⊕

λ∈Λ

N i
λ, Bi

λ · N
j
µ ⊂ N

i+ j

λ+µ
,

dN : N i
λ→ N i+1

λ
, d2

N (n) = cn, dN (bn) = dB(b)n+ (−1)|b|bdN (n), b ∈ B, n ∈ N .

Note that for a Λ-graded right curved differential graded module we have N
j
µ · B

i
λ
⊂ N

i+ j

λ+µ

and d2
N (n) = −cn.

We let ComΛ(B, d , c) be the category of these curved differential graded modules, with the

evident morphisms. When c = 0, we have simply a differential graded module.



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 518

Example 3. The following example is due to Floystad [3]. Let k be a field and let U be the

following filtered quadratic algebra,

U = k[x]/(x2− (a− b)x + ab) = k[x]/(x − a)⊕ k[x]/(x − b).

The dual of U will be A! = k[ξ] which is the CDGA with differential

d(ξn) =

(
−(a+ b)ξn+1 if n is odd

0 if n is even

and curvature c = abξ2. More details of this CDGA will be provided in Sections 5 and 6.

4. Koszul Algebras

Let A be a quadratic algebra over a field k, so A = T (V )/(R). Its dual A! is the quadratic

algebra over k given by A! = T (V ∗)/(R⊥) with R⊥ ⊂ V ∗⊗ V ∗ = (V ⊗ V )∗.

We define the Koszul Complex of A to be the complex (isomorphic to)

. . .→ A⊗ (A!
2)
∗→ A⊗ (A!

1)
∗→ A

where (A!)i = A!
i
. We can note that A!

1 = V ∗ and A!
2 = (V

∗ ⊗ V ∗)/(R⊥) = R∗ so (A!
2)
∗ = R.

Before defining the differentials, recall the canonical isomorphism W ∗ ⊗k V = Homk(W, V )

defined by φ(λ⊗k v)(w) = λ(w)v. Now if A= ⊕ j∈ZA j, then

A⊗k (A
!
i)
∗ =
⊕

j∈Z

A j ⊗k (A
!
i)
∗ =
⊕

j∈Z

Homk(A
!
i,A j).

Since A!
i

is finite dimensional, we know that

⊕

j∈Z

Homk(A
!
i,A j) = Homk(A

!
i,
⊕

j∈Z

A j) = Homk(A
!
i,A)

via the isomorphism A⊗k (A
!
i
)∗ = Homk(A

!
i
,A). The differential δ : A⊗k (A

!
i+1
)∗ → A⊗k (A

!
i
)∗

carries over to a differential d : Homk(A
!
i+1,A) → Homk(A

!
i,A). We define d as follows. Let

f ∈ Homk(A
!
i+1,A) then d f ∈ Homk(A

!
i,A) is given by

d f (ǎ) =
∑

α

f (v̌αǎ)vα for ǎ ∈ A!
i,

where
�

vα
	

is any basis of V = A1, and
�

v̌α
	

is the corresponding dual basis of V ∗ = A!
1. In

fact, this formula for d f does not depend on the choice of the basis
�

vα
	
. If
�

wα
	

is another

basis for V , i.e. wα = Σgβαvβ , where gβα is an invertible matrix with entries in k, then we

may also define the dual basis of
�
wα
	

as
�

w̌α
	
= Σhεα v̌ε. Now,

δαβ =< v̌α, vβ >=< w̌α, wβ >=< w̌α,
∑

γ

gγβ vγ >=
∑

γ

gγβ < w̌α, vα > .



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 519

If we substitute in for w̌α, we have

δαβ =
∑

γ

gγβ <
∑

ε

hεα v̌ε, vγ >=
∑

γ,ε

gγβhεα < v̌ε, vγ >=
∑

γ,ε

gγβhεαδεγ.

Since δεγ vanishes unless ε= γ we may rewrite the last term above as

δαβ =
∑

γ

gγβhγα

which can be rewritten as

δαβ =
∑

γ

(t g)βγhγα = (
t gh)βα.

This shows that t gh= 1, i.e., h= (t g)−1. So, if we have another basis for V ,

wα =
∑

β

gβαvβ ,

we know that

w̌α =
∑

γ

(t g)−1
γα v̌γ

and we would like to show that

d f (ǎ) =
∑

α

f (v̌αǎ)vα =
∑

α

f (w̌αǎ)wα.

Observe that ∑

α

f (w̌αǎ)wα =
∑

α

f (
∑

γ

(t g)−1
γα v̌γǎ)
∑

β

gβαvβ

allowing us to simplify to

∑

α

f (w̌α ǎ)wα =
∑

α,β ,γ

(t g)−1
γα gβα f (v̌γǎ)v̌β =

∑

α,β ,γ

gβαg−1
αγ f (v̌γǎ)v̌β .

Since Σgβα(g
−1)αγ = δ

γ

β
we may conclude that

∑

α

f (w̌αǎ)wα =
∑

β ,γ

(
∑

α

gβα(g
−1)αγ) f (v̌γǎ)v̌β =

∑

β

f (v̌γǎ)v̌β .

Now we will define δ so that the following commutes:

A⊗ (A!
i+1)

∗ δ
−−−→ A⊗ (A!

i)
∗

φ

y φ

y

Hom(A!
i+1

,A)
d

−−−→ Hom(A!
i
,A).



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 520

Since an element of A⊗ (A!
i+1)

∗ is a sum of tensors a⊗λ, a ∈ A, λ ∈ (A!
i+1)

∗ = Homk(A
!
i+1, k)

it is enough to define δ(a⊗λ). Set

δ(a⊗λ) =
∑

α

avα⊗λα,

where λα = λ(v̌α · −) ∈ (A
!
i
)∗ = Homk(A

!
i
, k) is the map ǎ 7→ λ(v̌αǎ). Let f = φ(a ⊗ λ) and

recall that φ(a⊗λ)( b̌) = λ( b̌)(a) giving us,

dφ(a⊗λ)(ǎ) = d f (ǎ) =
∑

α

f (v̌αǎ)vα =
∑

α

φ(a⊗λ)(v̌αǎ)vα,

so we have

dφ(a⊗λ)(ǎ) =
∑

α

φ(a⊗λ)(ǎv̌α)vα =
∑

α

λ(ǎv̌α)avα.

Now to verify that the diagram commutes, we must check that

dφ(a⊗λ)(ǎ) = φδ(a⊗λ)(ǎ). Given that δ(a⊗λ) =
∑
α avα⊗λα we know the following,

φδ(a⊗λ)(ǎ) =
∑

α

φ(avα ⊗λα)(ǎ) =
∑

α

λα(ǎ)avα =
∑

α

λ(v̌αǎ)avα

showing that the diagram commutes. Now we will need to show that d2 = δ2 = 0. If we can

show that δ2 = 0 then by duality we will know that d2 = 0. The map δ is defined as follows

δ(a⊗λ) =
∑

α

avα⊗λ(v̌α · −),

where δ : A⊗k (A
!
i+1
)∗→ A⊗k (A

!
i
)∗. Now define δ′ : A⊗k (A

!
i
)→ A⊗k (A

!
i+1
) by δ′ = xe where

e =
∑

v̌α ⊗ vα ∈ A! ⊗ A. Now we want to show that A⊗ A!
i+1
= [Homk(A

!
i+1

,A)]∗. Since any

k-linear map can be extended canonically to an A-linear map, and any A-linear map comes

from a k-linear map, we know that

(A!
i)
∗⊗ A= HomA(A

!
i ⊗ A,A) = Homk(A

!
i,A).

So A⊗ A!
i+1

and A⊗ (A!
i+1
)∗ are dual in the A-linear sense. Now since (δ′)2 x = xe2 and

assuming that e2 = 0 that implies that (δ′)2 = 0 implying that δ2 = 0 which, by duality, tells

us that d2 = 0. Now to show that e2 = 0 first note that

A!
2 ⊗ A2 = ((V

⊗2)∗/R⊥)⊗ (V⊗2/R)

= R∗ ⊗ (V⊗2/R)

= Hom(R, V⊗2/R).

Now consider the following diagram:

(A!
1⊗ A1)⊗ (A

!
1 ⊗ A1)

m
−−−→ A!

2 ⊗ A2

p

y q

y

Hom(V⊗2, V⊗2)
φ

−−−→ Hom(R, V⊗2/R),



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 521

where m is ring multiplication defined by

m[(ǎ1 ⊗ a1)⊗ (ǎ2 ⊗ a2)] = ǎ1ǎ2 ⊗ a1a2

with canonical isomorphisms p and q. Now given e =
∑

v̌α ⊗ vα ∈ A! ⊗ A, we know that

e2 = (
∑

v̌α ⊗ vα)(
∑

v̌β ⊗ vβ) =
∑
α,β v̌α v̌β ⊗ vαvβ by definition of the ring multiplication m.

We will show that
∑
α,β v̌α v̌β ⊗ vαvβ = 0.

Given f ∈ Hom(V⊗2, V⊗2), then g = φ( f ) is the composite

R ,→ V⊗2 f
→ V⊗2→ V⊗2/R.

Now we will check that the diagram commutes. To define p explicitly first recall,

(A!
1⊗ A1)⊗ (A

!
1 ⊗ A1) = (V

∗ ⊗ V )⊗ (V ∗ ⊗ V )

so we know that

p : (V ∗ ⊗ V )⊗ (V ∗ ⊗ V )→ Hom(V⊗2, V⊗2).

Now for (v̌α⊗ vβ)⊗ (v̌γ⊗ vδ) ∈ (V
∗⊗ V )⊗ (V ∗⊗ V ) and (vε ⊗ vθ ) ∈ V⊗2 define p as follows

p((v̌α⊗ vβ)⊗ (v̌γ⊗ vδ))(vε⊗ vθ ) = δ
α
εδ
γ

θ
vβ ⊗ vδ.

Let us check that p(e⊗ e) = id ∈ Hom(V⊗2, V⊗2), where

e⊗ e =
∑
(v̌α⊗ vα)⊗
∑
(v̌β ⊗ vβ).

We evaluate e⊗ e on vε ⊗ vθ and we have

(e⊗ e)(vε⊗ vθ ) =
∑

α,β

δαεδ
β

θ
vα⊗ vβ

which will equal zero when either α= ε or β = θ meaning that

(e⊗ e)(vε ⊗ vθ ) = (vε⊗ vθ ),

so e⊗ e is the identity map. The map φ clearly takes id ∈ Hom(V⊗2, V⊗2) to the zero map in

Hom(R, V⊗2/R). To define the isomorphism q recall that

A!
2 ⊗ A2 = ((V

∗)⊗2/R⊥)⊗ (V⊗2/R)

= R∗ ⊗ (V⊗2/R).

If we can show that the diagram commutes, that will prove that q(e2) = 0. For an equivalence

class of elements v̌α⊗ v̌γ⊗ vβ ⊗ vδ ∈ (V
∗)⊗2/R⊥⊗ (V⊗2/R) and for an element x ∈ R define q

as follows

q(v̌α⊗ v̌γ⊗ vβ ⊗ vδ)(x) = (v̌α⊗ v̌γ(x)) · vβ vδ = (v̌α⊗ v̌γ(x)) · vβ ⊗ vδ,



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 522

where v̌α ⊗ v̌γ ∈ V ∗ ⊗ V ∗ = ((V⊗2)∗ is a linear functional. To see that q is well defined, note

that v̌α⊗ v̌γ is unique up to (v̌α ⊗ v̌γ(x) + ρ) where ρ ∈ R⊥. Since ρ ∈ R⊥, ρ(x) = 0 and

in the quotient (V⊗2/R) so the map q is well defined. Note that x ∈ R is also contained in

(V⊗2) meaning that any linear functional on V⊗2 restricts to R. So to see that the diagram

commutes, observe that

φp[(v̌α ⊗ vβ)⊗ (v̌γ⊗ vδ)](x) = [(v̌α⊗ v̌γ)(x)] · vβ ⊗ vδ

= [(v̌α⊗ v̌γ)(x) · (v̌β ⊗ vδ)]

= qm[(v̌α⊗ vβ)⊗ (v̌γ⊗ vδ)](x).

We have shown that the diagram commutes proving that q(e2) = 0, so e2 = 0 since q is an

isomorphism.

Thus we may define the Koszul complex K i = A⊗ (A!
i)
∗. The notation K(A) will also be

used to indicate the Koszul complex of a quadratic algebra A.

Proposition 1. [Proposition 2.9.1, [1]]. Let A be a Koszul ring. Then A! is Koszul as well.

Proposition 2 (Koszul Algebra). Let A be a quadratic k-algebra, then following conditions are

equivalent:

(a) H i(K(A))n = 0 if i > 0 and if i = 0, n 6= 0, we have H0(K(A))0 = k.

(b) H i
j
(A, M), (Hochschild Homology) where A is considered as an A-bimodule, vanishes for

any Z+-graded A-bimodule M and i < − j.

(c) E x t i
j
(k, k) = 0 for all i 6= j, where E x t i(k, k) is taken in the category of left A-modules.

(d) K• is a resolution of A in the category of A-bimodules.

(e) K• ⊗A k is a resolution of k in the category of left A-modules.

(f) There exists a free resolution of k such that the i’th syzygies are all generated in degree i.

Example 4. Let us consider the special case where A is the symmetric algebra on V , S(V ). The

quadratic dual A! will be the exterior algebra on V ∗, E(V ∗). Therefore we may represent K(A) in

the following way

. . .
d2−→ A⊗∧2(V )

d1−→ A⊗ V
d0−→ A,

where d0 : a ⊗ v → av or more specifically for this example, d0 : p(x) ⊗ x i → x i p(x) with

p(x) ∈ S(V ) and x i ∈ ∧
i(V ). More generally, we may define di : A⊗∧p(V )→ A⊗∧p−1(V ) as

follows,

di : a⊗ vi1
∧ vi2

∧ · · · ∧ vip
→ Σavi j

(−1) j−1vi1
∧ · · · ∧ bvi j

∧ · · · ∧ vip
.

Note that the Koszul complex is graded by tensor degree since A and A! are both graded.

For a ⊗ v ∈ (A⊗ V )n, deg(a) = n − 1 and deg(v) = 1 so deg(a ⊗ v) = n. Also note that

H i(K∗)n = 0 for all i > 0 and for i = 0 with n 6= 0. The only nontrivial homology group is

H0(K∗)0 = k.



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 523

5. Duality

Let U = T (V )/ < P > be a Λ-graded filtered quadratic algebra. Let A = T (V )/ < R >,

where R= β(P) as in Section 2. We define a dual Λ-graded curved differential graded algebra

(A!, d , c) as follows:

A! = T (V ∗)/R⊥,

α : R
p−1

2−−−→ P
p1−−−→ V, β : R

p−1
2−−−→ P

p0−−−→ k,

with A!
1 = V ∗, d = α∗, and c = β∗(1).

Theorem 1. Assume A is Koszul. Then U is of PBW-type if and only if the map α∗ extends to

an antiderivation d on A! such that, letting c = β∗(1), (A!, d , c) is a curved differential graded

algebra. In particular, when c = 0, giving A! the structure of a differential graded algebra is

equivalent to giving a subspace P ⊂ V ⊕ (V ⊗ V ) with p2(P) = R such that grU = A.

Theorem 2. Assume A is Koszul. Then U is of PBW-type if and only if

1. im(α⊗ id − id ⊗α) ⊆ R⊆ V ⊗ V (this map is defined on (R⊗ V )∩ (V ⊗ R)).

2. α ◦ (α⊗ id − id ⊗α) = β ⊗ id − id ⊗ β .

3. β ◦ (α⊗ id − id ⊗α) = 0.

These two theorems amount to giving A! the structure of a CDGA. The following lemma

will be made use of in Section 6.

Lemma 1. The element in U ⊗k A!

∑
xαxβ ⊗ x̌β x̌α+

∑
xα⊗ d( x̌α) + 1⊗ c (1)

and the element in A!⊗k U
∑

x̌β x̌α⊗ xαxβ + d( x̌α)⊗ xα+ c ⊗ 1

are both zero.

Proof. Consider the pairing

(U ⊗ A!
2)⊗ (A

!
2)
∗→ U .

Denoting the element in (1) as m, we show that 〈m,−〉 : (A!
2)
∗→ U is zero. Note that

〈m, r〉 =
∑¬

xαxβ ⊗ x̌β x̌α, r
¶
+
∑


xα⊗ d( x̌α), r
�
+ 〈1⊗ c, r〉 .

Also note that for an element r in R= (A!
2)
∗ we have

∑
xαxβ
¬

x̌β x̌α, r
¶
= r,
∑

xα


d( x̌α), r
�
= α(r), 〈c, r〉 = β(r),

and since r +α(r) + β(r) = 0 in U , the lemma is proven.



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 524

Example 5. We may first consider the case where U = A is a Λ-graded filtered algebra. The dual

will be the cdga (A!, d = 0, c = 0).

Example 6. Let U = Ug be the universal enveloping algebra of a Lie algebra g. Then the dual

(A!, d , c = 0) is the Chevalley-Eilenberg complex of the Lie algebra g.

Example 7. The symmetric algebra on V , S(V ), is defined by R = (x ⊗ y − y ⊗ x)x ,y∈V . The

quadratic dual algebra of S(V ) is the exterior algebra E(V ∗) defined by the relations (x⊗ x)x∈V∗

in V ∗ ⊗ V ∗.

Example 8. Continuing Example 3, we can show explicitly that Theorems 1 and 2 amount to

giving A! the structure of a CDGA. We have the following algebra

U = k[x]/(x2− (a− b)x + ab) = k[x]/(x − a)⊕ k[x]/(x − b).

Given U, we know that A! = k[ξ] assuming < x ,ξ >= 1. By definition we know

α(x2) = −(a+ b)x and β(x2) = ab,

so now to calculate the differential we have

< α∗(ξ), x2 >=< ξ,α(x2)>=< ξ,−(a+ b)x >= −(a+ b).

This tells us that α∗(ξ) is the element of A!
2 for which < α∗(ξ), x2 >= −(a+ b), so α∗(ξ) = ξ2.

This tells us that dξ= −(a+ b)ξ2. Now let us calculate d(ξ2). We have

d(ξ2) = dξ ·ξ− ξ · dξ= −(a+ b)ξ2 · ξ+ ξ · (a+ b)ξ2 = 0.

Therefore we have the differential

d(ξn) =

(
−(a+ b)ξn+1 if n is odd

0 if n is even.

It is easy to see that d2 = 0.

6. The Duality Functors F and G

We let U be a Λ-graded filtered quadratic algebra such that A = gr(U) is Koszul, and we

let (A!, d , c) be the dual cdga. Let T = U ⊗ A!. This is a U − A! bimodule and we give it the

grading of A!. Let d be the endomorphism defined by

u⊗ a 7−→
∑

uxα⊗ x̌αa+ u⊗ d(a)

where xα is a basis for V and x̌α is the dual basis for V ∗. This definition is independent of

the choice of this basis and one can check that this gives U ⊗ A! the structure of a right cgd

module over (A!, d , c).



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 525

Lemma 2. For T = U ⊗ A! a Λ-graded (U ,A!)-bimodule, T is a right A!-CDGM.

Proof. It is clear that T is a Λ-graded (U ,A!)-bimodule. We only need to check that

d2(u⊗ a) = −u⊗ ac. We define d as follows

d(u⊗ a) =
∑

uxα⊗ x̌αa+ u⊗ d(a).

We have

d2(u⊗ a) =
∑

uxαxβ ⊗ x̌β x̌αa+
∑

uxα⊗ x̌αd(a)+
∑

uxα⊗ d( x̌αa) + u⊗ d2(a)

=
∑

uxαxβ ⊗ x̌β x̌αa+
∑

uxα⊗ d( x̌α)a+ u⊗ ca− u⊗ ac

= −u⊗ ac.

The minus sign is as we would expect since T is a right A!-module.

The pair of adjoint functors

F : ComΛ(A
!, d , c)⇆ ComΛ(U) : G

is given by

F(N) = T ⊗A! N , G(M) = HomU(T, M).

Explicitly, we have

F(N)
p

λ
=
⊕

µ+ν=λ

Uµ ⊗k N p
ν

with

d(u⊗ n) =
∑

α

uxα⊗ x̌αn+ u⊗ dN (n). (2)

Since degΛ( x̌α) = −degΛ(xα) one can show that d preserves the Λ-grading. Since

axα ∈ Uµ+deg(xα)
and x̌αn ∈ N

p+1

ν−deg(xα)
, the sum µ+ ν = λ remains unchanged in the tensor.

The map d is also U-linear, this can easily be seen since

d(u1u⊗ n) =
∑

u1uxα⊗ xαn+ u1u⊗ dN (n)

= u1

∑
uxα⊗ xαn+ u⊗ dN (n)

= u1d(u⊗ n).

Lemma 3. d2 = 0 in Equation (2) and if N is in ComΛ(A
!, d , c) then F(N) is in ComΛ(U).

Proof. All of the axioms necessary to show that F(N) ∈ ComΛ(U) are obvious except for

that d2 = 0, so let us verify this. For u⊗ n ∈ F(N) we have

d2(a⊗ n) =
∑

α

(
∑

β

axαxβ ⊗ x̌β x̌αn+ axα⊗ dN ( x̌αn))+
∑

γ

axγ⊗ x̌γdN (n) + a⊗ d2
N (n).



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 526

The term
∑
α axα⊗ dN ( x̌αn) may be rewritten as

∑

α

axα⊗ d( x̌α)n− axα⊗ x̌αdN (n)

which allows us to simplify and apply Lemma 1 as follows,

d2(a⊗ n) =
∑

α

(
∑

β

axαxβ ⊗ x̌β x̌αn+ axα+ d( x̌α)n) + a⊗ cn

= a(
∑

α

(
∑

β

xαxβ ⊗ x̌β x̌α+ xα+ d( x̌α)) + 1⊗ c)n

= 0.

Next, G(M) = HomU(T, M) = Homk(A
!, M) has the structure of a graded A!-module, de-

fined as

(a · f )(b) = (−1)q(p+r) f (ba), a ∈ (A!)q, b ∈ (A!)r , f ∈ Hom
p

k
(A!, M).

One can verify that a1 · (a2 · f ) = (a1a2) · f . Also it carries an internal Λ-grading:

G(M)
p

λ
=
∏

r≥0

∏

µ

Homk((A
!)rµ, M

p+r

λ+µ
)

with d is given by

d( f )(a) = (−1)| f |+1
∑

xα f ( x̌αa) + (−1)| f |+1 f (dA!(a)) + dM ( f (a))). (3)

Note that our sign conventions differ from that in [3].

Example 9. Continuing Example 3 the algebra U has two simple modules of dimension 1 over k

namely

ka = k[x]/(x − a) and kb = k[x]/(x − b).

By definition, the CDG-module G(ka)
p =
∏

r≥0 Hom(A!
r , k

p+r
a ). Note that k

p+r
a is nonzero only

if p = −r. Now since Hom(A!
r , ka) = (x

r) we can check that for the CDG-module G(ka) we have

· · · (x3)
aξ
→ (x2)

bξ
→ (x)

aξ
→ (1)→ (0)→ · · · .

The module multiplication for v ∈ ka is defined as x · v = av. Now let us consider FG(ka) which

has the form

· · · → U ⊗ (A!
p)
∗⊗ ka→ U ⊗ (A!

p−1)
∗ ⊗ ka→ · · · .

The differential for FG(ka) is given by

d(u⊗ ξ̇p ⊗ 1) = ux ⊗ ξ̇p−1 ⊗ 1+ (−1)p+1(u⊗ ξ̇p−1 ⊗ a+ u⊗ d(ξ̇p)⊗ 1).

It is easy to check that d2 = 0 and the complex FG(ka) will be quasi-isomorphic to ka by Propo-

sition 4.



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 527

Lemma 4. d2 = c in Equation (3) and if M is in ComΛ(U) then G(M) is in ComΛ(A
!, d , c).

Proof. To verify that d2( f ) = c f we have

d(d( f )(a)) = d[(−1)| f |+1
∑

xα f ( x̌αa)] + d[(−1)| f |+1 f (dA!(a))] + d[dM ( f (a))].

Let us examine the first term of this differential. Let g(a) = (−1)| f |+1
∑

xα f ( x̌αa) and let�� f
�� = p so we have

d(g)(a) = (−1)p+2
∑

β

xβ g( x̌βa) + (−1)p+2g(dA!(a)) + dM (g(a))

which can be rewritten as

d(g)(a) = −
∑

β ,α

xβ xα f ( x̌α x̌βa)−
∑

γ

xγ f ( x̌γdA!(a))+ (−1)p+1
∑

δ

dM (xδ f ( x̌δa)).

Now recall from Lemma 1 that

∑

α,β

xαxβ ⊗ x̌β x̌α+
∑

xα⊗ d( x̌α) + 1⊗ c = 0 (4)

and since we know

dA!( x̌γa) = dA!( x̌γ)a− x̌γdA!(a)

we have ∑
xγ f ( x̌γdA!(a)) =

∑
xγ f (dA!( x̌γ)a)−

∑
xγ f (dA!( x̌γa)).

So we may rewrite d(g)(a) as

d(g)(a) = f (ca) +
∑

γ

xγ f (dA!( x̌γa)) + (−1)p+1dM (
∑

δ

xδ f ( x̌δa)).

Now let us examine the second term of d(d( f )(a)). Let h(a) = (−1)p+1 f (dA!(a)) so we have

dh(a) = (−1)p+2
∑

xαh( x̌αa) + (−1)p+2h(dA!(a)) + dM (h(a))

= −
∑

xα f (dA!( x̌αa))− f (dA!(dA!(a)))+ (−1)p+1dM ( f (dA!(a)))

= −
∑

xα f (dA!( x̌αa)) + (−1)p+1dM ( f (dA!(a)))− f (ca) + f (ac).

For the third term of d(d( f )(a)), let k(a) = dM ( f (a)). So we have

dk(a) = (−1)p+2
∑

xαk( x̌αa) + (−1)p+2k(dA!(a))+ dM (k(a))

= (−1)p+2
∑

xαdM ( f ( x̌αa)) + (−1)p+2dM ( f (dA!(a)))+ dM (dM ( f (a)))

= (−1)p+2
∑

xαdM ( f ( x̌αa)) + (−1)p+2dM ( f (dA!(a))).



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 528

Note that since dM is U-linear we have

d(
∑

xδ f ( x̌δa)) =
∑

xδd( f ( x̌δa)).

Now we can list all remaining terms of the differential d2( f )(a) as so

d2( f )(a) = f (ca) +
∑

γ

xγ f (dA!( x̌γa)) + (−1)p+1dM (
∑

δ

xδ f ( x̌δa))

−
∑

xα f (dA!( x̌αa)) + (−1)p+1dM ( f (dA!(a)))− f (ca) + f (ac)

+ (−1)p+1
∑

xαdM ( f ( x̌αa)) + (−1)p+2dM ( f (dA!(a)))

= f (ac) = (c · f )(a).

We will now check that d(a · f ) = da · f + (−1)|a|a · d f . Let
�� f
�� = p, |a| = q, and |b| = r. So

we have

d(a · f )(b) = (−1)p+q+1
∑

xα(a · f )( x̌αb) + (−1)p+q+1(a · f )(d(b))+ d(a · f )(b)

which may be rewritten as

d(a· f )(b) = (−1)qp+qr+p+1
∑

xα f (( x̌αba))+(−1)qp+qr+p+1 f ((d(b)a))+(−1)qp+qrd( f (ba)).

Recalling that

f (d(b)a) = f (d(ba))+ (−1)r+1 f (b(da)),

we have

d(a · f )(b) =(−1)qp+qr+p+1
∑

xα f (( x̌αba)) + (−1)qp+qr+p+1 f (d(ba))

+ (−1)qp+qr+p+r f (b(da))+ (−1)qp+qrd( f (ba)).

Let us now calculate (da · f )(b)+ (−1)qa · d( f (b)). For the first term, we have

(da · f )(b) = (−1)qp+qr+p+r f (b(da))

and for the second term, we have

(−1)q(a·d f )(b) = (−1)qp+qr+p+1
∑

xα f (( x̌αba))+(−1)qp+qr+p+1 f (d(ba))+(−1)qp+qrd( f (ba))

which proves that d(a · f ) = da · f + (−1)|a|a · d f . These define a structure of a CDGM over

the CDGA (A!, d , c). More explicitly

(A!)pµ× G(M)qν → G(M)
p+q
µ+ν .

Note that when A! = E(V ∗) with dim V < ∞, the above direct products are direct sums

since dim A! <∞.



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 529

Proposition 3. We have:

1. For N in ComΛ(A
!, d) and M in ComΛ(U) there is a canonical isomorphism of differential

graded Λ-graded vector spaces

HomU(F(N), M) = HomA!(N , G(M)).

2. The functors F and G are adjoint, i.e.,

HomComΛ(U)
(F(N), M) = HomComΛ(A

!,d)(N , G(M)).

3. The functors F and G are exact.

Proof. Since F(N) = T ⊗A! N and G(M) = HomU(T, M), the first statement follows directly

from the adjointness of Hom and tensor product. Explicitly both complexes have (p,λ)’th

term equal to ∏

r∈Z

∏

µ

Homk(N
r
µ , M

p+r

λ+µ
)

with differential d given as follows for f ∈
∏

r∈ZHomk(N
r , M p+r) and n ∈ N r ,

d( f )(n) = (−1)rdM f (n)+ (−1)r+1 f dN (n)+ (−1)r+1
∑

α

xα f ( x̌αn).

To see that the functors F and G are adjoint we note that the two sides are the cycles of degree

0 in the complexes described in the first statement. The functor F is exact since as a module,

F(N) = T ⊗A! N = U ⊗k N and k is a field. For the same reason, the functor G is exact since

G(M) = HomU(T, M) = Homk(A
!, M).

Lemma 5. Let M be a U-module considered as a complex situated in degree 0. Then FG(M)→ M

is a quasi-isomorphism.

Note that FG(M) is the complex

· · · → U ⊗ (A!
p)
∗ ⊗M → U ⊗ (A!

p−1)
∗ ⊗M → · · · → U ⊗M

and the differential is given by

d(u⊗ a∗ ⊗m) =
∑

uxα⊗ x̌αa∗ ⊗m±
∑

u⊗ a∗ x̌α⊗ xαm± u⊗ d∗(a∗)⊗m.

Proof. Proof is from [3]. The complex FG(M) has a filtration FiG(M) defined as

· · · → Fi−2U ⊗ (A!
2)
∗ ⊗M → Fi−1U ⊗ (A!

1)
∗⊗M → FiU ⊗M .

We claim that for i ≥ 0, we have Hp(FiG(M)) = M for p = 0 and zero otherwise. This follows

by induction from the exact sequence

0→ Fi−1G(M)→ FiG(M)→ FiG(M)/Fi−1G(M)→ 0



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 530

by noting that the term FiGM/Fi−1G(M) is a homogeneous part of the Koszul complex for A

and A! tensored with M (over k) i.e.

A0 ⊗ (A
!
i)
∗⊗M → · · · → Ai ⊗ (A

!
0)
∗⊗M

with differential

d(u⊗ a∗ ⊗m) =
∑

uxα⊗ a∗ x̌α⊗m.

We want to show that FiG(M)→ M is a quasi-isomorphism, that is, Hp(FiG(M)) = 0 for

all i and with p < 0, and H0(FiG(M)) = M for all i. We know that FG(M) = lim
−→

Fi G(M) so

therefore

Hp(FG(M)) = Hp(lim
−→

FiG(M)) = lim
−→

Hp(FiG(M))

and thus FG(M)→ M is a quasi-isomorphism provided that FiG(M)→ M is for all i. Now we

will prove by induction that Hp(FiG(M)) = 0 for all p < 0 and H0(FiG(M)) = M for all i. The

case when i = 0 is trivial since F0G(M) = M . Now assume i > 0 and consider the following

commutative diagram of exact sequences,

0 −−−→ FiG(M) −−−→ Fi+1G(M) −−−→ Fi+1G(M)/FiG(M) −−−→ 0y α

y β

y γ

y
y

0 −−−→ M −−−→ M −−−→ 0 −−−→ 0.

The map α is a quasi-isomorphism by the induction hypothesis and the map γ is a quasi-

isomorphism since A is a Koszul algebra. Now by applying the 5-Lemma we know that the

map β is also a quasi-isomorphism. We know that FG(M) is lim
−→

FiG(M) and since taking the

filtered direct limit is an exact functor it commutes with cohomology so we get the lemma.

Corollary 1. For M• a bounded complex indexed as follows

0→ M b → M b+1→ · · · → M t−1→ M t → 0.

Hp(FiG(M
•)) = 0 for all i and for p < b.

Proposition 4. Assuming that A and A! are Koszul, the natural morphisms coming from the

adjunction

FG(M)→ M , N → GF(N)

are quasi-isomorphisms.

Proof. Assume M is bounded and indexed as follows

0→ M b → M b+1→ · · · → M t−1→ M t → 0.

Let σ>bM be the truncation M b+1 → M b+2 → · · · and so M b[−b] will just be a module

considered as a one term complex. We may now form the following short exact sequence

0→ σ>b M → M → M b[−b]→ 0.



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 531

Consider the following commutative diagram of exact sequences

0 −−−→ FG(σ >bM)) −−−→ FG(M) −−−→ FG(M b[−b]) −−−→ 0y α

y β

y γ

y
y

0 −−−→ σ >bM −−−→ M −−−→ M b[−b] −−−→ 0.

The map α is a quasi-isomorphism by induction on the length of the truncation, the map γ

is a quasi-isomorphism by Lemma 5 so by the 5-Lemma we know that the map β is also a

quasi-isomorphism.

Now let us assume that the complex M is bounded above so M = lim
−→
σ≥pM where the

σ≥pM are bounded. Since we know that

G(M)
p

λ
=
∏

r≥0

∏

µ

Homk((A
!)rµ, M

p+r

λ+µ
)

we can now show that G commutes with direct limit. We know that M = lim
−→
σ≤sM and even

[M]l = [lim
−→
σ≤sM]l . Now note that [lim

−→
σ≤sM]l = 0 if l < s and equals M l if l ≥ s. Since M

is bounded above, we know that M l = 0 for l >> 0 which tells us that there exists an s0 such

that for all s ≥ s0 we have (σ≤sM)p+r = M p+r for all p and r ≥ 0. Therefore

G(lim
−→
σ≥pM) = G(M).

Since F is a left adjoint it also commutes with the direct limit so we have that

FG(M) = FG(lim
−→
σ≥pM) = lim

−→
FG(σ≥p M)→ lim

−→
σ≥p M = M

is a quasi-isomorphism since lim
−→

is exact in the category of vector spaces.

Next suppose that M is bounded below e.g. M = σ>b M and indexed as follows

0→ M b → M b+1→ · · · .

We know by Lemma 5 that for a module M over U , FiG(M) is exact in cohomological degrees

< 0. First we must define the filtration FiG(M) for when M is a bounded complex, not just a

module. Let

[Fi G(M)]
a = Fi+aU ⊗
∏

p≥0

Hom(A!
p, M p+a).

If M is bounded above, in particular bounded, then

Gr F
i FG(M) =
⊕

p≥0

Ai+a ⊗ (A
!
p)
∗⊗M p+a.

Let M be a complex bounded below i.e. M• = 0 for i < b. We want to show that FG(M)→ M

is a quasi-isomorphism in degrees < b. Also we know that H i(M) = 0 for i < b. Since we



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 532

know that FG(M) = lim
−→

FiG(M) and since we know that taking cohomology commutes with

taking direct limits, it suffices to show that H i(FνG(M)) = 0 for i < b and for all ν >> 0

which amounts to showing that H i(FG(M)) = 0 for i < b. By our proof of Lemma 5 we

know that H i(FνG(M)) = 0 for i < b and for all ν >> 0 is true for the case where M is a

module considered as a one term complex situated in deg b. Now if we allow for M to be a

bounded complex, again by our proof of Lemma 5 we can induct on the length of the complex

and again show our intended result. Now let M = lim
←−
σ≤pM and note that each σ≤pM is

bounded since M is bounded below. We know that

FνG(M) = FνG(lim
←−
σ≤p M) = Fν (lim←−

G(σ≤p M)) = lim
←−
(FνG(σ≤p M)).

The first equality is easy to see since M = lim
←−
σ≤pM , the second equality follows from the

fact that G is a right adjoint and the third equality follows from the fact that each FνG(M) is

finite dimensional, and therefore commutes with inverse limit. Now we would like to show

that even though inverse limit does not usually commute with cohomology, in our case we do

have

H i(lim
←−

FνG(σ≤p M)) = lim
←−

H i(FνG(σ≤pM)).

This equality follows from the fact that each FνG(σ≤pM) and H i(FνG(σ≤pM)) satisfy the

Mittag-Leffler condition (See Proposition 5). We also know that for i < b the following is

true,

H i(lim
←−
σ≤p M) = lim

←−
H i(σ≤pM),

since M = 0 for i < b and we know by our proof of Lemma 5 that since each σ≤p M is bounded

we have

lim
←−

H i(FνG(σ≤pM)) = lim
←−

H i(σ≤pM),

proving that H i(FνG(M)) = 0 for i < b and for all ν >> 0.

Now let M be an arbitrary complex. Consider the following diagram

0 −−−→ FG(σ>p)M −−−→ FG(M) −−−→ FG(σ≤p M) −−−→ 0y α

y β

y γ

y
y

0 −−−→ σ>pM −−−→ M −−−→ σ≤p M −−−→ 0

and the resulting cohomology diagram:

H i(FG(σ>p M)) −−−→ H i(FG(M)) −−−→ H i(FG(σ≤p(M)) −−−→ H i+1(FG(σ>p M))yα
yβ

yγ
yα′

H i(σ>pM) −−−→ H i(M) −−−→ H i(σ≤pM) −−−→ H i+1(σ>pM).

We know that the map γ is an isomorphism in all degrees i since σ≤p M is bounded above.

We have also shown that α is an isomorphism in degrees i ≤ p since σ>p M is bounded below.

Similarly, α′ is an isomorphism in degrees i + 1 ≤ p. The 5-Lemma now shows that β is an



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 533

isomorphism in degrees i ≤ p− 1. Since p is arbitrary, β is an isomorphism in all degrees i.

We now would like to show that N → GF(N) is a quasi-isomorphism. The complex GF(k)

is the complex

· · · → (A!
p)
∗⊗ U → (A!

p−1)
∗⊗ U → · · · → U .

By the same argument as in the proof of Lemma 5 the map k→ GF(k) is a quasi-isomorphism.

So if N = N0 we have GF(N) = GF(k)⊗k N and so N → GF(N) is also a quasi-isomorphism.

By induction on the length of the truncations, we know that N → GF(N) is a quasi-

isomorphism for bounded N . Now let N be bounded above. We know that N = lim
−→
σ>pN for

p→−∞ and we also know that for N bounded above,

σ>pN → GF(σ>pN)

is a quasi-isomorphism for all p since each σ>pN is bounded. Since we know that direct limit

commutes with taking cohomology, we would like to show that

GF(lim
−→
σ>pN) = G lim

−→
F(σ>pN) = lim

−→
GF(σ>pN).

The first equality is clear since F is a left adjoint but it remains to show that G commutes with

direct limit. We know

GF(σ>pN)i =
∏

r≥0

Homk((A
!)r , F(σ>pN)i+r)

and since p− i + 1≤ r ≤ t − i we have

GF(σ>pN)i =

t−i⊕

r=p−i+1

Homk((A
!)r , F(σ>pN)i+r).

If we take the direct limit of both sides as p→−∞ we have

lim
−→

GF(σ>pN)i =

t−i⊕

0

Homk((A
!)r , F(σ>pN)i+r)

which we would like to have equal to G lim
−→

F(σ>pN). We know

G lim
−→

F(σ>pN) =
∏

r≥0

Homk((A
!)r , [lim
−→

F(σ>pN)]i+r)

=
∏

r≥0

Homk((A
!)r , lim
−→

U ⊗ (σ>pN)i+r)

=
∏

r≥0

lim
−→

Homk((A
!)r , U ⊗ (σ>pN)i+r).



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 534

The third equality is due to the fact that each (A!)r is finite dimensional and since we know

that r is bounded, i.e., p− i + 1≤ r ≤ t − i we have for p→−∞

G lim
−→

F(σ>pN) = lim
−→

t−i⊕

r=0

Homk((A
!)r , F(σ>pN)i+r)

which eventually stabilizes in this range of r giving us

G lim
−→

F(σ>pN) = lim
−→

GF(σ>pN)

showing that G can commute with direct limit and showing that for N bounded above, we

have N → GF(N) is a quasi-isomorphism.

Now let N be arbitrary. We know for p→∞,

lim
←−
σ≤pN = N

and we know that each σ≤pN is bounded above so for all p we know that

σ≤pN → GF(σ≤pN)

is a quasi-isomorphism. By Lemma 7 we know

lim
←−
σ≤pN = lim

←−
GF(σ≤pN)

and we also know that

lim
←−

GF(σ≤pN) = G lim
←−

F(σ≤pN)

since G is a right adjoint. It remains to show that F can commute with inverse limit. We know

that F(N)i = U ⊗ N i by definition and we know that for a fixed i and for p→∞ we have

lim
←−

F(σ≤pN)i = lim
←−

U ⊗ (σ≤pN)i

and since p will become greater that i after finitely many steps, both sides will equal U ⊗ N i

showing our intended equality and thus proving that for arbitrary N , N → GF(N) is a quasi-

isomorphism.

Proposition 5. Let K∗p be a projective system of complexes of modules over a ring

R : . . .→ K∗p+1→ K∗p → . . .. Suppose that K∗p satisfies the Mittag-Leffler condition (ML) and that

each Ha(K∗p) satisfies the ML condition, for instance if it satisfies the descending chain condition

(dcc), (eg., if each is a finite dimensional vector space over the field R= k). Then

(lim
←−

p

)Ha(K∗p) = Ha(lim
←−

p

K∗p).



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 535

Lemma 6. For all sufficiently large i, the projective systems

p→ FiG(σ
≤pM) and p→ Ha(FiG(σ

≤p M))

satisfy the ML condition when M is bounded below, therefore

lim
←−

p

Ha(FiG(σ
≤pM)) = Ha(FiG(M)) = Ha(lim

←−
p

FiG(σ
≤p M)).

Proof. Since M is bounded below we know that σ≤p M is a bounded complex. By choosing

i large enough we know that FiG(σ
≤p)M → σ≤p M is a quasi-isomorphism. Note that the i

that works depends only on the lower bound of the complex σ≤pM and this is the same for

all p, so let us fix an i. Thus p→ Ha(FiG(σ
≤pM)) is ML since for large p (in fact p > a), these

values are constant and equal to Ha(M).

For FiG(σ
≤p)M note that in any degree, the transition maps

Fi G(σ
≤p+1M) j = Ui+ j ⊗

p+1− j∏

r=0

Hom((A!)r , M j+r )→ FiG(σ
≤p M) j

induced by the natural projection

p+1− j∏

r=0

Hom((A!)r , M j+r )→
p− j∏

r=0

Hom((A!)r , M j+r )

are surjective, hence we have the ML condition satisfied.

Lemma 7. For all sufficiently large i, the projective systems

p→ GF(σ≤pN) and p→ Ha(GF(σ≤pN))

satisfy the ML condition for any N, therefore

lim
←−

p

Ha(GF(σ≤pN)) = Ha(GF(N)) = Ha(lim
←−

p

GF(σ≤pN)).

Proof. The second of the two projective systems clearly satisfies the ML condition since

Ha(GF(σ≤pN)) = Ha(σ≤pN) and since each σ≤pN is bounded above, and since for all i the

projective system is constant for large values of p and it equals Ha(N). Now to show that the

first system satisfies the ML condition observe that

(GFσ≤pN) j =
∏

r≥0

Hom((A!)r , U ⊗ (σ≤pN) j+r)

and since σ≤pN equals N j+r when j+ r ≤ p and zero otherwise we have

(GFσ≤pN) j =

p− j∏

r≥0

Hom((A!)r , U ⊗ N j+r)



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 536

so the natural projection

p− j∏

r≥0

Hom((A!)r , U ⊗ N j+r)→
p− j−1∏

r≥0

Hom((A!)r , U ⊗N j+r)

is clearly surjective thus showing that the ML condition is satisfied.

7. Equivalences of Categories

Let KΛ(A
!, d , c) be the category ComΛ(A

!, d , c) with morphisms being chain homotopy

equivalence classes of maps. Similarly we may define the category KΛ(U).

Definition 7. The null system, N, of KΛ(A
!, d , c) is defined to be all of the complexes, X , such

that F(X ) is acyclic. The null system, N, of KΛ(U) is defined to be all of the complexes, Y , such

that G(Y ) is acyclic.

We define DΛ(A
!, d , c) to be the category KΛ(A

!, d , c)/N where N is the null system of

KΛ(A
!, d , c). We also define DΛ(U) to be the category KΛ(U)/N where N is the null system of

KΛ(U).

Theorem 3 (The Main Result). The adjunction F and G given in Proposition 3 induces an

equivalence of categories between the quotient categories DΛ(A
!, d , c) and DΛ(U).

Proof. The proof of this is exactly the same as in [3].

8. Relating Floystad’s Duality to the Classical Koszul Duality

Classical Koszul duality concerns the pair of positively graded algebras A and A!. Floystad’s

version of Koszul duality considers the case where U = A for a filtered algebra U , meaning

that the filtration arises from a grading of A. The case which is of interest to us is represented

in the following commutative diagram

Gb : C(B!)
←−
−−−→ C(B) : Fb

i

y j

y

G̃ : CZ(A)
←−
−−−→ CZ(A

!
•) : F̃


 φ ↑ ↓ ψ

G : ComZ(U = A)
←−
−−−→ ComZ(A

!, d = 0, c = 0) : F.

In this commutative diagram B = A!, CZ(A) is the category of chain complexes of graded left

A-modules and CZ(A
!
•) is the category of chain complexes of graded left A!-modules. The

functors Fb and Gb are the functors from [1] given by

(Fb M)pq =
⊕

p=i+ j

q=l− j

B!
l ⊗M i

j , (GbN)pq =
⊕

p=i+ j

q=l− j

Homk(B−l , N i
j ).



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 537

The functors F̃ and G̃ have been constructed such that F̃ = Fψ and G̃ = Gφ making the lower

half of the diagram commute. We would like to relate the following

ψ : CZ(A
!
•)⇆ ComZ(A

!, d = 0, c = 0) : φ

by defining φ and ψ such that they are well defined, inverses of each other, and make the

diagram commute. Now note the A!
• on the left hand side is a ring while the A! on the right

hand side will be regarded as a CDGA with d = c = 0. Namely we define

(ψA!
•) = (A

!, d = 0, c = 0) as the Z-graded CDGA by the rule,

(ψA!)ij =

(
0 if i + j 6= 0

A!
i if − j = i ≥ 0.

For any complex (M , dM ) ∈ CZ(A
!
i
) we define a CDG-A!-module (ψM) = (N , dN ) by the rule

(ψM)i
j
= N i

j
= M

i+ j

− j
with differential dN (x) = (−1)sdM (x) for x ∈ N s. One must now check

all the necessary axioms to show that ψ(M , dM ) = (N , dN ) is an element of

ComZ(A
!, d = 0, c = 0) for (M , dM ) ∈ CZ(A

!
•). Clearly d2

N = 0 since we know that d2
M = 0. We

also need to check A!-linearity of dN meaning that for x ∈ N s and α ∈ (A!)k−k
we have

dN (αx) = (−1)sαdN (x).

This is clearly true since both sides equal (−1)s+kαdM (x). This also verifies that

dN (αx) = dA!(α)(x)+ (−1)deg(α)αdN (x)

since dA!(α)(x) = 0. Now we need to check the module structure of (N , dN ). We know that

A!
p ×M r+s

−s ⊂ M r+s
p−s

and under ψ this corresponds to

(A!)
p
−p × N r

s ⊂ N
p+r
−p+s

and we know this is true by our definition of ψ verifying that ψ(M) is an A!-module. Now

we’d like to check that φ(N , dN ) = (M , dM ) for (M , dM ) ∈ CZ(A
!
•). Now let us define φ such

that φ(A!, d = 0, c = 0) = A!
•. Namely we define φ(A!, d = 0, c = 0) as the chain complex of

graded left A!-modules by the rule

φ(A!, d = 0, c = 0)i− j = (A
!
•) j

and we will define an A!-module by the rule

(φN)ij = M i
j = N

i+ j

− j

with differential dM (x) = (−1)i+ jdN (x) for x ∈ M i
j
. Clearly d2

M = 0 since dN = 0. We also

must show that dM (αx) = αdM (x) and this is clear since

dM (αx) = (−1)i+ jα(dN (x)) = α(dM (x)).



F. Hawwa, J. Hoffman, and H. Wang, / Eur. J. Pure Appl. Math, 5 (2012), 511-539 538

Finally it remains to show that

A!
r ×M i

j ⊂ M i
j+r

which, under φ, corresponds to

(A!)r−r ⊗ N
i+ j

− j
⊂ N

i+ j+r

− j−r

which we know is true by our definition of φ thus completing our verification that

φ(A!, d = 0, c = 0) = A!
•.

Now we need to verify that the diagram commutes. The lower half of the diagram com-

mutes by definition of F̃ and G̃. More explicitly, F̃ = Fψ and G̃ = Gφ by definition. What

remains is to verify that

Fψ(M , dM )
p
q = (Fb M)pq and φG(N , dN )

p
q = (GbN)pq .

We know that

Fψ(M , dM )
p
q =
⊕

r+s=q

Ar ⊗ψ(M)
p
s =
⊕

r+s=q

B!
r ⊗M

p+s
−s

and we also know that

(FbM)pq =
⊕

p=i+ j

q=l− j

B!
l
⊗M i

j

so Fψ(M , dM )
p
q = (FbM)

p
q as long as i = p + s, j = −s, and l = r which are all clearly true

since we know that p = i+ j, q = l − j and r + s = q. We also must check that the differential

on Fψ(M , dM )
p
q matches the differential on (Fb M)

p
q . For x ∈ N s we have the differential on

Fψ(M , dM ) is given by

dF (a⊗ n) = Σaxα⊗ x̌αn+ a⊗ dN (n) = Σaxα⊗ x̌αn+ (−1)sa⊗ dM (n).

The differential for Fb(M , dM ) with a⊗m ∈ B!
M ⊗M i

j
is given by

dFb
(a⊗ n) = (−1)i+ jΣav̌α⊗ vαm+ a⊗ dM (n).

Since i+ j = s we see that dF and dFb
differ by a sign:

dF (a⊗ n) = (−1)i+ jdFb
(a⊗ n).

Let us check that Gb = φG. We know that the following expression

(GbN)pq =
⊕

p=i+ j

q=l− j

Homk(B−l , N i
j )

will be equal to

φG(N , dN )
p
q = G(N , dN )

p+q
−q =
⊕

r≥0

Homk(Br , N
p+q+r
−q−r )



REFERENCES 539

only if r = −l, j = −q− r, and i = p+q+ r. These identities follow immediately by definition

of p and q proving that Gb = φG. To verify that the differentials agree consider the total

differential on (GbN)i
l
, for b ∈ B = A! given by

(dGb
f )(b) = (−1)iΣv̌α f (vαa) + dN ( f (b))

and the differential on φG(N , dN )
p
q = Homk((A

!)r−r , N
p+q+r
−q−r ) for a ∈ A! which is given by

(dG f )(a) = (−1)| f |+1Σxα f ( x̌αa) + (−1)| f |+1 f (dA!(a)) + dN ( f (a)).

= (−1)p+q+1Σxα f ( x̌αa) + (−1)p+q+1dN ( f (a))

= (−1)r+1Σxα f ( x̌αa) + dN ( f (a))

We can see that the two differentials, dGb
and dG, only differ by a sign.

References

[1] A. Beilinson, V. Ginzburg, and W. Soergel. Koszul duality patterns in representation theory,

Journal of the American Mathematical Society. 9), no.2 473-527. 1996.

[2] I. Bernstein, I. Gelfand, and S. Gelfand. Algebraic bundles over Pn and problems of linear

algebra, Funktsional’nyi Analiz i ego prilozheniya 12); English translation in Functional

analysis and its applications 12, 212-214. 1978.

[3] G. Floystad. Koszul duality and equivalences of categories, Transactions of the American

Mathematical Society. 358, p. 2373-2398. 2006.

[4] M. Kapranov. On the derived categories of coherent sheaves on some homogeneous spaces,

Inventions Mathematicae. 92, 479-508. 1988.

[5] A. Polishchuk and L. Positselski. Quadratic Algebras, University Lecture Series,37. Amer-

ican Mathematical Society, Providence, RI, 2005.