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ON KOSZULITY OF TWISTED TENSOR PRODUCTS
PASCUAL JARA, JAVIER LO´PEZ PEN˜A, AND DRAGOS¸ S¸TEFAN
Abstract. Let R be a semisimple ring. A pair (A,C) is called pre-Koszul if A is a connected graded
R-ring and C is a compatible connected graded R-coring. To a pre-Koszul pair we associate several
complexes and we show that one of them is exact if and only if all other are so. In this case we say
that (A,C) is Koszul. We investigate the connection between Koszul pairs and the Koszulity of A and
C. We use Koszul pairs to show that the twisted tensor product of two Koszul rings is Koszul.
Introduction
Koszul rings were introduced in the seminal paper [BGS96] as graded rings A := ⊕n≥0An such that:
A0 is a semisimple ring, and A0 regarded as a graded left A-module has a resolution P• → A0 → 0
by projective graded left A-modules, with the property that every Pn is generated by its homogeneous
component of degree n. In loc. cit. the authors show that many fundamental properties of Koszul
algebras hold for Koszul rings too: Koszul rings are quadratic, a ring is Koszul if and only if the Koszul
complex is exact, the dual of a Koszul ring is Koszul, etc.
It is well-known that the tensor product of two Koszul algebras is Koszul too, so it is natural to ask if
a similar property can be proved for Koszul rings. Of course, first of all we have to clarify what is meant
by the tensor product of two graded rings. In the classical situation, when A and B are two Koszul
algebras, the tensor product is performed over the base field K. Note that in this case both algebras
are connected, that is A0 = B0 = K. In the general case, when A and B are just graded rings, it might
happen that A0 and B0 to be different, so there is no good candidate for the ‘base ring’. It is easy to
overcome this difficulty, by fixing a semisimple ring R and dealing only with connected graded R-rings,
that is graded rings with A0 = R. Equivalently, a connected graded R-ring is a connected graded
algebra in the monoidal category (RMR,⊗, R) of R-bimodules. Clearly, if A and B are connected
graded R-rings, then we may consider the graded R-bimodule A⊗B (throughout the paper the symbol
⊗ denotes the tensor product of two R-bimodules). Still, A⊗B do not inherit a ring structure from the
existing ones of A and B, as R is not commutative. In Noncommutative Geometry a good replacement
for the usual tensor product of two algebras is the twisted one, with respect to a graded twisting map
σ : B⊗A→ A⊗B (see §1.1 for the definition of a twisting map and of the corresponding twisted tensor
product A⊗σ B).
In view of the foregoing, we fix a semisimple ring R, two graded R-rings A and B and a graded
twisting map σ between A and B. If A and B are Koszul, now it makes sense to ask if A⊗σ B is Koszul
as well. The main aim of this paper is to show that this question has a positive answer, provided that
the twisting map is invertible.
To approach this problem we shall use as a main tool the dual notion of R-rings, the so called R-
corings. By definition, they are coalgebras in the monoidal category (RMR,⊗, R). Our interest for this
structure is explained by the fact that the Koszul complex of a quadratic R-ring A may be regarded
in a canonical way as a connected graded R-coring C (here by connected we mean that C0 = R).
Moreover, the differentials of the Koszul complex may be written by using the multiplication in A and
the comultiplication in C only. In the first part of the paper we show that something like the Koszul
complex exists for an arbitrary pair (A,C), where A is a connected graded R-ring and C is a connected
Date: November 19, 2010.
1

2 PASCUAL JARA, JAVIER LO´PEZ PEN˜A, AND DRAGOS¸ S¸TEFAN
graded R-coring that are compatible in following sense. First we ask that C1 and A1 are isomorphic as
R-bimodules via a map θC,A : C1 → A1. Since both structures are graded, the multiplication in A and
the comultiplication in C induce maps mp,q : Ap⊗Aq → Ap+q and ∆p.q : Cp+q → Cp⊗Cq. The second
condition that we impose on A and C is that
m1,1 ◦ (θC,A ⊗ θC,A) ◦∆1,1 = 0.
We call such a pair pre-Koszul. In the first section of the paper we investigate the main properties of
pre-Koszul pairs. We start by associating six complexes to each pre-Koszul pair (A,C). The first one is
a chain complex in the category of left A-module. By symmetry we obtain another one in the category
of right A-modules, and by combining these constructions we get a new chain complex in the category of
A-bimodules. The other three complexes are constructed by duality. Thus, they are cochain complexes
in appropriate categories of (bi)comodules.
In the case when A1 generates A, we prove that (A, T (A)) is pre-Koszul, where the coring structure
on T (A) := Tor A• (R,R) is defined using the normalized bar resolution of R = A0. Again by duality, if
C is a connected graded R-coring and C is cogenerated in degree one, then (E(C), C) is a pre-Koszul
pair, where E(C) := Ext•C(R,R).
In Theorem 2.4 we prove that one of the six complexes that are associated to a pre-Koszul pair (A,C)
is exact, then the other five are also exact. In this case we say that (A,C) is a Koszul pair.
For a graded R-ring A, we prove in Theorem 2.13 that there is a graded R-coring C such that (A,C)
is Koszul if and only if A is generated by A1 and (A, T (A)) is Koszul, if and only if A is a Koszul ring.
By duality one proves a similar result for a graded R-coring C. We end this section by showing that a
component of a Koszul pair uniquely determines the other one. As a consequence of this fact,
E(T (A)) ∼= A and T (E(C)) ∼= C,
for any Koszul ring A and any Koszul coring C (see §2.15 for the definition of Koszul corings). Therefore
T (A) and E(C) may be seen as the Koszul dual of A and of C, respectively. Note that these isomorphisms
hold without any finiteness condition on A and C.
In the third section of the paper we prove that A⊗σ B is Koszul, provided that A and B are Koszul,
and that the twisting map σ is invertible. In order to do that, we consider the Koszul pairs (A, T (A))
and (B, T (B)), and show that there is a twisting map τˆ : T (A) ⊗ T (B) → T (B) ⊗ T (A) such that
(A⊗σ B, T (A)⊗τˆ T (B)) is a Koszul pair.
1. Pre-Koszul pairs
In this section we introduce Koszul pairs and then we investigate their properties. First let us fix the
terminology and the notation that we use. Throughout this paper R will denote a semisimple ring.
1.1. R-rings. The category of R-bimodules will be denoted by RMR. It is a monoidal category with
respect to the tensor product of bimodules, that will be denoted by the symbol ⊗.
An R-ring is by definition an associative and unital algebra in the monoidal category RMR. One can
see easily that an R-ring is uniquely determined by an associative and unital ring A and a morphism
of unital rings u : R → A. We say that an R-ring A is graded if it is equipped with a decomposition
A = ⊕n∈NAn in RMR, such that the multiplication maps Ap⊗Aq to Ap+q. Note that the multiplication
m in A induces an R-bilinear map mp,q : Ap ⊗ Aq → A. A graded R-ring A is connected if A0 = R.
We denote by A¯ = ⊕n>0An the ideal generated by the homogeneous elements of positive degree. The
multiplication of A induces a bimodule map m : A¯ ⊗ A¯ → A¯. The projection of A onto An will be
denoted by pinA.
Let Rop be the opposite ring. If V and W are R-bimodules, then they become Rop-bimodules by
interchanging the left and right module structures. Moreover, there is a unique isomorphism of abelian
groups ΛV,W : V ⊗Rop W → W ⊗R V that maps v ⊗Rop w to w ⊗R v. If m denotes the multiplication

ON KOSZULITY OF TWISTED TENSOR PRODUCTS 3
of an R-ring A, then mop := m ◦ ΛA,A defines a new associative multiplication on A, regarded as an
Rop-bimodule. Together with the unit of A, the map mop defines an Rop-ring, that will be called the
opposite Rop-ring of A, and it will be denoted by Aop.
1.2. R-corings. An R-coring is a coassociative and counital coalgebra in RMR. Thus, an R-coring
is an R-bimodule C together with a coassociative comultiplication ∆ : C → C ⊗ C and a counit
ε : C → R, which are morphisms in RMR. Coassociativity can be written, using for ∆ the Σ-notation
∆(c) := ∑ c(1) ⊗ c(2), as
∑
c(1)(1) ⊗ c(1)(2) ⊗ c(2) =
∑
c(1) ⊗ c(2)(1) ⊗ c(2)(2) ,
for every c ∈ C. The map ε is the counit of C if, for c ∈ C,
∑
ε(c(1))c(2) = c =
∑
c(1)ε(c(2)).
A graded R-coring is a coring (C,∆, ε) together with a decomposition C = ⊕n∈N Cn in RMR, such
that ∆(Cn) in included into ⊕p+q=nCp ⊗ Cq, for all n. Hence, the comultiplication of C is given by a
family of R-bilinear maps ∆p,q : Cp+q → Cp ⊗ Cq. For a graded coring (C,∆, ε) we adapt the above
Σ-notation so that ∆p,q(c) is written as a sum ∆p,q(c) = ∑xp(1) ⊗ x
q
(2). Therefore, in a graded coring
∑(
cp+q(1)
)p
(1)
⊗
(
cp+q(1)
)q
(2)
⊗ cr(2) =
∑
cp(1) ⊗
(
cq+r(2)
)q
(1)
⊗
(
cq+r(2)
)r
(2)
, (1)
for all p, q, r ∈ N and c ∈ Cp+q+r. It is not hard to see that, for n > 0, the restriction of ε to Cn is 0.
In the graded case the counit satisfies the relations
∑
ε(c0(1))cn(2) = c =
∑
cn(1)ε(c0(2)), (2)
for every c ∈ Cn. We say that a graded R-coring C is connected if C0 = R. We denote the projection
of C onto Cn by pinC .
The comultiplication factors through a map ∆ : C → C ⊗C, where C := C/C0 denotes the quotient
R-bimodule. Note that (C,∆) is a coassociative coalgebra, without unit. We write pC for the canonical
projection of C onto C.
If (C,∆, ε) is an R-coring, then ∆op := Λ−1C,C ◦∆ together with ε define an Rop-coring Cop. We shall
say that Cop is the opposite coring of C.
1.3. Pre-Koszul pairs. Let C be a graded R-coring and let A be a graded R-ring. The pair (A,C) is
called pre-Koszul if A and C are connected, and there is an isomorphism θC,A : C1 → A1, such that
m1,1 ◦ (θC,A ⊗ θC,A) ◦∆1,1 = 0. (3)
Using the Σ-notation, this identity is equivalent to
∑
θC,A(c1(1))θC,A(c1(2)) = 0, for every c ∈ C2.
Remark 1.4. Let (A,C) be a pre-Koszul pair, where A and C are an R-ring and an R-coring,
respectively. Then (Aop, Cop) is also a pre-Koszul pair in the category of Rop-bimodules. Note that Aop
and Cop are an Rop-ring and an Rop-coring, respectively.
1.5. The normalized bar resolution of R in MA. We now want to show that, for every connected
graded R-ring A, which is generated by A1, there is a graded coring C such that (A,C) is pre-Koszul.
The groups TorA• (R,R) may be computed by using the bar resolution of R, cf. [Wei97, p. 283]. In
the graded case, the normalized version of the bar resolution may be used as well.
The right normalized bar resolution β¯r•(A) is the exact sequence in MA
0←− R←− A←− A¯⊗A←− · · · ←− A¯⊗n−1 ⊗A δ¯n←− A¯⊗n ⊗A←− · · · . (4)
where δ¯0 = 0 and
δ¯n(a1 ⊗ · · · ⊗ an ⊗ an+1) :=
∑n
i=1
(−1)ia1 ⊗ · · · ⊗ aiai+1 ⊗ · · · ⊗ an+1.

4 PASCUAL JARA, JAVIER LO´PEZ PEN˜A, AND DRAGOS¸ S¸TEFAN
By [Wei97, p. 283], the sequence of right A-modules β¯r•(A) is exact. Moreover, as A¯⊗n is projective as
a right R-module, it follows that A¯⊗n ⊗ A is projective in MA. Hence TorA• (R,R) is the homology of
normalized bar complex Ω•(A)
0←− R←− A¯←− A¯⊗ A¯←− · · · ←− A¯⊗n−1 d¯n←− A¯⊗n ←− · · · , (5)
where d¯1 = 0 and d¯n is defined by the same formula as δn−1, for any n > 0 and all a1, . . . , an ∈ A¯. The
homology of Ω•(A) will be regarded as the definition of TorA• (R,R). The homology class of x ∈ A in
T 1(A) will be denoted by [x].
The left version of the normalized bar resolution may be defined by β¯l•(A) := β¯r•(Aop). Note that
β¯rn(Aop) ∼= A
⊗n ⊗A.
Lemma 1.6. There is a canonical structure of differential graded R-coring on Ω(A) := ⊕nA¯⊗n. In
particular, T (A) := TorA• (R,R) is a connected graded R-coring, and T 1(A) is the cokernel of m.
Proof. We set Ωn(A) := A¯⊗n, where A¯⊗0 = R. On Ω(A) we put the comultiplication induced by the
canonical identifications A¯⊗(p+q) ≃ A¯⊗p ⊗ A¯⊗q. Clearly, Ω(A) becomes a connected graded R-coring.
The fact that Ω(A) is a DG-coring with respect to the differentials d¯• is proved in [PP05, Chapter 1.1],
in the particular case when R is a field and A is an R-algebra. It is easy to see that the same proof
works for R-rings as well. The homology of a DG-coring has a natural structure of graded R-coring, so
we conclude that T (A) is a graded R-coring. This coring is connected, since d1 = 0. The last assertion
is obvious, as d2 = m. 
Proposition 1.7. If A is a connected graded R-ring generated by A1, then (A, T (A)) is pre-Koszul.
Proof. We have already seen that T (A) is a connected graded R-coring. Since T 1(A) = A¯/A¯2, the
canonical projection pi1A of A onto A1 induces a map θT (A),A : T 1(A)→ A1. It follows that θT (A),A is an
isomorphism, as A¯/A¯2 = A1 (recall that A1 generates A). Furthermore, an element ω ∈ T 2(A) can be
written as the homology class of a certain ζ ∈ Ker m¯. Hence ζ = ∑ni=1 xi ⊗ yi, for some x1, . . . , xn and
y1, . . . , yn in A¯, such that
∑n
i=1 xiyi = 0. By definition,
∆1,1(ω) =
∑n
i=1
[xi]⊗ [yi] .
On the other hand, xiyi − pi1 (xi)pi1 (yi) belongs to
∑
n>2An, so
∑n
i=1
θT (A),A([xi])θT (A),A([yi]) =
∑n
i=1
pi1 (xi)pi1 (yi) = pi2
(∑n
i=1
xiyi
)
= 0.
Thus the relation (3) holds true. 
Our goal now is to associate to a pre-Koszul pair (A,C) three cochain complexes: one in the category
of graded left C-comodules and, symmetrically, one in the category of graded right C-comodules. By
combining the previous two constructions, we shall get the third cochain complex, that lives in the
category of graded C-bicomodules.
1.8. The categories MC , CM and CMC . Let C be an R-coring. The pair (M,ρM ) is a right C-
comodule if M is a right R-module and ρM : M →M ⊗C is a morphisms of right R-modules such that,
using the Σ-notation ρ(m) = ∑m〈0〉 ⊗m〈1〉, the relations below hold true for every m ∈M .
∑
m〈0〉〈0〉 ⊗m〈0〉〈1〉 ⊗m〈1〉 =
∑
m〈0〉 ⊗m〈1〉
(1)
⊗m〈1〉(2) ,
∑
m〈0〉ε
(
m〈0〉
)
= m,
The category MC is Grothendieck, as C is flat as a left R-module; see [Brz09, p. 264]. A right comodule
M is graded if M := ⊕nMn, and ρM (Mn) ⊆ ⊕p+q=nMp ⊗ Cq, for all n. Then the structure map of a
graded comodule M is uniquely defined by a family of R-bilinear maps ρp,qM : Mp+q →Mp ⊗ Cq.

ON KOSZULITY OF TWISTED TENSOR PRODUCTS 5
The category CM of left C-comodules is constructed in a similar way. For a left C-comodule (N, ρN )
we use the Σ notation ρN (n) =
∑n〈−1〉 ⊗ n〈0〉.
A C-bicomodule is a triple (M,ρlM , ρrM ) such that M is an R-bimodule, (M,ρlM ) is a left comodule,
(M,ρrM ) is a right comodule, and for every m ∈M∑
m〈−1〉 ⊗m〈0〉〈0〉 ⊗m〈0〉〈1〉 =
∑
m〈0〉〈−1〉 ⊗m〈0〉〈0〉 ⊗m〈1〉.
A morphism of C-bicomodules is a map which is left and right C-colinear. We write CMC for the
category of C-bicomodules.
Lemma 1.9. Let V be a right R-module. Then V ⊗ C is injective as a right C-comodule. A similar
result holds for left C-comodules.
Proof. The natural transformation
ΘM,V : HomR(M,V )→ HomC(M,V ⊗ C), ΘM,V (f) := (f ⊗ IdC) ◦ ρ (6)
is an isomorphism for any C-comodule M. Its inverse maps a morphism of C-comodules g : M → V ⊗C
to (V ⊗ ε)◦g. Therefore the functors HomC(−, V ⊗C) and HomR(−, V ) are isomorphic. As V is injective
(R is semisimple), the latter functor is exact. Then the other one is also exact. 
1.10. The complexes K•l (A,C) and K•r(A,C). Let (A,C) be a pre-Koszul pair. For an integer n ≥ 0
we set
Knl (A,C) := C ⊗An,
and we regard Knl (A,C) as a left C-comodule with structure map ∆⊗An. It is easy to see that Knl (A,C)
is a graded comodule, whose homogeneous component of degree r is Cr−n ⊗ An. The differential map
dnl : Knl (A,C)→ Kn+1l (A,C) is taken to be zero on C0 ⊗An. For p > 0 and c⊗ a ∈ Cp ⊗An,
dnl (c⊗ a) :=
∑
cp−1(1) ⊗ θC,A(c1(2))a,
where θC,A : C1 → A1, is the underlying isomorphism of the pre-Koszul pair (A,C). Note that, by
definition, dnl maps Cp⊗An to Cp−1⊗An+1, so dnl respects the gradings on Knl (A,C) and Kn+1l (A,C).
Recall that (Aop, Cop) is also a pre-Koszul pair, cf. Remark 1.4. Hence we may introduce
K•r(A,C) := K•l (Aop, Cop).
In view of the canonical isomorphism Cop ⊗Rop (Aop)n ∼= An ⊗R C, we regard K•r(A,C) as the complex
whose component of degree n is the R-bimodule Knr (A,C) := An ⊗R C. The differential dnr is zero on
An ⊗ C0. On the other hand, for p > 0 and a⊗ c ∈ An ⊗ Cp,
dnr (a⊗ c) =
∑
aθC,A(c1(1))⊗ c
p−1
(2) .
Proposition 1.11. If (A,C) is a pre-Koszul pair then (K•l (A,C), d•l ) is a complex in the category
of graded left C-comodules. Analogously, (K•r(A,C), d•r) is a complex in the category of graded right
C-comodules.
Proof. We only prove the first assertion. For the second one proceeds similarly. We first claim that
dn+1l ◦ dnl = 0. We may assume that p ≥ 2, otherwise the relation is trivially satisfied. Let θ := θC,A,
and pick up c⊗ a ∈ Cp ⊗An. Since the comultiplication in C is coassociative, and using (3), we get
(
dn+1 ◦ dn
)
(c⊗ a) =
∑
(cp−1(1) )
p−2
(1) ⊗ θ
(
(cp−1(1) )
1
(2)
)
θ(c1(2))a =
∑
cp−2(1) ⊗ θ
(
(c2(2))1(1)
)
θ
(
(c2(2))1(2)
)
a = 0.
Hence K•l (A,C) is a complex. Furthermore, for c⊗ a ∈ Cp ⊗An, we have
(C ⊗ dn)(ρ(c⊗ a)) =
∑
r+s=p
cr(1) ⊗ dn(cs(2) ⊗ a) =
∑
r+s=p
∑
cr(1) ⊗ (cs(2))s−1(1) ⊗ θ
(
(cs(2))1(1)
)
a.
On the other hand,
ρ(dn(c⊗ a)) = ρ
(∑
cp−1(1) ⊗ θ(c1(2))a
)
=
∑
u+v=p−1
(cp−1(1) )u(1) ⊗ (c
p−1
(1) )v(2) ⊗ θ(c1(2))a.

6 PASCUAL JARA, JAVIER LO´PEZ PEN˜A, AND DRAGOS¸ S¸TEFAN
Taking into account that the comultiplication is coassociative, by comparing the above two relations,
we deduce that dnl is a morphism of comodules. 
1.12. The complex K•(A,C). We are going to construct a cochain complex (K•(A,C), d•) in the
category CMC of C-bicomodules. By definition, Kn(A,C) = C ⊗An ⊗ C and
dn = dnl ⊗ C + (−1)n+1C ⊗ dnr . (7)
Since d•l is left C-colinear and d•r is right C-colinear it follows that d• is a morphism in CMC . Moreover,
as dn+1l dnl = dn+1r dnr = 0, we get
dn+1dn = (−1)n+1
[(
dn+1l ⊗ C
)
(C ⊗ dnr )−
(
C ⊗ dn+1r
)
(dnl ⊗ C)
]
.
By the definition of d•l and d•r , and by the fact the multiplication in A is associative, it results that
dn+1dn = 0, so (K•(A,C), d•) is a complex indeed.
1.13. The subcomplexes K•l (A,C,m) and K•r(A,C,m). For any integer m ≥ 0 we have already
noticed that dnl maps Cm−n ⊗ An to Cm−n−1 ⊗ An+1. Then we obtain a subcomplex K•l (A,C,m) of
K•l (A,C) such that Knl (A,C,m) := Cm−n⊗An. Its components are trivial in degree n > m. Therefore,
it can be displayed as follows
0 −→ Cm ⊗A0 −→ · · · −→ Cm−n ⊗An d
n
l−→ Cm−n−1 ⊗An+1 −→ · · · −→ C0 ⊗Am −→ 0.
Obviously, K•l (A,C) = ⊕mK•l (A,C,m). The complex K•r(A,C) admits a similar decomposition as a
direct sum of subcomplexes K•r(A,C,m).
1.14. The augmented complexes K˜•l (A,C), K˜•r(A,C) and K˜•(A,C). All three cochain complexes
that we have introduced can be augmented in a canonical way. The inclusion of R = C0 into C
defines the augmentation for both K•l (A,C) and K•r(A,C). The augmentation map of K•(A,C) is, by
definition, the comultiplication in C. These augmented complexes will be denoted by K˜•l (A,C), K˜•r(A,C)
and K˜•(A,C). Note that K˜•l (A,C) ∼= ⊕m>0K•l (A,C)⊕ K˜•l (A,C, 0), where K˜•l (A,C, 0) is the augmented
complex
0 −→ R ∼=−→ C0 ⊗Am −→ 0 −→ · · · .
The complex K˜•r(A,C) admits a similar decomposition.
1.15. The normalized bar resolution of R in MC . We are going to explain briefly how the preceding
constructions and results can be dualized. Let C be an R-coring. We assume that C is connected and
graded. Then R is a right C-comodule with respect to the trivial coaction. The bar resolution β•r (C) of
R is the exact sequence
0 −→ R −→ C −→ C ⊗ C −→ · · · −→ C⊗n+1 δ
n
−→ C⊗n+2 −→ · · · , (8)
where δn =
∑n+1
i=1 (−1)i−1IdC⊗i−1 ⊗∆⊗ IdC⊗n−i+1 . Note that C⊗n⊗C is an injective right C-comodule,
as C⊗n is injective as a right R-module. One can prove directly that this sequence is exact, by showing
that (−1)nIdC⊗n ⊗ ε is a contracting homotopy for β•r (C).
Recall that C := C/C0 and that ∆¯ : C → C⊗C is the unique map such that ∆¯◦pC = (pC ⊗ pC)◦∆,
where pC is the canonical projection. The right normalized bar resolution β¯•r (C) of R is the resolution
0 −→ R −→ C −→ C ⊗ C −→ · · · −→ C⊗n ⊗ C δ¯
n
−→ C⊗n+1 ⊗ C −→ · · · , (9)
with δ¯n = ∑ni=1(−1)i−1IdC⊗i−1⊗∆¯⊗IdC⊗n−i⊗C+(−1)
nIdC¯⊗n⊗∆˜, where ∆˜ := (pC⊗IdC)◦∆. The left
bar resolution and the left normalized bar resolution will be denoted by β•l (C) and β¯•l (C), respectively.
Both resolutions β•r (C) and β¯•r (C) can be used to compute Ext•C(R,R). The complex that corresponds
to β¯•r (C) will play an important role in our work, so we are going to write it down explicitly. Applying
HomC(R,−) to β¯•r (C) and using the isomorphisms ΘR,−, we obtain the normalized bar complex Ω•(C)
0 −→ R −→ C −→ · · · −→ C⊗n d¯
n
−→ C⊗n+1 −→ · · · . (10)

ON KOSZULITY OF TWISTED TENSOR PRODUCTS 7
with the differential morphisms d¯0 = 0 and d¯n =
∑n
i=1(−1)i−1IdC⊗i−1 ⊗ ∆¯⊗ IdC⊗n−i , if n > 0.
Lemma 1.16. There is a canonical connected differential graded R-ring structure on Ω(C) := ⊕nC
⊗n.
In particular, E(C) := Ext•C(R,R) is a connected graded R-ring, and E1(C) = Ker∆¯.
Proof. Let Ωn(C) := C⊗n, where by convention we take C⊗0 = R. Obviously, the canonical identifica-
tions Ωp(C) ⊗ Ωp(C) ∼= Ωp+q(C) define an R-ring structure on Ω(C) := ⊕nC
⊗n which is compatible
with the differentials. 
1.17. Cogenerated corings. Let ∆p,q : Cp+q → Cp ⊗ Cq denote the components of the comultipli-
cation of a connected graded coring C. The maps ∆(n) : Cn → C1 ⊗ · · · ⊗ C1 are defined inductively
by
∆(n) = (IdC1 ⊗∆(n− 1)) ◦∆1,n−1, (11)
where ∆(1) = IdC1 and ∆(2) = ∆1,1. For ∆(n) we use the Σ-notation
∆(n)(c) =
∑
c1(1) ⊗ c1(2) ⊗ · · · ⊗ c1(n).
By the definition of ∆(n) and coassociativity, we have ∆(p+ q) = (∆(p) ⊗∆(q)) ◦∆p,q, so
∆(n)(c) =
∑
(cp(1))
1
(1) ⊗ · · · (c
p
(1))
1
(p) ⊗ (c
q
(2))
1
(1) ⊗ · · · (c
q
(2))
1
(q).
In the case when all ∆(n) are injective we shall say that C is cogenerated in degree one. Note that if C
is cogenerated in degree one, then ∆p,q is injective for all p and q, and conversely.
Proposition 1.18. If C is a connected graded R-coring and C is cogenerated in degree one, then
(E(C), C) is a pre-Koszul pair.
Proof. The differential of Ω•(C) is trivial in degree zero, so E•(C) is connected. Let Cn := pC(Cn).
Hence C = ⊕n>0C
n and pC is injective on each component of positive degree. In particular, Cn ∼= C
n.
Let θ : C1 → C denote the restriction of pC to C1. The image of θ is included into Ker∆¯ = E1(C),
so we may regard θ as a map that takes values in E1(C). We claim that the pair (E(C), C) satisfies the
identity (3) with respect to θC,E(C) := θ. If B2(C) is the group of 2-coboundaries in the normalized bar
complex Ω•(C) and c ∈ C2, then
∑
pC(c1(1)) · pC(c1(2)) =
∑
pC(c1(1))⊗ pC(c1(2)) +B2 = ∆¯(pC(c)) +B2(C) = d¯1(pC(c)) +B2(C) = 0.
Note that the first equality is a consequence of the definition of the multiplication in E(C). It remains
to prove that θ is bijective. Let c be a class in Ker∆¯. Let cn denote the homogeneous component of
degree n of c, so c =
∑d
n=0 cn. We claim that cn = 0 for n > 1. We have
∑d
n=2
∑n−1
r=1
(pC ⊗ pC)(∆r,n−s(cn)) = ∆¯(pC(c)) = 0.
Since ωr,n−r := (pC ⊗ pC)(∆r,n−r(cn)) belongs to C
r ⊗ Cn−r, in view of the foregoing remarks, we
deduce that ωr,n−r = 0, for every r such that 0 < r < n. As C is flat, it follows that pC ⊗ pC is injective
on Cr ⊗ Cs. Hence ∆r,n−r(cn) = 0. Since C is cogenerated in degree one we get cn = 0, for any n > 1.
In conclusion, the kernel of ∆¯ is included into C1. The other inclusion is obvious, so E1(C) = C1. We
deduce in particular that θ is an isomorphism. 
1.19. The cotensor product C . Recall that, for X ∈ MC and Y ∈ CM, the cotensor product
XCY is defined as kernel of ρX ⊗ IdY − IdX ⊗ ρV , where ρX and ρY are the comodule structure maps
of X and Y . If C is a connected graded coring, X is a right R-module and M is a left C-comodule, then
(X ⊗ C)CM ∼= X ⊗M. (12)
It is not difficult to show that the above isomorphism is induced by the map IdX ⊗ ε ⊗ IdM , whose
inverse is IdX ⊗ ρM .

8 PASCUAL JARA, JAVIER LO´PEZ PEN˜A, AND DRAGOS¸ S¸TEFAN
In the case when C is graded and connected, we know that R is a left and right C-comodule with
respect to the trivial coactions. For such a coring C, there is a canonical isomorphism XCR ∼= XcoC ,
where the set XcoC of coinvariant elements contains all x ∈ X such that ρX(x) = x⊗ 1.
Proposition 1.20. Let (A,C) be a pre-Koszul pair. The complexes K•(A,C)CR and RCK•(A,C)
are isomorphic to K•l (A,C) and K•r(A,C), respectively. There are similar isomorphisms for the aug-
mented complexes.
Proof. We have just seen that Kn(A,C)CR can be identified with the set of coinvariant elements in
Kn(A,C). Clearly, a coinvariant element can be written as a sum of tensor monomials c ⊗ a ⊗ 1, with
c ∈ C and a ∈ An. Thus the application that maps x ∈ C ⊗An to x⊗ 1 defines an isomorphism in CM
between Kn(A,C) and Kn(A,C)CR. For a and c as above, a straightforward computation yields
dn(a⊗ c⊗ 1) = dnl (a⊗ c)⊗ 1.
Therefore, the above isomorphisms commute with the differentials of K•(A,C)CR and K•l (A,C). 
1.21. The complexes Kl•(A,C), Kr•(A,C) and K•(A,C). Let (A,C) be a pre-Koszul pair. By duality,
we shall construct a chain complex in each of the categories AM, MA and AMA. We first construct the
complex Kr•(A,C), whose nth component is the right A-module Krn(A,C) := Cn ⊗ A. The differential
drn : Cn ⊗A→ Cn ⊗A is defined, for every n > 0, by the relation
drn(c⊗ a) =
∑
cn−1(1) ⊗ θC,A(c
1
(2))a.
Applying the previous construction to the opposite pre-Koszul pair (Aop, Cop) we obtain a complex
(Kl•(A,C), dl•) in AM. Explicitly, Kln(A,C) := A⊗ Cn and
dln(a⊗ c) =
∑
aθC,A(c1(1))⊗ cn−1(2) .
We combine these chain complexes to obtain another one (K•(A,C), d•) in the category of A-bimodules.
By definition, Kn(A,C) := A⊗ Cn ⊗A. In degree n the differential of K•(A,C) is the map
dn = dln ⊗A+ (−1)nA⊗ drn.
1.22. The subcomplexes Kr•(A,C,m) and Kl•(A,C,m). The chain complex Kr•(A,C) can be writ-
ten as a direct sum of subcomplexes ⊕mKr•(A,C,m), where Krn(A,C,m) := Cm−n ⊗ An. Therefore,
Kr•(A,C,m) is the complex
0←− C0 ⊗Am ←− · · · ←− Cn−1 ⊗Am−n+1 d
r
n←− Cn ⊗Am−n ←− · · · ←− Cm ⊗A0 ←− 0,
with the differentials inherited from Kr•(A,C). The complex Kl•(A,C) has a similar description. The
corresponding subcomplexes will be denoted in this case by Kl•(A,C,m).
1.23. The coaugmented complexes K˜l•(A,C), K˜r•(A,C), and K˜•(A,C). All chain complexes that
we have introduced can be coaugmented in a canonical way. The projection of A onto R = A0 defines
the augmentation for Kl•(A,C) and Kr•(A,C). The augmentation map of K•(A,C) is given by the
multiplication in A. The augmented complexes will be denoted by K˜l•(A,C), K˜r•(A,C) and K˜•(A,C).
Note that K˜l•(A,C) ∼= ⊕m>0Kl•(A,C) ⊕ K˜l•(A,C, 0), where K˜l•(A,C, 0) is the coaugmented complex
0←− R ∼=←− C0 ⊗A0 ←− 0←− · · · .
The complex K˜•r(A,C) admits a similar decomposition.
Proposition 1.24. Let (A,C) be a pre-Koszul pair. The complexes K•(A,C)⊗AR and R⊗AK•(A,C)
are isomorphic to Kl•(A,C) and Kr•(A,C), respectively. There are similar isomorphisms for the corre-
sponding coaugmented complexes.
Proof. Omitted, being similar to that one of Proposition 1.20. 

ON KOSZULITY OF TWISTED TENSOR PRODUCTS 9
2. Koszul pairs
In this section we shall investigate the exactness of the augmented complex that we associated to a
pre-Koszul pair (A,C). Roughly speaking, we shall show that this property is equivalent to the fact
that A is a Koszul algebra and C is a Koszul coalgebra.
Proposition 2.1. Let (A,C) be a pre-Koszul pair. The following conditions are equivalent.
(a) The complex K˜•l (A,C) is exact.
(b) The complex K˜•r(A,C) is exact.
(c) The complex K˜•(A,C) is exact.
Proof. Since K˜•r(A,C) ∼= K˜•l (Aop, Cop) and K˜•(Aop, Cop) ∼= K˜•(A,C), it is enough to prove that (a) and
(c) are equivalent. If K˜•(A,C) is exact, then it is split exact in the category of right C-comodules, as its
components are injective C-comodules. Thus the complex obtained by applying the functor (−)CR is
also exact. We conclude that (c) implies (a) by applying Proposition 1.20.
Conversely, let us assume that K˜•l (A,C) is exact. We denote, for simplicity, the complex K˜•(A,C)
by K˜•. If Xn,p := Cn−p⊗An, then K˜n = ⊕pXn,p⊗C. Moreover, for any n and any p, the map dnl ⊗ IdC
maps Xn,p ⊗ C to Xn+1,p ⊗ C, while Id⊗ dnr maps the same set to Xn+1,p+1 ⊗ C. In particular,
dn(Xn,p ⊗ C) ⊆ Xn+1,p ⊗ C +Xn+1,p+1 ⊗ C. (13)
Hence K˜•i is a subcomplex of K˜•, where
K˜ni := ⊕p≥iXn,p ⊗ C.
We first claim that L•i := K˜•i /K˜•i+1 is exact. Note that Xn+1,i+1 ⊗ C is a subset of K˜n+1i+1 . We have
already remarked that Id⊗dnr maps Xn,p⊗C to K˜n+1i+1 , for every p ≥ i. Thus the differential of L•i maps
the class x⊗ c+ K˜ni+1 in Lni to dnl (x)⊗ c+ K˜n+1i+1 . On the other hand, as R-bimodules, we have
Lni ∼= K˜•l (A,C, i) ⊗ C.
In view of this isomorphism, and of the foregoing remarks, the complexes L•i and K˜•l (A,C, i) ⊗ C are
isomorphic. Since K˜•l (A,C, i) is a direct summand of K˜•l (A,C), it is exact. Hence L•i is exact too, as C
is flat as a left R-module.
Our aim now is to show that all quotients K˜•/K˜•i are exact. For, we proceed by induction. Clearly
K˜•0 = K˜•, so the quotient is trivially exact. Assume that K˜•/K˜•i is exact, and consider the following
short exact sequence
0 −→ K˜•i /K˜•i+1 −→ K˜•/K˜•i+1 −→ K˜•/K˜•i −→ 0.
We have seen that K˜•i /K˜•i+1 is exact. Thus, a fortiori, the middle term has the same property. Now we
can prove that K˜• is exact. Let ω be a cocycle of degree n in K˜•. We choose a big enough i such that ω
belongs to Mni := ⊕p≤iXnp ⊗C. The part of degree n of the complex K˜•/K˜•i+1 and Mni are isomorphic
via the canonical projection. There is a unique differential ∂n : Mni −→Mn+1i which commute with the
projections. By construction ∂n and dn (the differential of K˜•) agree on Xnp⊗C, for any p ≤ i. Clearly,
M•i ∼= K˜•/K˜•i+1 and ω is a cocycle in M•i . Consequently, M•i is exact, and there is ζ ∈Mn−1i such that
ω = ∂n(ζ) = dn(ζ). Thus the proposition is proved. 
Proposition 2.2. Let HomC(R, K˜•r(A,C)) denote the complex that is obtained by applying the functor
HomC(R,−) to K˜•r(A,C). Then, HomC(R, K˜•r(A,C)) is isomorphic to (A•, 0).
Proof. The map ΘR,An is an isomorphism between HomC(R, K˜nr (A,C)) and HomR(R,An). Hence we
may identify HomC(R, K˜nr (A,C)) and An via the map f 7→ (IdC ⊗ ε) (f(1)). Its inverse maps a ∈ An
to the C-comodule morphism fa, which is uniquely defined by fa(1) = a ⊗ 1. Let ∂n be the map that

10 PASCUAL JARA, JAVIER LO´PEZ PEN˜A, AND DRAGOS¸ S¸TEFAN
corresponds to HomC(R, dnr ) through the above identifications. Thus HomC(R, K˜nr (A,C)) ∼= (A•, ∂•).
For a ∈ An we have
∂n(a) = (An+1 ⊗ ε) ((dnr ◦ fa)(1)) = (An+1 ⊗ ε)(dnr (a⊗ 1)) = 0.
Therefore ∂ = 0, so the proposition is proved. 
Proposition 2.3. Let (A,C) be a pre-Koszul pair. The following three conditions are equivalent.
(a) The complex K˜l•(A,C) is exact.
(b) The complex K˜r•(A,C) is exact.
(c) The complex K˜•(A,C) is exact.
Proof. We have remarked that K˜l•(A,C) and K˜•(A,C)⊗AR are isomorphic. Therefore, in view of [BG98,
A3, Lemma], it results that (a) and (c) are equivalent. Analogously, (b) and (c) are equivalent. 
Theorem 2.4. If one of the complexes K˜•l (A,C), K˜•r(A,C), K˜•(A,C), K˜l•(A,C), K˜r•(A,C) or K˜•(A,C)
is exact, then all other five are so.
Proof. In view of Proposition 2.1 and Proposition 2.3, it is enough to prove that K˜•l (A,C) is exact if,
and only if, K˜r•(A,C) is exact. Recall that both complexes admit decompositions
K˜•l (A,C) = ⊕mK˜•l (A,C,m) and K˜r•(A,C) = ⊕mK˜r•(A,C,m),
so that form > 0 the complexes K˜•l (A,C,m) and K˜r•(A,C,m) are equal to K•l (A,C,m) and Kr•(A,C,m),
respectively. By construction, they are concentrated in degree up to p, and Kpl (A,C,m) = Krm−p(A,C,m)
and dpl = drm−p, for any 0 ≤ p ≤ m. It follows that, for such values of p,
Hp(K•l (A,C,m)) = Hm−p(Kr•(A,C,m)).
All other (co)homology groups vanishes. Hence K•l (A,C,m) is exact if, and only if, Kr•(A,C,m) is so.
Clearly, K˜•l (A,C, 0) and K˜r•(A,C, 0) always are exact. We end the proof by remarking that a direct sum
of complexes is exact if, and only if, each summand is exact. 
Corollary 2.5. If one of the six (co)augmented complexes associated to a pre-Koszul pair (A,C) is
exact then
(i) The complex K˜•l (A,C) is a resolution of R by injective graded left C-comodules.
(ii) The complex K˜•r(A,C) is a resolution of R by injective graded right C-comodules.
(iii) The complex K˜l•(A,C) is a resolution of R by projective graded left A-modules.
(iv) The complex K˜r•(A,C) is a resolution of R by projective graded right A-modules.
Under the additional assumptions that A and C are projective R-bimodules, then
(i) If R is a separable algebra over a field K, the complex K˜•(A,C) is a resolution of C by injective
graded C-bicomodules.
(ii) If C is , then the complex K˜•(A,C) is a resolution of A by projective graded A-bimodules.
Remark 2.6. Since R is semisimple, every R-bimodule is projective as left and right module. Though,
in general, the category RMR is not semisimple. For example, in the case when R is an algebra over a
fieldK, one can see that RMR is semisimple if and only if R itself is projective in this category, if and only
if the multiplication map R⊗K R→ R splits in RMR. By definition such an algebra is called separable.
The semisimplicity of CMC may be characterized in a similar way. This discussion explains why, in the
above corollary, we added the assumption that A and C are projective as R-bimodules, in order that
K˜•(A,C) and K˜•(A,C) be projective resolutions. Otherwise, we do not know if A⊗Cn⊗A (respectively
C ⊗An ⊗ C) is projective (respectively injective) as an A-bimodule (respectively C-bicomodule).
Definition 2.7. A pre-Koszul pair (A,C) is called Koszul if the complexes in Theorem 2.4 are exact.
Corollary 2.8. If (A,C) is Koszul then (Aop, Cop), the opposite pre-Koszul pair, is exact too.

ON KOSZULITY OF TWISTED TENSOR PRODUCTS 11
Proof. We have noticed that K˜•(Aop, Cop) ∼= K˜•(A,C), so one of these complexes is exact if, and only
if, the other one is exact too. 
To simplify the notation, we shall write c′ instead of pi1C(c), for all c ∈ C.
Proposition 2.9. Let (A,C) be a pre-Koszul pair. Let θ := θC,A. The R-bilinear maps
ψn : C⊗n ⊗ C → An ⊗ C, ψn(c1 ⊗ · · · ⊗ cn ⊗ c) =
∑
θ(c′1) · · · θ(c′n)⊗ c
define a morphism ψ• of chain complexes from β•r(C) to (K˜•r(A,C), (−1)•d•r).
Proof. To simplify the notation, we shall denote the homogeneous component of degree one of c ∈ C by
c′. Therefore, c′ := pi1C(c). It is not difficult to see that the relations below hold for x ∈ C and c ∈ Cp.
∑
θ(x′(1))θ(x′(2)) = 0 and
∑
θ(c′(1))⊗ c(2) =
∑
θ(c1(1))⊗ c
p−1
(2) , (14)
Then, for ci ∈ C, and c as before, we obtain
ψn+1 ◦ d¯nr (c1 ⊗ · · · ⊗ cn ⊗ c) =
∑n
i=1
∑
(−1)i−1θ(c′1) · · · θ((ci(1) )′)θ((ci(2) )′) · · · θ(c′n)⊗ c+
+ (−1)n
∑
θ(c′1) · · · θ(c′n)θ(c′(1))⊗ c(2).
Taking into account the first relation in (14) one shows that the double sum is zero. On the other hand,
(−1)ndnr ◦ ψn(c1 ⊗ · · · ⊗ cn ⊗ c) = (−1)n
∑
θ(c′1) · · · θ(c′n)θ(c1(1))⊗ cp−1(2) .
To conclude the proof of the proposition one uses the second identity in (14). 
Proposition 2.10. Suppose that (A,C) is Koszul. Then A is generated in degree one and C is cogen-
erated in degree one. In particular, (A, T (A)) and (E(C), C) are pre-Koszul pairs.
Proof. In view of Theorem 2.4, we may assume that K˜•r(A,C) is exact. The part in low degrees of this
sequence is
0 −→ R σ
0
−→ C d
0
r−→ A1 ⊗ C,
where R is placed in degree −1. Let θ := θC,A. By definition, d0r and (θ ⊗ IdCn) ◦∆1,n agree on Cn+1.
As θ is bijective, Ker∆1,n = Kerd0r |Cn+1. On the other hand, since the sequence is exact in C, we get
Kerd0r = C0. Then, a fortiori, Kerd0r|Cn+1 = 0. Thus ∆1,n is injective for every n. By induction on n,
using relation (11), we deduce that ∆(n) is injective, for all n > 0.
Let us prove that A is generated by A1. The morphism ψ• that we defined in Proposition 2.9 lifts
the identity of R. By hypothesis (K˜•r(A,C), (−1)•d•r) and β¯•r (C) are resolutions of R in MC . It follows
that λ• := HomC(R,ψ•) is a quasi-isomorphism of complexes. On the other hand, we have proven that
HomC(R, K˜•r(A,C)) ∼= (A•, 0). By applying the functor HomC(R,−) to β¯•r (C), we obtain the complex
Ω•(C), so λ• may be regarded as a morphism of complexes from Ω•(C) to (A•, 0) satisfying
λn(c1 ⊗ · · · ⊗ cn) = θ(c′1) · · · θ(c′n),
where all c¯i’s are in C and c′ = pi1(c), for every c ∈ C. Passing to cohomology, we get an isomorphism,
λ˜n : En(C)→ An. Let us pick up a ∈ An. Then a is a cocyle in (A•, 0), so there is a cohomology class
ω = ∑ni=1 c1,i ⊗ · · · ⊗ cn,i +Bn(C) in En(C) such that λ˜n(ω) = a. Equivalently,
a =
∑n
i=1
θ(c′1,i) · · · θ(c′n,i).
This means that a ∈ (A1)n, i.e. A is generated by A1. 
Proposition 2.11. Let (A,C) be a pre-Koszul pair. Let θ := θC,A. The R-bilinear maps
φn : A⊗ Cn → A⊗A
⊗n, φn := IdA ⊗
(
θ⊗n ◦∆(n)
)
define a morphism φ• of complexes from K˜l•(A,C) to the normalized bar resolution β
l
•(A).

12 PASCUAL JARA, JAVIER LO´PEZ PEN˜A, AND DRAGOS¸ S¸TEFAN
Proof. We first compute the value of φn−1 ◦ dln at a⊗ c ∈ A⊗Cn. By the definitions of φn and dln, and
using (11), we get
φn−1 ◦ dln(a⊗ c) =
∑
aθ(c1(1))⊗ θ(c1(2))⊗ · · · ⊗ θ(c1(n)).
On the other hand, using again the definitions of φn and δn, we get
δn ◦ φn(a⊗ c) =
∑
aθ(c1(1))⊗ θ(c1(2))⊗ · · · ⊗ θ(c1(n))+
+
∑n−1
i=1
∑
(−1)i a⊗ θ(c1(1))⊗ · · · ⊗ θ(c1(i))θ(c1(i+1))⊗ · · · ⊗ θ(c1(n)).
By coassociativity and the condition (3) it follows that, for a given i, the corresponding term in the
above sum is zero. By comparing the results of the above computations we deduce that φ• is a morphism
of cochain complexes. 
Let (A,C) be a Koszul pair. We have seen that A is generated by A1 and (A, T (A)) is pre-Koszul.
We are going to prove that (A, T (A)) is Koszul, and that this property characterizes Koszul algebras, a
class of algebras that has been intensively studied in the literature; see for example [BGS96, Pri70] and
the references therein.
Let V denote an R-bimodule. On TR(V ) := ⊕nV ⊗n one defines a graded R-coring structure such
that ∆p,q is the canonical identification V ⊗p+q ∼= V ⊗p ⊗ V ⊗q. Its counit is IdR on R := V ⊗0, and zero
on all other summands. To make distinction between the usual ring structure on TR(V ) and the coring
structure that we have just introduced, we denote them by T aR(V ) and T cR(V ), respectively.
2.12. The pre-Koszul pair (AW , CW ). If W ⊆ V ⊗ V is a subbimodule, then we set C0W := R,
C1W := W and
CnW =
⋂n−2
p=0
V ⊗p ⊗W ⊗ V ⊗n−p−2,
for every n ≥ 2. The R-bimodule CW :=
⊕
n∈NCnW . We claim that CW is a graded sub-coring of T cR(V ).
To show this, we have to prove that ∆p,q(x) ∈ CpW ⊗ C
q
W , for all p and q, and any x ∈ C
p+q
W . In the
case when p = 0 or q = 0, this property is obvious. Let us assume that p > 0 and q > 0. Clearly
∆p,q(x) ∈ CpR ⊗ T q, if p = 1. In the case when p ≥ 2 we have ∆p,q(x) = x, where in the right hand side
of this identity one regards x as an element in V ⊗p ⊗ V ⊗q. Thus
∆p,q(x) ∈
⋂p+q−2
i=0
V ⊗i ⊗W ⊗ V ⊗p+q−i−2 ⊆
⋂p−2
i=0
V ⊗i ⊗W ⊗ V ⊗p+q−i−2 = CpW ⊗ V ⊗q.
Similarly, ∆p,q(x) ∈ V ⊗p ⊗ CqW . Hence ∆p,q(x) ∈ C
p
W ⊗ C
q
W , that means that our claim is proved.
We define AW to be the quotient graded R-algebra of T aR(V ) modulo the two-sided ideal generated
by W. By construction, AW and CW are graded and connected, and A1W = C1W . If x ∈ C2W = W, then
∆1,1(x) = x belongs to V ⊗ V = A1W ⊗A1W . On the other hand the multiplication on T aR(V ) is defined
by the canonical isomorphisms V ⊗p⊗V ⊗q ∼= V ⊗p+q. Henceforth, m1,1(∆1,1(x)) = x+W = 0. Thus the
conditions that are required in the definition of pre-Koszul pairs are trivially satisfied for (AW , CW ).
Theorem 2.13. Let A be a connected graded R-ring. The following assertions are equivalent.
(a) The R-ring A is Koszul in the usual sense, cf. [BGS96, Definition 1.2.1].
(b) There is a graded R-coring C such that (A,C) is a Koszul pair.
(c) The R-ring A is generated by A1 and (A, T (A)) is Koszul.
Proof. Let us assume that A is Koszul in the usual sense. Any Koszul algebra is quadratic, cf. [BGS96].
In fact one may choose W ⊆ A1 ⊗ A1 such that A ∼= AW . By the very definition of the ring structure
on AW and the coring structure on CW , it follows that Kl•(AW , CW ) coincides with the Koszul complex
[BGS96, p. 483]. As A is Koszul the latter complex is exact, so (AW , CW ) is Koszul.
Let (A,C) be an arbitrary Koszul pair. We have already proven that, for such a pair, A is generated
by A1, and (A, T (A)) is pre-Koszul. We claim that (A, T (A)) is Koszul. Let φ• : K˜l•(A,C) → β
l
•(A)
be the morphism of complexes that we constructed in Proposition 2.11. Proceeding as in the proof of

ON KOSZULITY OF TWISTED TENSOR PRODUCTS 13
Proposition 2.10, one shows that φ˜• : (C•, 0)→ (Ω•(A), d¯•) is a quasi-isomorphism between the graded
R-bimodule C regarded as a complex with zero differentials and (Ω•(A), d¯•). This quasi-isomorphism
respects the coring structures so, passing to homology, it yields an isomorphism of graded corings, still
denoted φ˜•, between C and T (A). For c ∈ C it verifies the relation
φ˜n(c) := ∆(n)(c) +Bn(A),
where Bn(A) is the group of n-boundaries in Ωn(A). By a standard computation one checks that φ˜•⊗IdC
is an isomorphism of chain complexes from K˜l•(A,C) to K˜l•(A, T (A)), so the latter complex is exact.
Finally, let as assume that (A, T (A)) is Koszul. Hence K˜l•(A, T (A)) is a resolution of R by graded
left A-modules, such that each K˜ln(A, T (A)) is generated as a module by its homogeneous component
of degree n, so A is a Koszul algebra in the usual sense, cf. [BGS96, Definition 1.2.1]. 
Corollary 2.14. If A is Koszul in the usual sense, then Aop is Koszul too.
Proof. Let A be a graded ring, which is Koszul in the usual sense. Then (A, T (A)) is Koszul and A is
generated by A1, by the preceding theorem. Since (Aop, T (A)op) is Koszul too, and Aop is generated by
(Aop)1, it follows that Aop is Koszul in the usual sense. 
2.15. Koszul R-corings. The preceding theorem has a counterpart for corings. To define Koszul
R-corings we dualize [BGS96, Definition 1.2.1]. Let C be an R-coring. We say that a graded right
C-comodule M := ⊕nMn, with structure map ρM , is cogenerated in degree n if ρn,pM : Mn+p →Mn⊗Cp
is injective for all p ≥ 0.
The coring C is said to be (right) Koszul if there is an injective resolution 0 −→ R −→ I• of R by
injective graded right C-comodules, such that each In is cogenerated in degree n.
Theorem 2.16. Let C be a connected graded R-coring. The following assertions are equivalent.
(a) The R-coring C is Koszul.
(b) There is a graded R-ring A such that (A,C) is a Koszul pair.
(c) The R-coring C is cogenerated in degree one and (E(C), C) is Koszul.
Proof. By duality, from the proof of Theorem 2.13. The details are left to the reader. 
Lemma 2.17. Let (A,C′) and (A,C′′) be Koszul pairs. Then C′ ∼= C′′ as graded R-corings. Dually, If
(A′, C) and (A′′, C) are Koszul, then A′ ∼= A′′ as graded R-rings.
Proof. By the proof of Theorem 2.13, there canonical coring isomorphisms C′ ∼= T (A) and C′′ ∼= T (A).
For the proof of second part of the corollary, one uses the isomorphisms A′ ∼= E(C) and A′′ ∼= E(C)
that are induced by λ•; see the proof of Proposition 2.10. 
Proposition 2.18. If A is a Koszul R-coring, then E(T (A)) ∼= A. Dually, if C is a Koszul R-coring,
then T (E(C)) ∼= C.
Proof. Since the pair (A, T (A)) is Koszul, it follows that T (A) is cogenerated in degree one and the
pair (E(T (A)), T (A)) is Koszul; see the implication (b) ⇒ (c) in Theorem 2.16. Now we can apply the
previous lemma to deduce that A ∼= E(T (A)). The second isomorphism is proved in the same way. 
Remark 2.19. Under the additional assumption that A is left finite (i.e. the homogeneous parts of
A are finitely generated left R-modules), the first isomorphism in the above corollary is also proved in
[BGS96, Theorem 2.10.2], in a slightly different form. More precisely, in loc. cit. the authors work with
the Koszul dual of A, which is an R-ring, and not a coring.

14 PASCUAL JARA, JAVIER LO´PEZ PEN˜A, AND DRAGOS¸ S¸TEFAN
3. The pre-Koszul pair of a twisted tensor product
Throughout R is a ring. All objects that we shall work with are R-bimodules and all maps are
morphisms of R-bimodules, otherwise explicitly stated. Recall that the symbol ⊗ stands for the tensor
product in the monoidal category of R-bimodules. In this section we consider two connected graded
R-rings A and B that are generated in degree 1, and a graded twisting map (see definition below)
σ : B ⊗A→ A⊗B. Our first aim is to show that σ induces a graded twisting map of R-corings
τ : T (A)⊗ T (B)→ T (B)⊗ T (A)
such that (A ⊗σ B, T (A) ⊗τ T (B)) is a pre-Koszul pair. Of course, here A ⊗σ B denotes the twisted
tensor product R-ring with respect to σ. Similarly, T (A)⊗τ T (B) is the twisted tensor product of T (A)
and T (B) with respect to τ. Furthermore, if A and B are Koszul, then we shall show that this pair is
Koszul, which in particular means that A⊗σB is a Koszul algebra. In order to prove the last mentioned
result, for a twisting map σ as above, we shall also construct an invertible entwining map
λ : T (A)⊗B → B ⊗ T (A).
3.1. Graded twisting maps of R-rings. Let A and B be graded R-rings. A graded twisting map
between A and B is given by a morphisms of R-bimodules
σ : B ⊗A→ A⊗B
such that σ(Bp ⊗Aq) ⊆ Aq ⊗Bp, and the following relations hold:
σ ◦ (IdB ⊗mA) = (mA ⊗ IdB) ◦ (IdA⊗σ) ◦ (σ ⊗ IdA), (15)
σ ◦ (mB ⊗ IdA) = (IdA⊗mB) ◦ (σ ⊗ IdB) ◦ (IdB ⊗σ). (16)
In addition, σ(1B ⊗ a) = a⊗ 1B and σ(b ⊗ 1A) = 1A ⊗ b, for all a ∈ A and b ∈ B. For the restrictions
of σ to Bp ⊗Aq we shall use the notation σp,q.
In computations we shall denote σ(b⊗ a) ∈ A⊗B, as a symbolic sum ∑σ aσ ⊗ bσ. Thus, for instance
(IdA⊗σ) ◦ (σ ⊗ IdB)(b⊗ x⊗ y) =
∑
σ,σ′
xσ ⊗ yσ′ ⊗ (bσ)σ′ .
The occurrence of σ and σ′ in the right hand side of the above identity indicates that the twisting map
is used twice: the first time on the first and the second factors, and then once again on the second and
the third factors.
If σ is a twisting map then A⊗B inherits a graded R-ring structure from the existing ones in A and
B, with respect to the multiplication
(a′ ⊗ b′)(a′′ ⊗ b′′) =
∑
σ
a′a′′σ ⊗ b′σb′′.
and the unit 1A ⊗ 1B. The twisted tensor product will be denoted by A ⊗σ B. In the case when R
is commutative, and A and B are R-algebras, the twisted tensor product A ⊗σ B may be seen as a
deformation of the usual tensor product algebra.
3.2. Twisting maps of DG-rings. We now assume that (A, dA) and (B, dB) are differential graded
rings. A graded twisting map σ : B⊗A→ A⊗B is called a twisting map of DG-rings if σ is compatible
with the differential maps of A and B, in the sense that, for every q the family of morphisms σ•,q is
a map of complexes from (B ⊗ Aq, dB ⊗ IdAq ) to (Aq ⊗ B, IdAq ⊗dB). Symmetrically, σp,• must be a
morphism between (Bp ⊗A, IdBp ⊗dA) and (A⊗Bp, dA ⊗ IdBp), for all p.
Proposition 3.3. Let (A, dA) and (B, dB) be differential graded corings. Suppose that V is an R-
bimodule.

ON KOSZULITY OF TWISTED TENSOR PRODUCTS 15
(i) If σA : (V ⊗A, IdV ⊗dA)→ (A⊗V, dA⊗ IdV ) is a morphism of complexes that satisfies (15), with
B replaced by V, then H•(σA) still satisfies this relation (note that H•(A) is an R-ring, as A is a
DG-ring). Moreover, for every v ∈ V such that σA(v ⊗ 1) = 1⊗ v, then
H•(σA)(v ⊗ 1¯) = 1¯⊗ v.
(ii) If σB : (B ⊗ V, dB ⊗ IdV ) → (V ⊗ C, IdV ⊗dB) is a morphism of complexes that satisfies (16),
with A replaced by V, then H•(σB) still satisfies this relation. Moreover, for every v ∈ V such that
σB(1⊗ v) = v ⊗ 1, then
H•(σA)(1¯⊗ v) = v ⊗ 1¯.
(iii) If σ : B ⊗ A → A ⊗ B is a twisting map of differential graded corings, then σ induces a twisting
map of corings
σ¯ : H•(B)⊗H•(A)→ H•(B)⊗H•(A).
Proof. Let (σpA)p be the components of σ. They define maps
Hp(σA) : Hp(V ⊗A)→ Hp(A⊗ V ).
By assumption every left or right R-module is flat. Hence, σ¯ := Hp(σA) can be seen as a map from
V ⊗Hp(A) to Hp(A)⊗ V. For ω ∈ Zp(A) let ω¯ denote the cohomology class of ω. If v ∈ V, then
Hp(σA)(ω¯ ⊗ v) =
∑
vσA ⊗ ωσA , (17)
where in the above relation we used the same notation as in the case of twisting maps of R-rings. Since
σA satisfies the first identity in the definition of twisting maps of graded R-rings, we get
∑
σA
(ζξ)σA ⊗ vσA =
∑
σ,σ′
ζσAξσ′A ⊗ (vσA)σ′A .
By the definition of the multiplication in the cohomology R-ring, and using the relation (17), we get
∑
σ¯
(
ζ¯ ξ¯
)
σ¯ ⊗ vσ¯ =
∑
σ¯,σ¯′
ζ¯σ¯ ξ¯σ¯′ ⊗ (vσ¯)σ¯′ .
In conclusion, H•(σA) satisfies (15). Using (17), we get that H•(σA)(1¯ ⊗ v) = v ⊗ 1¯, for all v ∈ V.
To prove the second part of the proposition one proceeds similarly. The third part of the proposition
follows in two steps. We fix p ≥ 0. As σp,• is a morphism of complexes from Bp ⊗ A to A ⊗ Bp, that
satisfies (15), by the first part of the proposition we deduce that H•(σp,•) has the same property. Now
we consider the family of morphisms of complexes Hq(σ•,q) : B ⊗Hq(A) → Hq(A)⊗ B. One can easily
see that it verifies (16), as σ does. Thus, passing to cohomology, for each p and q, we get a map
σ¯p,q : Hp(B) ⊗Hq(A)→ Hq(A) ⊗Hp(B).
The family σ¯ = (σ¯p,q)p,q, in view of the foregoing remarks, is obviously a twisting map. 
3.4. Graded twisting maps of R-corings. Let C and D be two graded R-corings. A graded twisting
map is given by an R-bilinear map
τ : C ⊗D → D ⊗ C
such that τ(Cp ⊗Dq) ⊆ Dq ⊗ Cp and the following relations hold:
(∆D ⊗ IdC) ◦ τ = (IdD ⊗τ) ◦ (τ ⊗ IdD) ◦ (IdC ⊗∆D), (18)
(IdD ⊗∆C) ◦ τ = (τ ⊗ IdC) ◦ (IdC ⊗τ) ◦ (∆C ⊗ IdD). (19)
In addition, (IdD ⊗εC) ◦ τ = εC ⊗ IdD and (εD ⊗ IdC) ◦ τ = IdC ⊗εD. The restrictions of τ to Cp ⊗Dq
will be denoted by τp,q.
For a twisting map of corings we use the notation τ(c ⊗ d) = ∑τ dτ ⊗ cτ , for all c ∈ C and d ∈ D.
The tensor product C ⊗D inherits a graded R-coring structure, that will be denoted by C ⊗τ D. Its
comultiplication ∆ is defined by the formula
∆ := (C ⊗ τ ⊗D) ◦ (∆C ⊗∆D),

16 PASCUAL JARA, JAVIER LO´PEZ PEN˜A, AND DRAGOS¸ S¸TEFAN
and its counit is εC ⊗ εD. The twisted tensor product will be denoted by C ⊗τ D.
3.5. Twisting maps of DG-corings. We now assume that (C, dC) and (D, dD) are differential graded
corings. A graded twisting map τ : C ⊗ D → D ⊗ C is called a twisting map of DG-corings if τ is
compatible with the differentials of C and D in the sense that τp,• and τ•,q are morphisms of chain
complexes, for all p and q.
Reasoning as in the proof of Proposition 3.3, one can show that the following result holds true.
Proposition 3.6. Let (C, dC) and (D, dD) be differential graded corings. Suppose that V is an R-
bimodule.
(a) If τD : (V ⊗ D, IdV ⊗ dD) → (D ⊗ V, dD ⊗ IdV ) is a morphism of complexes that satisfies (18),
with C replaced by V, then H•(τD) still satisfies this relation (note that H•(D) is a coring, as C is
a DG-coring). Moreover, if (εD ⊗ IdV ) ◦ τD = IdV ⊗εD, then
(εH•(D) ⊗ IdV ) ◦H•(τD) = IdV ⊗εH•(D).
(b) If τC : (C ⊗ V, dC ⊗ IdV )→ (V ⊗ C, IdV ⊗dC) is a morphism of complexes that satisfies (19), with
D replaced by V, then H•(τC) still satisfies this relation. Moreover, if (IdV ⊗εC) ◦ τC = εC ⊗ IdV ,
then
(IdV ⊗εH•(C)) ◦H•(τC) = IdV ⊗εH•(C).
(c) If τ : C ⊗ D → D ⊗ C is a twisting map of differential graded corings, then τ induces a twisting
map of corings
τ¯ : H•(C) ⊗H•(D)→ H•(D)⊗H•(C).
3.7. Graded entwining maps. Let A be a graded R-ring, and let C be a graded R-coring. Let
λ : C ⊗A→ A⊗ C
be an R-bilinear map such that λ(Cp ⊗ Aq) ⊆ Aq ⊗Cp. We denote the restriction of λ to Cp ⊗ Aq by
λp,q. We say that λ is a graded entwining map if λ(c⊗ 1A) = 1A⊗ c and (IdA⊗εC) ◦ λ = εC ⊗ IdA, and
the following two relations hold:
λ ◦ (C ⊗mA) = (mA ⊗ C) ◦ (A⊗ λ) ◦ (λ⊗A), (20)
(A⊗∆C) ◦ λ = (λ ⊗ C) ◦ (C ⊗ λ) ◦ (∆C ⊗A). (21)
Let λ : C ⊗ A → A ⊗ C be a graded entwining map. We assume that, in addition, A is a differential
graded R-ring and C is a differential graded R-coring. We say that a graded entwining map λ is a
differential graded entwining map if λp,• and λ•,q are morphisms of chain complexes, for any p and q.
We state for future reference, without proof, the following proposition.
Proposition 3.8. If (A, dA) is a DG-ring, (C, dC) is a DG-coring and λ : C ⊗ A → A ⊗ C is a
differential graded entwining map, then λ induces an graded entwining map
λ¯ : H•(C) ⊗H•(A)→ H•(A)⊗H•(C).
Recall that, for every R-bimodule V, the graded R-ring T aR(V ) and the graded R-coring T cR(V ) have
the same homogeneous component of degree n, namely V ⊗n. The multiplication and the comultiplication
are induced by the canonical identification V ⊗p ⊗ V ⊗p ∼= V ⊗p+q.
Let α : W ⊗V → V ⊗W be a morphism of R-bimodules. We define αp,q : W⊗p⊗V ⊗q → V ⊗q⊗W⊗p
as follows. First, we take α0,p to be the canonical identification R⊗V ⊗p ∼= V ⊗p⊗R, and then we define
αq,0 analogously. For q > 0 we set
α1,q := (V ⊗q−1 ⊗ α) ◦ (V ⊗q−2 ⊗ α⊗ V ) ◦ · · · ◦ (α⊗ V ⊗q−1).
Note that α1,1 = α. Finally, if p > 0 we put
αp,q :=
(
α1,q ⊗W⊗p−1
) (
W ⊗ α1,q ⊗W⊗p−2
)
◦ · · · ◦
(
W⊗p−1 ⊗ α1,q
)
.

ON KOSZULITY OF TWISTED TENSOR PRODUCTS 17
Let α˜ : TR(W )⊗ TR(V )→ TR(V )⊗ TR(W ) denote the unique map induced by the family (αp,q)p,q∈N .
Here TR(V ) and TR(W ) are regarded just as R-bimodules, without any other additional structure.
The map α˜ provides important examples of twisting and entwining maps. The next result is ‘folklore’,
so we omit the proof.
Lemma 3.9. Let α : W ⊗V → V ⊗W be a morphism of R-bimodule. Let α˜ be the lifting of α as above.
(a) The map α˜ : T aR(W )⊗ T aR(V )→ T aR(V )⊗ T aR(W ) is a graded twisting map.
(b) The map α˜ : T cR(W )⊗ T cR(V )→ T cR(V )⊗ T cR(W ) is a graded twisting map.
(c) The map α˜ : T cR(W )⊗ T aR(V )→ T aR(V )⊗ T cR(W ) is a graded entwining map.
Let σ : B ⊗ A → A ⊗ B be a twisting map of graded connected R-rings. Recall that Ω•(A) is a
DG-coring, and Ω(A) ∼= T cR(A¯) as graded corings. Since σ respects the gradings on A and on B, it maps
B ⊗ A¯ to A¯⊗ B. Let α denote the restriction of σ to B ⊗ A¯. If (αp,q)p,q are the liftings of α as above,
then α˜ : Ω(B)⊗ Ω(A)→ Ω(A)⊗ Ω(B) is a graded twisting map of R-corings. Using the relations
αp+q,r = (B⊗p+1 ⊗ αq−1,r) ◦ (B⊗p−1 ⊗ α2,r ⊗B⊗q−1) ◦ (αp−1,r ⊗Bq+1),
together with the fact that σ is a twisting map of R-rings, one shows by induction that α•,q is a
morphism of complexes. In a similar way it follows that α•,q is morphism of complexes too. Therefore
α˜ is a twisting map of DG-corings, so we may apply the Proposition 3.6 for α˜ to get a twisting map of
corings τ : T (B)⊗ T (A)→ T (A)⊗ T (B).
For q > 0, let βp,q be the restriction of αp,1 to B⊗p⊗Aq.We take βp,0 to be the canonical identification
B⊗p⊗R ∼= R⊗B⊗p. Since (βp,q)p,q defines an entwining map between Ω(B) and A, which is compatible
with the differential maps (one regards A as a chain complex with trivial differentials), by Proposition
3.8 there is a graded entwining map λ : T (B)⊗A→ A⊗ T (B). Note that, by construction
(
θT (A),A ⊗ IdTp(B)
)
◦ τp,1 = λp,1 ◦ (IdTp(B) ⊗ θT (A),A), (22)
(IdAq ⊗ θT (B),B) ◦ λ1,q = σ1,q ◦ (θT (B),B ⊗ IdAq ), (23)
where θT (A), and θT (B),B are the isomorphisms corresponding to the pre-Koszul pairs (A, T (A)) and
(B, T (B)), respectively. In particular, taking p = q = 1 and composing the second relation to the right
with IdT 1(B) ⊗ θA we deduce that
σ1,1 ◦ (θT (B),B ⊗ θT (A),A) = (θT (A),A ⊗ θT (B),B) ◦ τ1,1. (24)
Of course, if σ is an invertible twisting map, then τ and λ are invertible too. Since σ−1 is a twisting map
we can use it to produce a new invertible graded twisting map of corings τ ′ : T (A)⊗T (B)→ T (B)⊗T (A)
and new graded invertible entwining map λ′ : T (A) ⊗ B → B ⊗ T (A). Clearly, τ ′ and λ′ satisfy the
above relations with A and B interchanged, and with σ1,1 and σ1,q replaced by
(
σ1,1
)−1 and
(
σq,1
)−1 ,
respectively. Later on in the paper, we shall use only σ, τ ′ and λ′. For this reason we shall change the
notation, writing τ and λ instead of τ ′ and λ′, respectively. We summarize up the foregoing remarks in
the next proposition.
Proposition 3.10. Let σ : B ⊗ A → A ⊗ B be an invertible twisting map of connected graded R-
corings. If A and B are generated in degree one, then there are an invertible graded twisting map
τ : T (A)⊗T (B)→ T (B)⊗T (A) and an invertible graded entwining map λ : T (A)⊗B → B⊗T (A) such
that the relations (22), (23) and (24) hold with A and B interchanged, and with σ1,1 and σ1,q replaced
by
(
σ1,1
)−1 and
(
σq,1
)−1 , respectively.
Our goal is to construct a pre-Koszul pair (A ⊗σ B, T (A) ⊗τˆ T (B)), where τˆ and τ are equal up to
a sign (see Remark 3.12 for the definition of τˆ ). Then we shall prove that this pair is Koszul, provided
that the algebras A and B are Koszul algebras. In fact we shall do that in a more general setting, that
we are now going to introduce.

18 PASCUAL JARA, JAVIER LO´PEZ PEN˜A, AND DRAGOS¸ S¸TEFAN
3.11. Notation and assumptions. From now on, (A,C) and (B,D) are two pre-Koszul pairs that
are equipped with the following extra data:
(i) An invertible graded twisting map σ : B ⊗A→ A⊗B .
(ii) An invertible graded twisting map τ : C ⊗D → D ⊗ C.
(iii) An invertible entwining map λ : C ⊗B → B ⊗ C
We impose on these data the following two conditions:
λp,1 ◦ (IdCp ⊗ θD,B) = (θD,B ⊗ IdCp) ◦ τp,1, (25)
(
σq,1
)−1 ◦ (θC,A ⊗ IdBq ) = (IdBq ⊗ θC,A) ◦ λ1,q, (26)
where θC,A : C1 → A1 and θD,B : D1 → A1 are the isomorphisms corresponding to the pre-Koszul pairs
(A,C) and (B,D), respectively.
Note that, for an invertible twisting map σ : B⊗A→ A⊗B, the pairs (A, T (A)) and (B, T (B)) fulfil
the conditions (25)–(26), where τ and λ are as in Proposition 3.10.
Let us remark that the identities (25) and (26) together yields
σ1,1 ◦ (θD,B ⊗ θC,A) ◦ τ1,1 = θC,A ⊗ θD,B.
Equivalently, for c ∈ C1 and d ∈ D1, we have
∑
σ,τ
θC,A(cτ )σ ⊗ θD,B(dτ )σ = θC(c)⊗ θD(d). (27)
Remark 3.12. If τ : C ⊗ D → D ⊗ C is a twisting map of graded coalgebras, then the map
τˆ : C ⊗D → D ⊗C defined by τˆp,q := (−1)pqτp,q is also a graded twisting map of coalgebras, to which
we can associate the twisted tensor product C ⊗τˆ D.
Proposition 3.13. Keeping the notation and assumptions in §3.11, then the pair (A⊗σ B,C ⊗τˆ D) is
pre-Koszul.
Proof. It is obvious that A⊗σ B and C ⊗τˆ D are connected. Also, by definition we have
(A⊗σ B)1 = (A0 ⊗B1)⊕ (A1 ⊗B0) = (R ⊗B1)⊕ (A1 ⊗R),
(C ⊗τˆ D)1 = (C0 ⊗D1)⊕ (C1 ⊗D0) = (R⊗D1)⊕ (C1 ⊗ R).
Then we define θ : (C ⊗τˆ D)1 → (A ⊗σ B)1 by θ = (θC,A ⊗ IdR) ⊕ (IdR ⊗ θD,B). Let ·σ denote the
multiplication on A⊗σ B. It remains to show that, for any x ∈ (C ⊗τˆ D)2,
∑
θ(x1(1)) ·σ θ(x1(2)) = 0. (28)
Let ∆ := ∆C⊗τˆD. First, recall that ∆1,1 : (C ⊗τˆ D)2 → (C ⊗τˆ D)1 ⊗ (C ⊗τˆ D)1, where
(C ⊗τˆ D)2 = (C2 ⊗R)⊕ (C1 ⊗D1)⊕ (R⊗D2).
Therefore, we can restrict ourselves to prove (28) for x belonging to each of the former three direct
summands. Let us consider in the first place the case x ∈ C2 ⊗R, so we can write x = c⊗ 1 for c ∈ C2.
By definition of the comultiplication on C ⊗τˆ D, we have
∆1,1
∣∣
C2⊗R = (C ⊗ τˆ
1,0 ⊗R) ◦ (∆1,1C ⊗∆
0,0
D ),
and since τˆ |C⊗R is just the flip, we get
∆1,1(c⊗ 1) =
∑
(c1(1) ⊗ 1)⊗ (c1(2) ⊗ 1).
Henceforth, for x = c⊗ 1 ∈ C2 ⊗R we have
∑
θ(x1(1)) ·σ θ(x1(2)) =
∑
(θC,A(c1(1))⊗ 1) ·σ (θC,A(c1(2))⊗ 1) =
∑
θC,A(c1(1))θC,A(c1(2))⊗ 1 = 0,
since (A,C) is a pre-Koszul pair. If x ∈ C0 ⊗D2, the computations are done in a similar way.

ON KOSZULITY OF TWISTED TENSOR PRODUCTS 19
Let us finally assume that x ∈ C1 ⊗ D1. We can just prove the condition for a generator of the
form x = c ⊗ d with c ∈ C1, d ∈ D1. Recall that, for any graded coring C and c ∈ C1 we have
∆c = 1⊗ c+ c⊗ 1. Thus
∆(c⊗ d) = (C1 ⊗ τˆ ⊗ C2)(∆C(c)⊗∆D(d))
= (c⊗ d)⊗ (1⊗ 1) + (c⊗ 1)⊗ (1 ⊗ d) + (1 ⊗ 1)⊗ (c⊗ d) + 1⊗ τˆ1,1(c⊗ d)⊗ 1
= (c⊗ d)⊗ (1⊗ 1) + (c⊗ 1)⊗ (1 ⊗ d) + (1 ⊗ 1)⊗ (c⊗ d)− 1⊗ τ1,1(c⊗ d)⊗ 1,
where in the last step we used that τˆ (c ⊗ d) = −τ(c ⊗ d). The component of the latest expression
belonging to (C ⊗τˆ D)1 ⊗ (C ⊗τˆ D)1 is precisely
∆1,1(c⊗ d) = (c⊗ 1)⊗ (1 ⊗ d)− 1⊗ τ1,1(c⊗ d)⊗ 1 = (c⊗ 1)⊗ (1⊗ d)−
∑
τ
(1⊗ dτ )⊗ (cτ ⊗ 1).
Henceforth, composing with the product we get
∑
θ(x1(1)) ·σ θ(x1(2)) = θ(c⊗ 1) ·σ θ(1⊗ d)−
∑
τ
θ(1⊗ dτ ) ·σ θ(cτ ⊗ 1)
= θC,A(c)⊗ θD,B(d)−
∑
τ
1⊗ θD,B(dτ ) ·σ θC,A(cτ )⊗ 1
= θC,A(c)⊗ θD,B(d)−
∑
σ,τ
θC,A(cτ )σ ⊗ θD,B(dτ )σ
= θC,A(c)⊗ θD,B(d)− θC,A(c)⊗ θD,B(d) = 0.
where for the penultimate equality we used (27). 
Theorem 3.14. Keeping the notation and the assumptions in §3.11, then K l•(A ⊗σ B,C ⊗τˆ D) is the
tensor product of the complexes K l•(A,C) and K l•(B,D). In the particular case when (A,C) and (B,D)
are Koszul, it follows that (A⊗σ B,C ⊗τˆ D) is Koszul two.
Proof. In Proposition 3.13 we have already proven that (A⊗σ B,C ⊗τˆ D) is a pre-Koszul pair. Then it
makes sense to consider the complex K• := K l•(A⊗σ B,C ⊗τˆ D). Let us show that K• is isomorphic to
the tensor product of K ′• := K l•(A,C) and K ′′• := K l•(B,D). Let
∂n : (A⊗σ B)⊗ (C ⊗τˆ D)n → (A⊗σ B)⊗ (C ⊗τˆ D)n−1
denote the differential map in K•. Let ∆ denote the comultiplication in C ⊗τˆ D. For any c ∈ Cp and
d ∈ Dq we have
∆(c⊗ d) =
∑p
r=0
∑q
u=0
(−1)(p−r)ucr(1) ⊗
(
du(1)
)
τ
⊗
(
cp−r(2)
)
τ
⊗ dq−u2 ,
so, the component of ∆(c ⊗ d) in (C ⊗τˆ D)1 ⊗ (C ⊗τˆ D)p+q−1 is obtained from the above equality by
dropping all summands but the ones with either r = 1 and u = 0, or r = 0 and u = 1. We get
∆1,p+q−1(c⊗ d) =
∑
c1(1) ⊗ 1⊗ c
p−1
(2) ⊗ d+ (−1)p
∑
τ
1⊗
(
d1(1)
)
τ
⊗ cτ ⊗ dq−1(2) .
and thus, in degree n, for any a ∈ A, b ∈ B, and any c ∈ Cp, d ∈ Dq with p+ q = n we have:
∂n ((a⊗ b)⊗ (c⊗ d)) =
∑
τ
(−1)p (a⊗ b) ·σ θ(1 ⊗ (d(1)1 )τ )⊗ cτ ⊗ d
(q−1)
2 +
+
∑
(a⊗ b) ·σ θ((c(1)1 )⊗ 1)⊗ c
(p−1)
2 ⊗ d
=
∑
τ
(−1)pa⊗ bθD,B((d(1)1 )τ )⊗ cτ ⊗ d
(q−1)
2 +
∑
aθC,A(c(1)1 )σ ⊗ b⊗ c
(p−1)
2 ⊗ d.
To make computations with morphisms in the category of R-bimodules we use string representation
of morphisms in a monoidal category, which is explained for example in [Kas95, Chapter XIV.1]. Each
morphism will be represented downwards, as a string together with a box. The name of the morphism
will be written inside the box. For the identity of a bimodule we shall draw a string without any box.
The tensor product and the composition of two morphisms will be represented by horizontal and vertical
juxtaposition, respectively. In conclusion, every string diagram may be interpreted as the representation

20 PASCUAL JARA, JAVIER LO´PEZ PEN˜A, AND DRAGOS¸ S¸TEFAN
∂n = (−1)p
τ
θD,B
A B Cp Dq
A B Cp Dq−1
+
θC,A
σ
A B Cp Dq
A B Cp−1 Dq
Figure 1. The definition of ∂n
of a composition f1 ◦ · · · ◦ fn, where each fi is a tensor product fi = IdXi ⊗ gi⊗ IdYi . The corresponding
diagrams will be drawn one under the other one, starting with f1 on the top.
As an example, let us have a look at the picture in Figure 1, which represents ∂n. The first term is
defined on Am ⊗ Bn ⊗ Cp ⊗Dq and takes values in Am ⊗Bn+1 ⊗ Cp ⊗Dq−1, and can be written (up
to a sign) as the composition of four maps
(IdAm⊗mB⊗IdCp⊗Dq−1)(IdAm⊗Bn⊗θD,B⊗IdCp⊗Dq−1)(IdAm⊗Bn⊗τ⊗IdDq−1 )(IdAm⊗Bn⊗Cp⊗∆1,q−1).
As usual, the multiplication of an R-ring is drawn by joining two strings. For the comultiplication of a
coring we shall use the ‘dual’ representation, obtained by turning up-side-down the string diagram of
the multiplication of a ring (∆ splits the string representing the domain into two strings). Note that
the special notation of σ as a crossing. For τ and λ we shall use the inverse crossing representation, to
put stress on the fact they were associated to σ−1, and not σ.
If we define δn := (IdA ⊗ λ ⊗ IdD) ◦ ∂n ◦ (IdA ⊗ λ−1 ⊗ IdD) then we have that δn = δ′n + δ′′n, where
δ′n is the first term of ∂n composed to the left by IdA ⊗ λ ⊗ IdD and to the right by IdA ⊗ λ−1 ⊗ IdD.
The map δ′′n is obtained similarly from the second term of ∂n. The computation of δ′n is performed in
Figure 2. For the first equality we are using (25). The second one means that λ is compatible with
the multiplication of B, while the third one is obvious, as λ and λ−1 are inverses each other. The
morphism δ′′n is computed, using the same method, in Figure 3. To deduce the first identity we are
using (26). The compatibility between the coring structure of B and the entwining map λ implies the
second equality. The third one is again trivial as λ−1 is the inverse of λ. The above computations shows
that δ′• = (−1)•IdK′• ⊗ d′′ and δ′′• = d′ ⊗ IdK′′• , where d′ and d′′ are the differentials of K′• and K′′• ,
respectively. Therefore,Kl•(A ⊗σ B,C ⊗τˆ D) ∼= Kl•(A,C) ⊗ Kl•(B,D). Let us assume that (A,C) and
(B,D) are Koszul. By definition, then the complexes Kl•(A,C) and Kl•(B,D) are acyclic and their
homology groups in degree zero are isomorphic to R. If we prove that Kl•(A⊗σB,C⊗τˆ D) has the same
δ′n = (−1)p
λ
τ
θD,B
λ−1
A Cp B Dq
A Cp B Dq−1
= (−1)p
λ
θD,B
λ
λ−1
A Cp B Dq
A Cp B Dq−1
= (−1)p
θD,B
λ
λ−1
A Cp B Dq
A Cp Bn Dq−1
= (−1)p θD,B
A Cp B Dq
A Cp B Dq−1
Figure 2. The computation of δ′n

ON KOSZULITY OF TWISTED TENSOR PRODUCTS 21
δ′′n =
λ
θC,A
σ
λ−1
A Cp B Dq
A Cp−1 B Dq
=
λ
λ−1
θC,A
λ−1
A Cp B Dq
A Cp−1 B Dq
=
λ
λ−1
θC,A
A Cp B Dq
A Cp−1 B Dq
= θC,A
A Cp B Dq
A Cp−1 B Dq
Figure 3. The computation of δ′′n
property, then the augmented complex K˜l•(A ⊗σ B,C ⊗τˆ D) would be exact. Since R is semisimple,
using Ku¨nneth formula [Wei97, Theorem 3.6.3], we get
Hn(Kl•(A⊗σ B,C ⊗τˆ D)) ∼=
⊕
p+q=n
Hp(Kl•(A,C))⊗Hq(Kl•(B,D)).
Thus Kl•(A⊗σ B,C ⊗τˆ D) is clearly acyclic, and
H0(Kl•(A⊗σ B,C ⊗τˆ D)) ∼= H0(Kl•(A,C)) ⊗H0(Kl•(B,D)) ∼= R⊗R ∼= R.
In conclusion, the theorem is proved. 
Corollary 3.15. Let σ : B ⊗A→ A⊗B be a twisting map between two Koszul R-rings. Then A⊗σ B
is Koszul and T (A⊗σ B) ∼= T (A)⊗τˆ T (B).
Proof. One applies the preceding theorem, taking τ and λ to be the twisting map and the entwining
structures that we constructed in Proposition 3.10. 
We conclude this section by providing an example of two pairs that satisfies the conditions in §3.11.
We start with a simple example of Koszul pair.
For every R-bimodule V let T := T aR(V ). Let D = R⊕ V be the graded coring with comultiplication
∆, such that ∆(v) = v ⊗ 1 + 1⊗ v, for any v ∈ V. We take θT,D to be the identity map.
Lemma 3.16. The pair (T,D) is Koszul.
Proof. Clearly T and D are graded and connected. The identity from the definition of Koszul pairs is
automatically verified, as D2 = 0. Thus (T,D) is pre-Koszul. Furthermore, K˜l•(T,D) is the complex
0←− R←− T ←− T ⊗ V ←− 0←− · · ·
whose non-zero arrows are the projection pi0T of T on T 0 and dl1. Since the multiplication in T is given by
juxtaposition of tensor monomials, and ∆1,0(v) = v⊗1, for every v ∈ V, we deduce that dl1(x⊗v) = x⊗v.
Therefore, dl1 identifies T ⊗ V and T . Hence K˜l•(T,D) is exact. 
From now on we assume that R is a field and that V has a finite basis {e1, e2, . . . , en}. For a given
pre-Koszul pair (A,C), we are going to characterize graded twisting maps σ : T ⊗ A → A ⊗ T, graded
twisting maps τ : C ⊗D → D ⊗ C and graded entwining maps λ : C ⊗ T → T ⊗ C.
3.17. The twisting maps between T and A. Let σ be a twisting map as above. On T 0 ⊗ A the
map σ is always the canonical identification T 0 ⊗A ∼= A⊗ T 0. Since σ maps V ⊗A to A⊗ V ,
σ(ei ⊗ a) =
∑n
j=1
σij(a)⊗ ej ,

22 PASCUAL JARA, JAVIER LO´PEZ PEN˜A, AND DRAGOS¸ S¸TEFAN
where σij is a graded R-linear endomorphisms of A, for every 1 ≤ i, j ≤ n. Since σ is compatible with
the multiplication of A, it follows that
σij(ab) =
∑n
p=1
σik(a)σkj(b), (29)
for all a, b ∈ A, and all 1 ≤ i, j ≤ n. The compatibility with the unit of A is equivalent to the relation
σij(1) = δi,j1. (30)
Writting down the condition expressing the compatibility between σ and the multiplication of T, we get
σ(ei1 · · · eip ⊗ a) =
n∑
j1,...jp=1
σi1j1 · · ·σipjp(a)⊗ ej1 · · · ejp . (31)
In conclusion to every twisting map σ corresponds a matrix (σij)i,j ∈ Mn(EA) such that the relations
(29)–(31) hold, where EA is the ring of graded linear endomorphisms of A. It is not difficult to see that,
conversely, a matrix (σij)i,j satisfying (29) and (30) defines in a unique way a graded twisting map σ.
Of course, given (σij)i,j , the definition of the corresponding σ is given in (31).
One can prove that σ is invertible if and only if the transposed of the corresponding matrix (σij)i,j
is invertible in Mn(EA) or, equivalently, (σij)i,j is invertible in Mn(EopA ). More precisely, σ is invertible
if and only if there is a matrix (σ′ij)i,j ∈ Mn(EA) such that
∑n
j=1 σjiσ′kj =
∑n
j=1 σ′jiσkj = δi,kIdA, for
all 1 ≤ i, k ≤ n. The matrix (σ′ij)i,j determines the inverse σ′ of σ by the formula
σ′(a⊗ ei1 · · · eip) =
n∑
j1,...jp=1
ej1 · · · ejp ⊗ σ′ipjp · · ·σ
′
i1j1(a). (32)
3.18. The twisting maps between C and D. Now let us fix a graded twisting map τ : C⊗D → D⊗C.
Since D = R⊕ V and τ(c ⊗ 1) = 1⊗ c, for all c ∈ C, we have to compute τ(c ⊗ ei). Let EC denote the
R-algebra of graded linear endomorphisms of C. Proceeding as in the case of twisting maps of algebras,
we see that there is a matrix (τij)i,j in Mn(EC) such that the following identities hold
τ(c⊗ ei) =
∑n
j=1
ej ⊗ τji(c), (33)
∆C (τij(c)) =
∑n
k=1
τik(c(1))⊗ τkj(c(2)), (34)
εC (τij(c)) = δi,jεC(c). (35)
Conversely, for a matrix that satisfies (34) and (35) , the formula (33) defines a twisting map of coalgebras
between C and D. The twisting map τ is invertible if, and only if, the corresponding matrix (τij)i,j is
invertible in Mn(EC). If the inverse is (τ ′ij)i,j then τ ′, the inverse of τ , is given by
τ ′(ei ⊗ c) =
∑n
j=1
τ ′ji(c)⊗ ej. (36)
3.19. The entwining map between C and T. Now we are looking for a graded entwining map
λ : C ⊗ T → T ⊗ C. The restriction of λ to C ⊗D obviously is a graded twisting map between C and
D. Hence, there is a matrix (λij)i,j in Mn(EC) such that
λ(c⊗ ei) =
∑n
j=1
ej ⊗ λji(c), (37)
∆C (λij(c)) =
∑n
k=1
λik(c(1))⊗ λkj(c(2)), (38)
εC (λij(c)) = δi,jεC(c). (39)
Furthermore, using the compatibility between λ and the multiplication in T, we get
λ(c⊗ ei1 · · · eip) =
n∑
j1,...jp=1
ej1 · · · ejp ⊗ λjpip · · ·λj1i1(c). (40)

ON KOSZULITY OF TWISTED TENSOR PRODUCTS 23
Conversely, if (λij)i,j is a matrix in Mn(EC) such that (38) and (39) hold, then the formula (40) defines
a graded twisting map λ between C and T, which is invertible if, and only if, the matrix that we start
with is so. Note that, if (λ′ij)i,j is the inverse of (λij)i,j , then
λ′(ei1 · · · eip ⊗ c) =
n∑
j1,...jp=1
λ′j1i1 · · ·λ′jpip(c)⊗ ej1 · · · ejp . (41)
3.20. The compatibility between σ, τ and λ. We fix a graded twisting map σ : T ⊗ A→ A⊗ T, a
graded twisting map τ : C ⊗ D → D ⊗ C and a graded entwining map λ : C ⊗ T → T ⊗ C as above.
Let (σij)i,j , (τij)i,j and (λij)i,j be the corresponding matrices. We want to explicit the conditions (25)
and (26) in this particular setting. For simplicity we assume that A1 = C1 and θC,A = IdC1 . Hence the
former relation becomes λp,1 = τp,1, for every p. Clearly this is equivalent to the fact that λ and τ are
defined by the same matrix in Mn(EC). On the other hand, (26) is equivalent to (σq,1)−1 = λ1,q, for
every q. Taking into the account the fact that the inverse of σ is given by (32), the above identities are
equivalent to
n∑
j1,...jq=1
ej1 · · · ejq ⊗ σ′iqjq · · ·σ
′
i1j1(a) =
n∑
j1,...jq=1
ej1 · · · ejq ⊗ λjqiq · · ·λj1i1 (c), (42)
for every q and all c ∈ C1. In the case when q = 1 we deduce that σ′ij(c) = λi,j(c), for all c ∈ C1 and
every 1 ≤ i, j ≤ n. Visibly, the fact that (σ′ij)i,j equals on C1 the transposed matrix of (λij)i,j implies
(42). In conclusion The compatibility relations (25) and (26) are equivalent in our particular setting to
the fact that (τij)i,j = (λij)i,j , and on C1 = A1 we have
∑n
j=1
σjiλjk =
∑n
j=1
λijσkj = δi,kIdA, (43)
Proposition 3.21. Let R be a field. Suppose that (A,C) is a pre-Koszul pair such that A1 = C1, and
that V is a vector space of dimension n. There is an one-to-one correspondence between the following
data:
(a) The pairs (Σ,Λ), with Σ := (σij)i,j ∈Mn(EA) and Λ := (λij)i,j ∈Mn(EA), so that:
(i) Both Σt (the transposed of Σ) and Λ are invertible.
(ii) The conditions (29) and (30) hold.
(iii) The conditions (38) and (39) hold.
(iv) The condition (43) hold on A1.
(b) The triples (σ, τ, λ) as in §3.11.
The correspondence is given by (Σ,Λ) 7→ (σ, τ, λ), where σ, τ and λ are defined as in (31), (33) and
(40), respectively, where for the definition of τ one takes τij = λij , for all i, j.
Acknowledgments
The third named author was financially supported by CNCSIS, Contract 560/2009 (CNCSIS code
ID 69).
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University of Granada, Department of Algebra, Granada E-18071, Spain
E-mail address: pjara@ugr.es
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United King-
dom
E-mail address: jlp@math.ucl.ac.uk
University of Bucharest, Faculty of Mathematics and Computer Science, 14 Academiei Street, Bucharest
Ro-010014, Romania
E-mail address: dragos.stefan@fmi.unibuc.ro