Further Properties and Applications of Koszul Pairs

Koszul pairs were introduced in [arXiv:1011.4243] as an instrument for the study of Koszul rings. In this paper, we continue the enquiry of such pairs, focusing on the description of the second component, as a follow-up of the study in [arXiv:1605.05458]. As such, we introduce Koszul corings and prove several equivalent characterizations for them. As applications, in the case of locally finite $R$-rings, we show that a graded $R$-ring is Koszul if and only if its left (or right) graded dual coring is Koszul. Finally, for finite graded posets, we obtain that the respective incidence ring is Koszul if and only if the incidence coring is so.


Introduction
By the definition in [BGS], an N-graded ring A := ⊕ n∈N A n is said to be Koszul if and only if A 0 is a semisimple ring and there is a resolution P * of A 0 by projective graded left A-modules such that each P n is generated by homogeneous elements of degree n. Koszul rings are natural generalizations of Koszul algebras which, in turn, were discovered by Priddy [Pr]. Koszul algebras and rings have proved to be very useful tools in various fields of Mathematics, such as: Representation Theory, Algebraic Geometry, Algebraic Topology, Quantum Groups and Combinatorics; a comprehensive read is, for example, [PP] and the references therein.
In [JPS], studying some cohomological properties of Koszul rings, the authors were led to a new notion, the so-called Koszul pairs. Let R be a semisimple ring. Recall that a graded R-ring is a graded algebra in the tensor category of R-bimodules with respect to the tensor product of bimodules. A graded R-ring A := ⊕ n∈N A n is conected if A 0 = R. Connected graded R-corings are defined by duality. By definition, an almost-Koszul pair consists of a graded connected R-ring A and a graded connected R-coring C, together with an R-bimodule isomorphism θ A,C : C 1 → A 1 . These data must be compatible in the sense that the composition of the following three maps is zero: where ∆ 1,1 and µ 1,1 denote the components of the structure maps of C and A, respectively. By [JPS], to an almost-Koszul pair correspond three chain complexes and three cochain complexes, which measure how far is A from being Koszul. More precisely, an important feature of an almost-Koszul pair is that one of the corresponding six complexes is exact if and only if all of them are so. In this case, the pair (A, C) is called Koszul. Furthermore, one proves that a connected graded R-ring A is Koszul if and only if there exists a connected graded R-coring C such that (A, C) is Koszul.
In this paper we continue the work on Koszul pairs and their applications, our purpose being two-fold. First, we show that the Koszulity of a connected graded R-ring A can be stated in terms of comodule theoretical properties of its connected graded coring T (A) := Tor A • (R, R). This is mainly done in Theorem 2.1, where we also recover in a unifying way other well-known equivalent characterizations of these rings. For instance, here we show that A is Koszul if and only if the coring T (A) is strongly graded, if and only if the primitive part of T (A) coincides with its homogeneous component of degree one, if and only if T (A), regarded in a canonical way as a bigraded coring, is diagonal. We also include

Preliminaries
In this section we recall some basic concepts and notations from [JPS] and then we shall prove some new preliminary results, which are needed later on.
1.1. Connected (co)rings. Throughout, R will denote a semisimple ring. Since we always work with algebras and coalgebras in the tensor category of R-bimodules, to ease the notation, the tensor product ⊗ R of two bimodules will be denoted by ⊗. If V is an R-bimodule, then we shall use the notation V (n) for the nth tensor power V ⊗ · · · ⊗ V . By convention we take V (0) = R. In a similar way, for any bimodule morphism f : M → N , the tensor product f ⊗ · · · ⊗ f with n factors will be denoted by f (n) .
By definition A = ⊕ n∈N A n is a connected R-ring if and and only if it is an N-graded algebra in the tensor category of R-bimodules and A 0 = R. Dually, a connected R-coring is a graded coalgebra C = ⊕ n∈N C n , in the category of R bimodules such that C 0 = R. The augmentation ideal of A will be denoted by A + = ⊕ n∈N A n . Sometimes we shall identify the quotient C + := C/R with ⊕ n∈N * C n . Note that the comultiplication induces a canonical coassociative map ∆ + : For a connected R-ring A the multiplication µ is defined by some maps µ p,q : A p ⊗ A q → A p+q . We say that A is strongly graded if these maps are surjective, for all p, q ≥ 0. Equivalently, A is strongly graded if and only if the iterated multiplication µ(n) : (A 1 ) (n) → A n is surjective for all n ≥ 2.
Similarly, the comultiplication of a connected R-coring C is defined by some R-bimodule maps ∆ p,q : C p+q → C p ⊗ C q and the coring C is called strongly graded if all of them are injective.
Let ∆(1) := I C1 and for n ≥ 2 we define ∆(n) : C n → C From the very definition of the maps ∆(n) and coassociativity, we get the equality: Note that C being strongly graded can be put in terms of ∆(n) being injective for all n. Equivalently, C is strongly graded if and only if ∆ 1,n is injective for all n, if and only if ∆ n,1 in injective for all n.
Let C be a connected R-coring, so C 0 = R. By definition of connected corings the unit of R is a group-like element, that is ∆(1) = 1 ⊗ 1, see [JPS]. Hence it makes sense to speak about the primitive elements of C, i.e. about those c ∈ C such that ∆(c) = c ⊗ 1 + 1 ⊗ c. The set of primitive elements in C includes C 1 and it will be denoted by P C. In general, the inclusion C 1 ⊆ P C is strict.
Indecomposable elements of an augmented R-ring correspond by duality to primitive elements. They were introduced by May in [Ma]. In the graded case the R-bimodule QA of indecomposable elements is defined by the exact sequence: There is a canonical morphism A 1 → QA, which maps a ∈ A 1 to its class a + A 2 + ∈ QA. This map has a left inverse QA → A 1 , which is induced by the projection A + → A 1 .
1.2. The R-ring V, W and the R-coring {V, W }. For an R-bimodule V and a sub-bimodule W ⊆ V ⊗V , we define the R-ring V, W to be the quotient T a R (V )/ W of the tensor algebra of the R-bimodule V by the two-sided ideal generated by W . Note that W = n∈N W n , where For V and W as above, one also constructs a graded R-coring {V, W } by taking {V, W } 0 = R and {V, W } 1 = V . For all n ≥ 2 define: As it is shown in [JPS], the direct sum {V, W } = n∈N {V, W } n is a graded subcoring of T c R (V ), the tensor coalgebra of the R-bimodule V .
If A is a connected graded R-ring and C is a connected graded R-coring then the graded coring {A 1 , Ker µ 1,1 } and the graded ring C 1 , Im ∆ 1,1 will be denoted by A ! and C ! , respectively. The homogeneous component of degree n of A ! will be denoted by A ! n . To simplify the notation, for the ring C ! we shall write C ! n instead of (C ! ) n . 1.3. Bigraded corings. We collect here some basic facts regarding bigraded corings, which hold in general and which we will be using throughout this article when discussing quadratic and Koszul Rcorings.
Let C be an R-coring. We say that C is bigraded if C has a decomposition as a direct sum of R-bimodules C = n,m∈N C n,m such that its comultiplication induces a collection of maps In this context, coassociativity translates as the commutativity of the diagram below, for all positive integers m, n, p, m ′ , n ′ and p ′ .
By definition, the counit must vanish on C n,m , provided that either n > 0 or m > 0.
To any bigraded coring, one can associate a graded coring gr(C), whose homogeneous component of degree n is gr n (C) = m C n,m . Thus, we define gr(C) := n gr n (C). In this paper we are interested only in connected bigraded corings, i.e. in those bigraded corings for which C 0,0 = R and C 0,m = 0, for m > 0. Note that, if C is connected, then gr(C) is connected as well.
If C is a (connected) bigraded coring and we put: then C ′ := n,m C ′ n,m is a (connected) bigraded coring. We denote the graded coring gr(C ′ ) by Diag(C).
For C and C ′ as above, there are canonical R-bimodule morphisms π n,m : C n,m → C ′ n,m which act as identity on C n,n and zero maps in rest. The collection π = {π n,m } n,m defines a morphism of bigraded corings. As Diag(C) = gr(C ′ ), the map π induces a morphism gr(π) : gr(C) − → Diag(C) of graded corings. Clearly, the n-degree component of the kernel of π coincides with n =m C n,m .
Note that one can define by duality the corresponding notions of bigraded R-rings. We omit the details here, since this case is better known than the one for corings.
Since the normalized bar complex has a canonical structure of DG-coalgebra in the category of Rbimodules with respect to the comultiplication ∆ p,q (a 1 ⊗· · ·⊗a p+q ) = (a 1 ⊗· · ·⊗a p )⊗(a p+1 ⊗· · ·⊗a p+q ) it follows that T (A) = ⊕ n∈N T n (A) has a canonical structure of connected R-coring.
The complex Ω • (A) decomposes as a direct sum Ω • (A) = ⊕ m≥0 Ω • (A, m) of subcomplexes. In order to define Ω • (A, m) we introduce the following terminology and notation. An n-tuple m = (m 1 , . . . , m n ) is called a positive n-partition of m if and only if n i=1 m i = m and all m i are positive. The set of all positive n-partitions of m will be denoted by P n (m). Furthermore, if m = (m 1 , . . . , m n ) is a positive n-partition and A is a connected R-ring then for the tensor product A m1 ⊗ · · · ⊗ A mn we shall use the notation A m . For a positive n-partition m = (m 1 , . . . , m n ) of m, the multiplication µ of A induces bimodule maps µ m : A m → A m and µ(m) : (A 1 ) (m1) ⊗ · · · ⊗ (A 1 ) (mn) → A m . Note that, by definition, µ(m) = µ(m 1 ) ⊗ · · · ⊗ µ(m n ). Hence, with this notation, Ω • (A, m) is the following subcomplex of Ω • (A): Of course, there is only one 1-partition of m, namely m 1 = (m). Thus the first direct sum in Ω • (A, m) coincides with A m . The homology in degree n of Ω • (A, m) will be denoted either by T n,m (A) or Tor A n,m (R, R). Clearly, we have T (A) = ⊕ m≥0 T n,m (A) and this decomposition is compatible with the coring structure, in the sense that T (A) is a bigraded coring with T n,m (A) as (n, m)-homogeneous component.
Dually, for a connected R-coring C, the normalized bar cochain complex ( Here C + := C/C 0 and ∆ + : C + → C + ⊗ C + is the map induced by the comultiplication of C. It is wellknown that E n (C) = Ext n C (R, R) is the nth cohomology group of Ω • (C) and that E(C) = n∈N E n (C) is a connected R-ring with respect to the multiplication induced by the DG-algebra structure (in the category of R-bimodules) on Ω • C which is defined by concatenation of tensor monomials.
The complex Ω • (C) is a direct sum of subcomplexes Ω • (C, m), that are defined as follows. For a positive n-partition of m let C m := C m1 ⊗ · · · ⊗ C mn . Hence Ω • (C, m) is the subcomplex Of course, Ω 1 (C, m) = C m . The homology in degree n of Ω • (C, m) will be denoted either by E n,m (C) or Ext n,m C (R, R). Clearly, we have E(C) = ⊕ m≥0 E n,m (C) and this decomposition is compatible with the ring structure in the sense that E(C) is a bigraded ring with E n,m (C) as (n, m)-homogeneous component. For more details the reader is referred to [JPS,§1.15].
Now we can prove our first preliminary results. We start with the following lemma, where we extend a well known criterion for the injectivity of a morphism between two connected ordinary coalgebras.
Lemma 1.5. Let C be a connected coring.
(1) The coring C is strongly graded if and only if P C = C 1 .
(2) If C is strongly graded and f : C → D is a morphism of graded corings such that the components f 0 and f 1 are injective, then f is injective. (3) Let C = n,m≥0 C n,m be a bigraded R-coring. If gr(C) is strongly graded and C n,m = 0 for n = 0, 1 and all m = n, then the component C n,m is also trivial, for all n ≥ 2 and m = n.
Proof. Let us prove the first part of the Lemma. We first assume that C is strongly graded. Let We claim that x k = 0 for any k = 1. Because Let us fix k ≥ 2. In the rightmost term of the above equation, ∆ p,k−p (x k ) ∈ C p ⊗ C k−p . On the other hand, x k ⊗ 1 + 1 ⊗ x k ∈ C k ⊗ C 0 + C 0 ⊗ C k , for any k. Hence, ∆ p,k−p (x k ) = 0. Since C is strongly graded, all ∆ p,k−p are injective. It follows that x k = 0, so our claim was proved. In conclusion, x = x 1 ∈ C 1 , that is we have P C ⊆ C 1 . The other inclusion always holds, completing the proof of the direct implication.
Conversely, let us assume that P C = C 1 and prove that all maps ∆(n) are injective. Since the kernel of ∆(2) = ∆ 1,1 coincides with P C C 2 = 0, it follows that this map is injective. We assume that ∆(k) is injective for all k ≤ n. Hence, the map ∆(p) ⊗ ∆(n + 1 − p) is injective, for all 0 < p < n + 1. Using the relation (2) we deduce that ∆ p,n+1−p (x) = 0, for any x ∈ Ker ∆(n + 1) and all p as above. Thus x ∈ P C C n+1 = 0, that is ∆(n + 1) is injective.
To prove (2) we assume, for an inductive argument, that the component f k is injective for k = 0, . . . , n. As f is a morphism of graded corings, the diagram: is commutative. By the standing assumptions, ∆ C n,1 and f n ⊗ f 1 are injective. It follows that f n+1 is injective. Thus, by induction, all components of f are injective, so f itself is injective.
It remains to prove the last part of the lemma. Let Diag C = n≥0 C n,n . We denote the canonical morphism of graded R-corings by gr(π) : gr(C) → Diag C. Since we have gr 1 (C) = m C 1,m = C 1,1 , the component gr 1 (π) coincides with the identity map of C 1,1 = P (gr C) = Diag 1 (C). On the other hand, as C 0,0 = gr 0 (C) = Diag 0 (C), the component gr 0 (π) coincides with the identity map of C 0,0 .
Using the first part of the lemma, it follows that gr(π) is injective. As Ker gr n (π) = m =n C n,m , we get C n,m = 0, for all n and m, with m = n.
Remark 1.6. In the particular case of ordinary coalgebras, the first part of the lemma is also proved in [MS,Lemma 2.3].
There is a dual version of the previous lemma, in the case of bigraded rings.
Lemma 1.7. Let A be a graded and connected R-ring.
(1) A is strongly graded if and only if the canonical map QA → A 1 is injective, if and only if the canonical map A 1 → QA is surjective. (2) If A is strongly graded, B is a connected graded R-ring and g : B → A is a morphism of graded R-rings such that its components g 0 and g 1 are surjective, then g is surjective.
(3) Let A = n,m∈N A n,m be bigraded. If grA is strongly graded and A n,m = 0 for n = 0, 1 and all m = n, then A n,m = 0 for all n ≥ 2, m = n.
Proof. Since QA → A 1 is a left inverse of A 1 → QA it follows that the former map is injective if and only if the latter is surjective. On the other hand, A is strongly graded if and only if A n ⊆ A 2 + , for all n ≥ 2. Consequently, the first statement holds, as the above inclusions are true if and only if the map A 1 → QA is surjective.
The proof of the second statement is similar to the one of Lemma 1.5(b), using the diagram below.
For proving the last part of the lemma we define Diag(A) to be graded subring of gr(A) given by Diag(A) = ⊕ n∈N A n,n . Let ι : Diag(A) → gr(A) denote the inclusion map. Obviously, by the standing assumptions, ι 0 and ι 1 are surjective maps. Hence, by the second part of the lemma, ι is surjective. Thus A n,m must be zero for all n and m, with n = m.
1.8. Almost-Koszul pairs. As mentioned in the introduction, an almost Koszul pair (A, C) consists of a connected and graded R-ring A and a connected graded R-coring C, together with an isomorphism of R-bimodules θ A,C : C 1 → A 1 which satisfies the relation (1). Using the graded version of Sweedler's notation for the comultiplication of C, the above equation is equivalent to where c is an arbitrary element of C 2 . If (A, C) and (B, D) are almost-Koszul pairs, then a morphism of almost-Koszul pairs is a couple (φ, ψ), where φ : A → B is a morphism of graded R-rings and ψ : C → D is a morphism of graded R-corings such that they commute with the isomorphisms θ A,C and θ B,D . Henceforth, the diagram below is commutative: A first example of an almost-Koszul pair is given in [JPS,Proposition 1.8]. For any connected and strongly graded R-ring A, the pair A, T (A) is almost-Koszul. Here, the coring structure of T (A) is defined as in §1.4, and the map θ A,T (A) : T 1 (A) → A 1 is induced by the projection A + → A 1 . Note that T 1 (A) = A + /A 2 + ∼ = A 1 , as A is strongly graded, so θ A,T (A) is an isomorphism.
The couple (A, A ! ) is another example of an almost-Koszul pair. Recall that A ! 1 = A 1 , so we can take θ A,A ! := I A 1 . The condition (1) is trivial in this particular case as, by construction, A ! 2 = Ker µ 1,1 . Dually, if C is a connected and strongly graded R-coring, the pair (E(C), C) is almost-Koszul, cf. [JPS,Proposition 1.18]. In this example the R-ring structure of E(C) is defined in §1.4. Since E 1 (C) = Ker ∆ + , the map θ E(C),C : C 1 → E 1 (C), given by θ E(C),C (c) := c + C 0 ∈ C/C 0 , is well defined and it is an isomorphism of R-bimodules.
The couple (C ! , C) can be seen as an almost-Koszul pair with respect to the map θ C ! ,C = I C1 . The relation (1) is verified since C ! 2 := (C 1 ⊗ C 1 )/ Im ∆ 1,1 and the component C ! 1 ⊗ C ! 1 → C ! 2 of the multiplication on C ! coincides with the canonical projection from C 1 ⊗ C 1 onto C ! 2 . 1.9. Koszul pairs. Following [JPS] we shall briefly recall the definition of Koszul pairs, which are our main tool to investigate Koszul R-(co)rings. For any almost-Koszul pair (A, C) and n ≥ 0 we define a complex of graded right C-comodules by Note that, by convention, C p = 0 for p < 0, so K n r (A, C, m) is trivial if either n < −1 or n > m.
Using the fact that (A op , C op ) is an almost-Koszul pair over R op , cf. [JPS,Remark 1.4]), a complex of left C-comodules is obtained by setting K • l (A, C) = K • r (A op , C op ). By combining the complexes K • l (A, C) and K • r (A, C), in [JPS] one constructs another cochain complex K • (A, C) in the category of C-bicomodules. Since we do not use it in this paper, we omit its definition.
By duality, to an almost-Koszul pair correspond also three chain complexes K l • (A, C), K r • (A, C) and K • (A, C). For instance, K l • (A, C) is the complex of left A-modules: K l −1 (A, C) = R and K l n (A, C) = A ⊗ C n , whose differential d l 0 is given by the left action of A on R = C 0 . For n > 0, the map d l n acts as d l n (a ⊗ c) = aθ C,A (c 1,1 ) ⊗ c 2,n−1 .
The complex K l • (A, C) also decomposes as a direct sum ⊕ m≥0 K l • (A, C, m) of subcomplexes, where K l n (A, C, m) = A m−n ⊗ C n . Note that, K l n (A, C, m) = 0, for n > m. We conclude this subsection by recalling that the exactness of any of these six complexes implies the exactness of all the others, cf. [JPS,Theorem 2.3]. Whenever this is the case, (A, C) is called Koszul. Thus, for a Koszul pair (A, C), the complexe K l • (A, C) is a resolution of R by graded projective left A-modules. Similarly, for such a pair, is Koszul for all m > 0. 1.10. Examples of morphisms of almost-Koszul pairs. We start by defining a morphism (I A , φ A ) from (A, A ! ) to A, T (A) . The graded coring morphism φ A is obtained applying [JPS,Proposition 1.24] as follows. We keep the notation from the above mentioned result. Deleting the component of degree −1 and then applying the functor . Note that the last two complexes are concentrated in non-negative degrees. By [JPS,Proposition 1.23] the former complex is isomorphic to (A ! , 0), the complex with trivial differential maps and whose module of n-chains coincides with A ! n . On the other hand, by definition, the latter complex equals Ω • (A). Since I R ⊗ φ • is compatible with the coring structure of A ! and Ω • A , by applying the homology functor we get the desired coring map φ A : A ! → T (A). Let us remark that so any x in A ! n is an n-cycle in Ω • (A) and φ A (x) as the homology class of x. It is easy to check now that (I A , φ A ) is a morphism of almost-Koszul pairs.
In a similar way one defines a canonical morphism of almost Koszul-pairs (φ C , and then applying the functor Hom C (R, −) to the resulting map of complexes we get a morphism φ • from Ω • C to the cochain complex with trivial differential maps (C ! , 0), cf. [JPS, Proposition 1.23 and Proposition 1.24]. Henceforth, we can define the coring map φ C : . Let us notice that, for any x 1 , . . . , x n ∈ C + , we have 1 and π 1 : C + → C 1 denotes the map induced by the projection C → C 1 . Thus, φ C maps the cohomology class of an n-cocyle ω to φ n (ω).

Koszul R-rings revisited
By [BGS, Definition 1.1.2], a connected strongly graded R-ring is called Koszul if there is a resolution P * of R by projective graded left A-modules such that each P n is generated by homogeneous elements of degree n. The main goal of this section is to characterize Koszul R-rings in terms of properties of the R-coring T (A). In particular, we shall recover the well known result that A is Koszul if and only if T n,m (A) = 0 for n = m. We also include here a new proof of the fact that a Koszul R-ring is quadratic, cf. [BGS,Proposition 1.2.3]. Our approach will allow us to show in the next section that similar results holds true for connected graded R-corings.
Theorem 2.1. Let A be a connected strongly graded R-ring. The following are equivalent: (1) The R-ring A is Koszul.
Let us assume now that A is Koszul. We fix a resolution (P • , d • ) of R such that each P n is graded and generated by P n,n , the homogeneous component of degree n. Hence, The differential maps of this complex are induced by those of the resolution (P • , d • ). Since P n is generated by P n,n it follows that K n (m) = 0 for n = m. Thus (1) =⇒ (7).
We are going to prove that (7) =⇒ (3). By hypothesis, T n,m (A) = 0, for all n and m, with n = m. Our aim is to show that K l • (A, A ! ) is exact. We proceed by induction. Clearly, the sequence Via these identities, the above exact sequence becomes where the leftmost arrow is induced by the multiplication map of A. Using the fact A is strongly graded, it follows that this sequence is exact. Let us assume that the sequence We denote the module of n-cycles by Z n . Thus we have an exact sequence Since the homogeneous components of A ⊗ A ! n are trivial in degree less than or equal to n and d n is injective on R ⊗ A ! n , the i-degree component Z n,i of Z n is trivial, for any i ≤ n. We complete the sequence (4) to a resolution by projective graded left A-modules: In particular, we obtain a resolution of Z n−1 by projective graded left A-modules: By comparing the resolutions (6) and (7), we get an isomorphism Tor A n+1,m (R, R) ≃ Tor A 1,m (R, Z n−1 ), for every m. But, taking into account the sequence (5), it follows that n contains only homogeneous elements of degree n. We fix m > n + 1. Since, by hypothesis, Tor A n+1,m (R, R) = 0 it follows that Tor A 1,m (R, Z n−1 ) vanishes as well. Moreover, using the above exact sequence we get (R ⊗ A Z n ) m = 0. As In particular, Z n,n+2 = A 1 Z n,n+1 . Since A n−m−1 = A 1 A n−m−2 , it is not difficult to see by induction on m > n that Z n,m = A m−n−1 Z n,n+1 . Thus Z n is generated by Z n,n+1 , so we conclude the proof of this implication by noticing that d l n+1 induces an isomorphism between the component of degree n + 1 of K l n (A, A ! ), which coincides with R ⊗ A ! n+1 , and Z n,n+1 . Let us prove that (3) =⇒ (4). By §1.10, there is a morphism of complexes φ , which lifts the identity map of R. Since both K l • (A, A ! ) and β l • (A) are resolutions of R by projective left A-modules, by the Comparison Theorem, the morphism {φ n } n∈N is invertible up to a homotopy in the category of complexes of left A-modules. Hence {I R ⊗ A φ n } n∈N is invertible up to a homotopy in the category of complexes of R-modules. It follows that H n ( is strongly graded, as A ! is always so. Thus we have proved that (4) =⇒ (5).
It remains to prove that (3) =⇒ (2). Since (3) and (4) are equivalent, the canonical map In particular, the latter complex is exact.
Next we give a new proof of the fact that a Koszul R-ring is quadratic, which is based on Lemma 2.2 and Proposition 2.3. These results may be of interest in their own right. Lemma 2.2. For any connected strongly graded R-ring A = ⊕ n≥0 A n , the following sequence is exact where A ≥m denotes the ideal ⊕ p≥m A p and Tor Proof. Since A is strongly graded, for any positive 2-partition m of m, we have that A m d m 2 − − → A m is surjective, so the following sequence is exact: Using the complex Ω • (A, m) we get: On the other hand, to work with the algebra A/A ≥m , we use the complex Ω • (A/A ≥m , m). Since in degree 1 this complex is trivial, for any m, we have: Putting all the information together, we get the sequence: which is also exact due to (8) and gives the desired sequence, because of the relations (9) and (10).
If m is a positive integer such that the components π i are bijective, for all 0 ≤ i < m, and Tor A 2,m (R, R) = 0, then π m is bijective as well.
Proof. By the relations (9) and (10), it follows that π induces the maps: . By construction, the squares of the following diagram are commutative.
Since the homogeneous components of the graded R-rings A/A ≥m and B/B ≥m are zero in degree i ≥ m, and π i is bijective for i < m, we deduce that π induces an isomorphism Ω In particular, we get that π m is an isomorphism of R-bimodules. By the Snake Lemma, the kernel of π m is isomorphic to Coker π m . As Tor A 2,m (R, R) = 0 it follows that π m is injective. Clearly, Coker π m = 0 To a connected graded R-ring A we associate the R-ring A 1 , K A , where K A denotes the kernel of µ 1,1 : There is a unique R-ring morphism φ A : A 1 , K A → A which lifts µ 1,1 . This morphism is surjective, provided that A is strongly graded. Following [BGS], we say that A is quadratic if and only if A is a connected strongly graded R-ring such that φ A is an isomorphism.
We now fix a connected strongly graded R-ring A. It is easy to see that α m (x) belongs to Z 2,m , as x is an element in the kernel of µ(m) = µ m • µ(m), for any positive 2-partition m of m.
. For all other values of i, the proof of the fact that α m (x) is a boundary is similar, so it will be omitted.
Summarizing, α m defines a map from K m A to B 2,m , still denoted by α m . In other words, the diagram is commutative, where the vertical arrows denote the canonical inclusions.
Note that y is a 2-cycle as well. Our goal is to show that x is a boundary. It is enough to prove that y ∈ B 2,m . As A is strongly graded, the map f = V ⊗ µ(m − 1) is surjective. Hence, there exists y ′ ∈ V (m) such that f (y ′ ) = y. Since y is a cycle, we also get that y ′ belongs to K A m , the kernel of µ(m).

By assumption, K
. Note that f (y ′ i ) = 0, for any i > 1. Thus y = f (y ′ 1 ). As µ(m)(y ′ 1 ) = 0, for any positive 2-partition m = (1, m − 1), it follows that y = α m (y ′ 1 ). We conclude the proof that y is a boundary by remarking that α m maps K m A to B 2,m . Proposition 2.5. Let A be a strongly graded R-ring. Then A is quadratic if and only if Tor A 2,m (R, R) vanishes for all m ≥ 3.
Proof. Let B := A 1 , K A . We first prove that A is a quadratic R-ring, provided that Tor A 2,m (R, R) = 0, for all m ≥ 3. It is enough to show that the components φ m A of the canonical map φ A : B → A are all isomorphisms. We proceed by induction on m. The maps φ 0 A and φ 1 A are obviously injective, because they coincide with the identity maps of R and A 1 , respectively. Let us assume that φ k A is an isomorphism, for all k ≤ m − 1, where m ≥ 3. By assumption, Tor A 2,m (R, R) = 0 for all m ≥ 3. Hence, using Proposition 2.3, it follows that φ m A is bijective. The converse follows directly by the preceding lemma, as K m A = K A m , for all m, by the definition of quadratic R-rings.

Koszul R-corings
In this section we introduce and investigate the properties of Koszul corings, the dual notion of Koszul R-rings. Here we shall also show that any Koszul coring is quadratic.
Strongly graded comodules, the substitute of strongly graded rings when one works with graded corings and not with graded rings, will play an important role in this part of the paper. Hence, we start by discussing their main properties.
Let (X, ρ X ) be a graded right comodule over a graded R-coring C. For every n, the map ρ X defines a morphism of graded C-comodules ρ X n from X to X n ⊗ C, whose component of degree k is ρ X n,k−n . Note that the component of degree k of X n ⊗ C is X n ⊗ C k−n , where C i = 0, for all i < 0. Thus, in particular, ρ X n,k−n = 0 for k < n.
Definition 3.1. Let X = ⊕ n≥0 X n be a graded right C-comodule, with structure map ρ X . We say that X is strongly graded in degree n if the morphism of graded C-comodules ρ X n : X → X n ⊗ C is injective.
Let us remark that, by definition, a graded right C-comodule is strongly graded in degree n if and only if X p = 0 for all p < n and ρ n,q : X n+q → X n ⊗ C q is injective for all q ≥ 0.
Some useful properties of graded comodules are collected in the following lemma.
Lemma 3.2. Let C be a strongly graded R-coring. Let (X, ρ X ) denote a graded right C-comodule. ( ) X is strongly graded in degree n if and only if X i = 0, for i < n, and ρ X i,1 is injective for i ≥ n. (4) Let f : X → Y be a morphism of graded right C-comodules. If X is strongly graded in degree n and f n : X n → Y n is an injective map then f is injective as well.
Proof. To prove (1), consider the following commutative diagram: Using the fact that I Xj ⊗ ∆ C i−j,m−i is injective, as C is strongly graded, it follows that ρ X j,m−j (x) is zero, hence x ∈ Kerρ X j,m−j , as required. (2) Let X and Y denote two graded C-comodule. By definition, f ∈ Hom C (X, Y ) m if and only if f is a morphism of C-comodules and f (X k ) ⊆ Y m+k , for all k. In particular, for a C-colinear map f : R → X of degree m, we have f (1) ∈ X m and ρ X (f (1)) = f (1) ⊗ 1. The latter relation is equivalent to the equations ρ X i,m−i (f (1)) = 0, for all 0 ≤ i ≤ m − 1. In conclusion, the required isomorphism is given by the map f → f (1). The equation from the statement follows by the first part of the lemma.
(3) In view of the remark just after the Definition 3.1, the condition X i = 0 for i < n is part of the hypothesis for both implications. We must prove that ρ X i,1 is injective for all i ≥ n if and only if ρ X n,q : X n+q → X n ⊗ C q is injective for all q ∈ N.
To prove that ρ X n,q is injective for q ∈ N we proceed by induction on q. The map ρ M n,0 coincides with the canonical isomorphism M n ∼ = M n ⊗ R and ρ M n,1 is injective by assumption. Let us assume that ρ X n,q is injective. Thus, the horizontal arrow on the bottom of the following commutative diagram is injective.
Since, by hypothesis, the leftmost vertical map is injective it follows that ρ X n,q+1 is also injective.
For the converse one uses the commutative diagram: The component ρ X n,i+1−n is injective by assumption, while I Xn ⊗ ∆ C i−n,1 is so as C is strongly graded. We conclude that ρ X i,1 is injective, as required. (4) We have to prove that all components f k are injective. For k < n this property trivially holds as, by preceding part of the lemma, we have X k = 0. The map f n is injective by assumption, so we can suppose that k > n. The following diagram is commutative, as f is a morphism of graded comodules.
By hypothesis, f n ⊗ I C k−n is injective. On the other hand, ρ X n,k−n is injective, as X is strongly graded in degree n. We conclude that x = 0.
We can now introduce Koszul corings by dualizing [BGS,Definition 1.2.1]. Several characterizations of this class of graded corings are given in Theorem 3.4.

Definition 3.3. A connected R-coring C is called
Koszul if R has an resolution 0 → R → Q • by injective graded right C-comodules such that every term Q n is strongly graded in degree n. Proof. Note that, for any Koszul pair (A, C), the complex K • r (A, C) is a resolution of R satisfying the conditions from the definition of Koszul corings. In particular it follows that C is Koszul, so the implications (2) =⇒ (1) and (3) =⇒ (1) are obvious.
To prove (1) =⇒ (7), assume that C is Koszul. That is, following the definition, R has an injective resolution 0 → R → Q • such that Q n is strongly graded in degree n. Note that E n,m (C) = Ext C n,m (R, R) is the cohomology in degree n of the complex obtained by applying the functor Hom C (R, −) m to this resolution. Thus, to conclude the proof of this implication, it is enough to show that Hom C (R, Q n ) m is trivial for m = n. By Lemma 3.2(2) we have Hom C (R, Q n ) m = Kerρ Q n m−1,1 . Thus, the claim follows from the fact that Q n is strongly graded since, in view of Lemma 3.2, we have Q n,m = 0 for m < n, and the component ρ Q n m−1,1 is injective for all m > n. For (7) =⇒ (3), let us assume that E(C) is diagonal. We shall prove by induction on n that K • r (C ! , C) is exact in degree n. The sequence exact by the definition of K • (C ! , C) and the fact that C is strongly graded, cf. [JPS,Lemma 2.1].
We now assume that K • (C ! , C) is exact in degree i, for all 0 ≤ i ≤ n − 1, and prove that it is exact in degree n as well. We consider the exact sequence: where Y := Coker d n−1 r and α denotes the canonical projection. We claim that Y is strongly graded in degree n + 1. Since d n−1 r ( x ⊗ c) = x ⊗ c ⊗ 1, for all x ∈ C n−1 1 and c ∈ C 1 , the homogeneous component (C ! n ⊗ C) n is included into the image of d n−1 r . Thus Y has no non-zero elements of degree n. On the other hand, if Y k denotes k-degree component of Y , then Y k = 0 for all k < n, as Y is the quotient of C ! n ⊗ C by a graded subcomodule. In conclusion, for proving our claim, it remains to show that ρ Y m−1,1 is injective for all m > n + 1. Let X := Ker α = Im d n−1 r . We complete 0 → X → C ! n ⊗ C to a resolution 0 → X → C ! n ⊗ C → Q n+1 → Q n+2 → . . . by injective graded right C-comodules, which combined with (12) yields us a new injective resolution: Thus we can compute Ext n+1,m C (R, R) as the cohomology in degree n + 1 of the complex: Recall that X is the kernel of α, so the long exact sequence connecting the functors of Ext •,m C (R, −) m can be written as follows: From the standing assumption, E(C) is diagonal, so using the above isomorphism between the Extgroups, it follows that Ext 1.m C (R, X) = 0, for all m > n + 1. Moreover, Hom C (R, C ! n ⊗ C) is isomorphic to Hom R (R, C ! n ) as graded R-modules, where the latter module is concentrated in degree n. It follows that Hom C (R, Y ) m = 0, for all m > n + 1. By Lemma 3.2 (2) it follows that ρ Y m−1,1 is injective for all m > n + 1. Since we already know that Y k = 0 for k < n + 1, we conclude by Lemma 3.2 (3) that Y is strongly graded in degree n + 1, that is our claim has been proved.
We can now prove that K • (C ! , C) is exact in degree n. We must show that the kernel of the map ∂ : Y → C ! n+1 ⊗ C induced by d n r is trivial. In view of Lemma 3.2 (4), as Y is strongly graded in n + 1, it is enough to prove that the component ∂ n+1 : Let y be an element in the kernel of ∂ n+1 . Hence there are y 1 , . . . , y p ∈ C (n) and c 1 , . . . c p ∈ C 1 such that y is the equivalence class of . Since ∂ n+1 (y) = 0 and using the fact that d n r is induced by the multiplication in C ! , it follows that y ′ = p i=1 y i ⊗ c i belongs to the submodule W n+1 ⊆ C (n+1) from the construction of C ! n+1 . Note that W 2 = Im ∆ 1,1 . Thus y ′ = x + q j=1 y ′′ j ⊗ c j 1,1 ⊗ c j 2,1 , for some x ∈ W n ⊗ C 1 , y ′′ 1 , . . . , y ′′ q ∈ C (n−1) 1 and c 1 , . . . , c q ∈ C 2 . The relation y = 0 now follows by the computation: For the implication (3) =⇒ (4) we use the morphism of complexes φ • : β • r (C) → K • r (C ! , C), which lifts the identity of R, see §1.10. Since both K • r (C ! , C) and β • r (C) are resolutions of R by injective right C-comodules, by the Comparison Theorem, the morphism {φ n } n∈N is invertible up to a homotopy in the category of complexes of right C-comodules. Hence Hom C (R, φ • ) is invertible up to a homotopy in the category of complexes of right R-modules. It follows that H n (Hom C (R, φ • )) is an isomorphism for all n ≥ 0. Thus φ n C : E n (C) → C ! n is an isomorphism, as it coincides with H n (Hom C (R, φ • )), see §1.10.
We conclude the proof by remarking, for the implication (3) =⇒ (2), that the complex K • r (C ! , C) is isomorphic to K • r (E(C), C), which makes the latter exact.
Remark 3.5. By the proof of [JPS,Theorem 2.14], the complex K l • (A, A ! ) is isomorphic to the Koszul complex of A, which was introduced in [BGS,p. 483]. By analogy, K • (C ! , C) will be seen as the Koszul complex associated to a connected R-coring C.
Using the method from the previous section we are going to show that any Koszul coring is quadratic. This property of Koszul corings will follow as a direct application of a cohomological criterion for bijectivity of a morphism between two strongly graded corings. Lemma 3.6. Let C be a strongly graded R-coring. Then the following sequence is exact: Proof. The differential d 1 m of Ω • (C, m) maps c ∈ C m to the family {∆ m (c)} m∈P2(m) where, by notation, ∆ m is the component ∆ m1,m2 : C m1+m2 → C m1 ⊗ C m2 , for any positive 2-partition m = (m 1 , m 2 ) of m. By hypothesis ∆ m is injective for all m ∈ P 2 (m), so d 1 m is injective as well. Thus the sequence On the other hand, the complex Ω • (C <m , m) coincides in small degrees with so we conclude the proof by remarking that Ext 2,m C<m (R, R) ≃ Ker d 2 m .
Proposition 3.7. Let π : C → D denote a morphism of strongly graded connected R-rings. If m is a positive integer such that Ext 2,m C (R, R) = 0 and the components π i are bijective, for all 0 ≤ i < m, then π m is bijective as well.
Proof. As in the case of graded R-rings, π induces morphisms π m : Ext 2,m C (R, R) → Ext 2,m D (R, R) and π m : Ext 2,m C<m (R, R) → Ext 2,m D<m (R, R), such that the following diagram is commutative.
The map π m is an isomorphism, as π i is bijective, for all i < m. Therefore, by Snake Lemma, we have Ker π m = 0 and Coker π m ∼ = Ker π m = 0.

Quadratic corings.
For a connected R-coring C, the family ∆(n) n∈N of iterated comultiplications ∆(n) : C n → C (n) 1 defines a morphism φ C : C → T c R (C 1 ). The image of φ C is a subcoring C of C := {C 1 , Im ∆ 1,1 }. Hence we may regard φ C as a coring morphism from C to C. We shall say that C is quadratic if and only if this morphism is a bijection. Therefore, C is quadratic if and only if it is strongly graded and C = C.
Let C be a strongly graded R-coring. As in the case of R-rings, we can relate Z 2,m = Ker d 2 m , the set of all 2-cocycles in Ω • C, m , and C m . To do this we first notice that, for any 2-cocycle x = {x m } m∈P2(m) , the element ∆(m)(x m ) does not depend on m ∈ P 2 (m), as the components of the (iterated) comultiplication are all injective. Recall that ∆(m) = ∆(m 1 ) ⊗ ∆(m 2 ), for any positive 2partition m = (m 1 , m 2 ). Since ∆(m) = ∆(m)•∆ m , for any positive 2-partition of m, it is not difficult to see that ∆(m)(x m ) ∈ C m . Hence, we can define the function α m : Z 2,m → C m by α m (x) = ∆(m)(x m ).
Let B 2,m := Im d 1 m denote the set of 2-coboundaries. If x = {x m } m∈P2(m) is a 2-coboundary, then there is c ∈ C m such that x m = ∆ m (c), for all positive 2-partitions m. Thus α m (x) ∈ C m . Henceforth, α m induces a map, still denoted by α m , from B 2,m to C m . We get the following commutative diagram where the vertical arrows represent the canonical inclusions.
Lemma 3.9. Let C be a strongly graded R-coring. Proof. In view of the preceding Lemma, the vanishing of Ext 2,m C (R, R) is a necessary condition. To prove that it is a sufficient condition as well, using Proposition 3.7 and proceeding as in the proof of Proposition 2.5, one shows by induction that the components of φ C : C → C are all isomorphisms.
Proof. The conclusion follows by combining the results in Proposition 3.10 and the fact the statements (1) and (7) of Theorem 3.4 are equivalent.

Locally finite Koszul R-(co)rings
In this section, we shall prove that a left (right) locally finite R-ring is Koszul if and only if its left (right) graded dual R-coring is Koszul as well. This result will allow us to show that the incidence R-ring of a graded finite poset is Koszul if and only if its incidence R-coring is also Koszul.
Recall that a graded left R-bimodule V = ⊕ n∈N V n is called left (right) locally finite if and only if its components V n are finitely generated as left (right) modules. In the case when V is both left and right locally finite, we will simply say that V is locally finite. Throughout this section we keep the assumption that R is a semisimple ring, so all R-bimodules are projective as left (and right) R-modules.

The left dual of an
. This becomes a bimodule over the opposed ring R op with respect to the actions defined, for r ∈ R and α ∈ * V , by: In the case that V = ⊕ n∈N V n is a graded R-bimodule we define the left graded dual of V to be the R op -bimodule * -gr V := ⊕ n∈N * V n . It is well known that the dual of the tensor product of two finite dimensional vector spaces is the tensor product of their duals. A similar property holds for R-bimodules. More precisely, if V and W are R-bimodules, there exists a bi-additive map: Thus φ ′ is R op -balanced, so it induces a morphism of abelian groups: As a matter of fact, φ is an R op -bimodule map, as φ((r·α)⊗ R op β)(v⊗ R w) = (r·α)(vβ(w)) = α(vβ(w))r.
On the other hand, Right-linearity is proved analogously. We claim that, under the additional assumption that W is finitely generated as a left R-module, the map φ is a bijection. Indeed, since R is semisimple, W is projective as a left R-module. So, there are finite dual bases on W , that is two sets {w 1 , . . . , w n } ⊆ W and { * w 1 , . . . , * w n } ⊆ * W such that In the above formula γ denotes an element in * (V ⊗ R W ) and the application γ The only thing left to show is that ψ and φ are mutual inverses.
Let γ =: φ(α ⊗ R op β), for some α ∈ * V and β ∈ * W . Thus γ(v ⊗ R w) = α(vβ(w)), for all v ∈ V and w ∈ W . Hence, by the definition of ψ, we get: By the definition of the left R-action on * W and the definition of dual bases, together with the fact that β is a morphism of left R-modules, so ψ is a left inverse of φ. On the other hand, if γ belongs to the left dual of V ⊗ R W then: Thus the computation below implies that ψ is a right inverse of φ as well.

4.2.
The graded dual of an R-(co)ring. The left (right) dual of a left (right) finitely generated R-ring was first introduced in [BW,§17.9]. This construction can be easily adapted for a left locally finite connected graded R-ring A = ⊕ n A n . First, we define * -gr A = ⊕ n * (A n ), which is an R op -bimodule with respect to the actions defined in the previous subsection. When there is no risk of confusion, we shall drop the parentheses to avoid unnecessary clutter. As such, we will write * A n instead of * (A n ).
Furthermore, for making * -gr A a graded connected R op -coring, we consider the following diagram: * The leftmost vertical arrow denotes the transposed map of µ n,m : A n ⊗ A m → A n+m , the component of the multiplication of A. The lower morphism ψ is the isomorphism described in the previous subsection. Then ∆ n,m := ψ • * µ n,m is a morphism of R op -bimodules and the family {∆ n,m } n,m∈N induces a map ∆ : * -gr A → * -gr A ⊗ * -gr A that respects the gradings on * -gr A and * -gr A ⊗ * -gr A.
Let α ∈ * A n+m . One can show that the relation: holds true for some α ′ 1 , . . . , α ′ p and α ′′ 1 , . . . , α ′′ p if and only if we have: for all a ′ ∈ A n and a ′′ ∈ A m . Using this equivalence it is easy to see now that ∆ defines a coassociative comultiplication on the left graded dual of A, which respects the grading on * -gr A. Let us note that we have an isomorphism of rings Hom R ( R R, R R) ∼ = R op , so we can identify * A 0 and R op as R op -bimodules. This isomorphism can be extended in a unique way to an R op -bimodule morphism ε : * -gr A → R op so that it vanishes on all other homogeneous components of * -gr A. Obviously, ( * -gr A, ∆, ε) is a connected graded R op -coring, which will be called the graded left dual R op -coring of A. For the comultiplication of * -gr A we will use the Sweedler type notation: Thus, for α ∈ * A n+m , a ′ ∈ A n and a ′′ ∈ A m , the relation (16) can be rewritten as: α(a ′ a ′′ ) = α 1,n (a ′ α 2,m (a ′′ )).
Dually, to any graded connected R-coring C corresponds a graded connected R op -ring * -gr C that we will call the graded left dual of C. By definition we have ( * -gr C) n = * (C n ). To simplify the notation we shall write * C n instead of * (C n ). The graded convolution product of α ∈ * C n and β ∈ * C m is given by the relation α * β = µ n,m (α ⊗ β). Hence, for c ∈ C n+m , we have: The unit of the graded left dual of C coincides with the counit of C. Note that the graded left dual makes sense for a graded R-coring C which is not necessarily left locally finite.
In a similar way we can define the graded right dual A * -gr of a right locally finite connected graded R-ring A and the graded right dual C * -gr of a right locally finite connected graded R-coring C. One can prove that, for a locally finite R-ring A, there are canonical isomorphisms ( * -gr A) * -gr ∼ = A ∼ = * -gr (A * -gr ) of graded R-rings. Similar isomorphisms can be proved for a locally finite R-coring C.
Note that the first and the second equalities follow by the definitions of the product of * -gr C and of the transposed map, respectively. Taking into account the equivalence between the relations (17) and (18) we get the third equality. The ultimate equation holds as θ satisfies the relation (3). The fact that (A * -gr , C * -gr ) is almost-Koszul can be proved in a similar way.
We can take this result a step forward and prove that a Koszul pair corresponds by left duality to a Koszul pair.
As an application of Corollary 4.5 and Corollary 4.6 we are going to show that the incidence ring of a finite graded poset is Koszul if and only if the incidence coring of that poset is Koszul.
This class of examples was considered initially in [Wo]. They were also studied even in a more general (nongraded) setting in [RS].
4.8. Incidence (co)rings. Let (P, ≤) be a finite poset. If x ≤ y are comparable elements in P, then one defines the interval [x, y] to be the set [x, y] := {z ∈ P | x ≤ z ≤ y}. In the case when x < y and [x, y] = {x, y} we shall say that x is a predecessor of y or that y is a successor of x. By definition, a chain in [x, y] is an increasing sequence x 0 < x 1 < · · · < x l such that x 0 = x and x l = y. The chain is said to be maximal, provided that x i is a predecessor of x i+1 , for all i. The number l will be called the length of the chain.
In this paper we are interested only in graded posets, that is in those posets satisfying the property that for every interval all of its maximal chains have the same length. We write l([x, y]) for the length of a maximal chain in [x, y], and we refer to this number as the length of [x, y]. The set of intervals of length p will be denoted by I p .
Let k be a field. For a finite poset P we denote the k-linear space having a basis B := {e x,y | x ≤ y} by k [P]. With respect to the product given by: e x,y · e z,u = δ y,z e x,u , k[P] becomes an associative and unital k-algebra, that will be called the incidence algebra of P, and it will be denoted by k a [P]. Note that {e x,x } x∈P is a set of orthogonal idempotents, so it spans a subalgebra R, which is isomorphic to k n , where n = |P|. Hence k a [P] can be regarded as an R-ring, which is graded and connected, provided that P is graded. Note that its homogeneous component of degree p is the linear space generated by all elements e x,y such that l([x, y]) = p. The R-ring k a [P] will be called the incidence R-ring of P.
By duality, to every finite poset corresponds a coalgebra, namely its incidence coalgebra k c [P], which as a linear space coincides with k[P]. The comultiplication is given by the formula: The counit ε is uniquely defined such that ε(e x,y ) = δ x,y . Note that k c [P] is an R-bimodule, where R = e x,x | x ∈ P is regarded as a k-algebra as above and the actions are defined by the relation: e x,x · e y,z · e u,u = δ x,y δ z,u e y,z .
is the canonical map. The counit of this coring maps e x,y to δ x,y e x,x . The comultiplication and the counit of k c [P], regarded as a coring, will still be denoted by ∆ and ε. Let us notice that ∆(e x,y ) = where, as usual in this paper, ⊗ = ⊗ R . The relation between k a [P] and k c [P] is explained in the following result. . Therefore, we have to prove that there exists an isomorphism of graded R-rings A ∼ = * -gr C. For two comparable elements x ≤ y we define the k-linear map f x,y : C p → R by f x,y (e u,v ) = δ x,u δ y,v e u,u , for all u ≤ v such that l([u, v]) = p. An easy computation shows that f x,y is left R-linear. Moreover, if f ∈ * C p and [x, y] ∈ I p , then there is a a certain element α x,y in k such that f (e x,y ) = α x,y e x,x . It follows that f = [x,y]∈Ip α x,y f x,y . Thus f can be written in a unique way as linear combination of the elements of {f x,y | [x, y] ∈ I p }, so this set is a linear basis of * C p .
For x ≤ y, z ≤ t and v ≤ w one proves, by a straightforward computation, that: On the other hand, by definition, f x,t (e v,w ) = δ x,v δ t,w e v,v . Thus f x,y * f z,t = δ y,z f x,t , for all x, y, z, v and w as above. Summarizing, the k-linear map χ p : A p → * C p , defined by χ p (e x,y ) = f x,y , for all x ≤ y such that l([x, y]) = p, is the component of degree p of an isomorphism of graded R-rings.
By Corollary 4.6, if C is a Koszul coring, then A ∼ = * -gr C is a Koszul ring. Both A and C are locally finite, being finite dimensional linear spaces. Thus, C ∼ = ( * -gr C) * -gr ∼ = A * -gr . Hence C is Koszul, provided that A is so, cf. Corollary 4.5.
Definition 4.10. We shall say that a graded poset P is Koszul if its incidence ring k a [P] is Koszul.
For a graded poset P let us denote the homogeneous component of degree 1 of its incidence ring by V . For every interval [x, y] of length 2 we define the element: ζ x,y := z∈(x,y) e x,z ⊗ R e z,y .
Let I P denote the ideal generated in T a R (V ) by the set {ζ x,y | l([x, y]) = 2}. With this notation in our hands, we have the following result.
Theorem 4.11. If A is the incidence ring of a Koszul poset P, then the R-ring T a R (V )/I P is Koszul. Proof. By Theorem 4.9 the incidence coring C := k c [P] is Koszul. Hence, in view of Theorem 3.4, the R-ring C ! is Koszul as well. We conclude by remarking that C ! = T a R (V )/I P , as ζ x,y = ∆ 1,1 (e x,y ), for any interval [x, y] of length 2.

Examples of Koszul posets
In this section, we shall provide some classes of posets which are Koszul. All of them will be obtained by adjoining a greatest element to a given subset of a Koszul poset. The Koszulity of the resulting poset is ensured by imposing homological restrictions to a certain module, canonically associated to the given subset.
We first prove that the direct (co)product of two Koszul (co)rings is still Koszul. This result will be used in §5.11 to show that a poset whose Hasse diagram is a planar tiling is Koszul.
Proposition 5.1. Let R and S be semisimple rings. We assume that A is a connected R-ring, B is a connected S-ring, C is a connected R-coring and D is a connected S-coring.
( Proof. For an R-bimodule V and an S-bimodule W we shall regard the abelian group V ⊕ W as an R × S-bimodule with respect to the actions: (r, s) · (v, w) = (rv, sw) and (v, w) · (r ′ , s ′ ) = (vr ′ , ws ′ ). Note that we have an isomorphism of left A × B-modules: where the A × B-action on the direct sum of A ⊗ R V and B ⊗ S W is also defined component-wise. This isomorphism is defined by the map (a, b) Recall that the comultiplication of the coproduct A ! ⊕ B ! is given by: Using the identification (19) and performing a similar computation as above one can show easily that: is Koszul if and only if both (A, A ! ) and (B, B ! ) are so. Therefore, the first part of the proposition follow by Theorem 2.1. The second part can be proved in a similar way.

Notation.
Let k be a field, and we fix a finite graded poset P, together with a maximal element t ∈ P. Let Q := P \ {t} and let F denote the set of predecessors of t in P. The incidence algebras of P and Q will be denoted by A and B, respectively. Note that A has a canonical structure of graded R-ring, where R = x∈P ke x,x . On the other hand, B is an S-ring, where S = x∈Q ke x,x .
Finally, let M denote the linear subspace of A generated by all elements e x,t , with x < t. Hence e x,t ∈ M if and only if x ≤ u for a certain u ∈ F. It is easy to see that M is a B-submodule of A and M = u∈F Be u,t .
As usual, we denote the elements of the canonical basis of A by e x,y , where x, y ∈ P and x ≤ y. Considering only the elements e x,y , with x, y ∈ Q, we get the canonical basis of B.
Recall that an n-chain in P is a sequence x = (x 0 , . . . , x n ) of elements in P such that x 0 < · · · < x n . For the element x n of x we shall use the special notation t(x) and we shall refer to it as the target of x.
The length of x is, by definition, the integer number l( , where l([x, y]) denotes the length of the interval [x, y]. For the set of n-chains in P of length m we shall use the notation P n,m .
Let P n denote the set of n-chains in P. For every x ∈ P n we define e x ∈ A ⊗ k n + by e x := e x0,x1 ⊗ k e x1,x2 ⊗ k . . . ⊗ k e xn−1,xn .
To investigate the Koszulity of A we need the following result. Recall that the index m on the Tor spaces corresponds to the internal grading, that is the grading induced by the fact that P is a graded poset. By §1.4, one can compute Tor A n,m (R, R) as the nth homology group of the complex Ω • (A, m). Note that A + is the direct sum of the R-bimodules ke x,y , where x and y are arbitrary elements in P such that x < y. Since ke x,y ⊗ R ke x ′ ,y ′ = 0, whenever y = x ′ , we get that Ω n (A, m) = (A ⊗Rn + ) m is the direct sum of the R-bimodules ke x0,x1 ⊗ R ke x1,x2 ⊗ R . . . ⊗ R ke xn−1,xn , where x = (x 0 , . . . , x n ) is a chain in P n,m . Moreover, it is not difficult to see that ke x0,x1 ⊗ R ke x1,x2 ⊗ R . . . ⊗ R ke xn−1,xn ∼ = ke x0,x1 ⊗ k ke x1,x2 ⊗ k . . . ⊗ k ke xn−1,xn = ke x .
for any n-chain of length m. Thus we can identify Ω n (A, m) with the k-linear subspace of the nth tensor power of A + which is spanned by all e x , with x ∈ P n,m . Via this identification, the differential of Ω • (A, m) is given by the formula: Clearly, we can identify in a similar way Ω n (B, m) with the subspace of A ⊗ k n + which is spanned by the elements e x , with x ∈ Q n,m . In this way, Ω • (B, m) can be seen as a subcomplex of Ω • (A, m).
Let Ω ′ n denote the linear subspace of A ⊗ k n + generated by e x , where x ∈ P n,m and t(x) = t. Note that x 0 , . . . , x n−1 are elements in Q. Obviously, Ω ′ • is a subcomplex of Ω • (A, m). Since t is a maximal element in P, it follows that Ω • (A, m) is the direct sum of Ω • (B, m) and Ω ′ • . In particular, computing the homology of Ω • (A, m) we get: To compute the homology group from the above relation, let us note that Ω ′ Therefore, proceeding as above we identify the homogeneous component of degree m of B ⊗S n−1 + ⊗ S M with the subspace of A ⊗ k n + generated by e x ⊗ k e t(x),t , where x ∈ Q n−1 and l(x) + l([t(x), t]) = m. Since the latter linear space is precisely Ω ′ n we have an isomorphism Ω ′ n ∼ = (β r n−1 (B) ⊗ B M ) m . It is easy to see that these maps are compatible with the differential maps, so they define an isomorphism of complexes. In conclusion, . This completes the proof of the isomorphism (20). The last claim, concerning the Koszulity of A, follows by Theorem 2.1.
We need another result which will be used in the following applications. Lemma 5.4. Keeping the notations and the definitions above, we have Tor A n (R, Ae u,u ) = 0, for all u ∈ P and n > 0. Even more, Tor A n,m (R, Ae u,u ) = 0, for all n ≥ 0 and m = n. Proof. Let n > 0. Remark that Ae u,u is a projective A-module, because e u,u is an idempotent in A, so A = Ae u,u ⊕ A(1 − e u,u ). Since Tor A n,m (R, Ae u,u ) is a direct summand in Tor A n (R, Ae u,u ) = 0, it follows that the former R-module is also zero, for all m.
We are left to proving that Tor A 0,m (R, Ae u,u ) = 0, for all m > 0. This follows by the relation Tor A 0,m (R, Ae u,u ) = (R ⊗ A Ae u,u ) m and the following isomorphisms: Remark 5.5. The relation Tor A n,m (R, Ae u,u ) = 0 holds for any poset P and every u ∈ P, provided that m = n. Then, keeping the notation from §5.2, we also have Tor B n,m (S, Be s,s ) = 0, for s ∈ Q and m = n.
Lemma 5.6. Keeping the notation from §5.2, we have Tor B n−1,m (S, Be u,t ) = 0, for every u ∈ F and for all m = n.
Proof. The sets {e x,u | x ≤ u} and {e x,t | x ≤ u} are bases of the vector spaces Be u,u and Be u,t , respectively. Hence the unique linear map given by e x,u → e x,t , for all x ≤ u, is bijective. On the other hand, one sees easily that this map is a morphism of B-modules. With respect to the internal grading, since deg(e x,t ) = deg(e x,u ) + 1, the above map induces an isomorphism of degree +1 between the graded modules Be u,u and Be u,t . This implies that: n−1,m (S, Be u,t ) ≃ Tor B n−1,m−1 (S, Be u,u ). We conclude the proof by applying Lemma 5.4 and taking also into account the preceding Remark.

The condition ( †).
Under the standing notations and assumptions §5.2, our next goal is to provide examples of graded posets P such that the R-ring A is Koszul. In order to do that we want to apply Theorem 5.3, so we need conditions that ensures the vanishing of Tor B n−1,m (S, M ), for all m = n. We shall say that the set F satisfies the condition ( †) if and only if either F is a singleton, or there is a common predecessor s of all elements in F such that s is the infimum of every couple of distinct elements in F. Note that s = inf{u, v} if and only if: Although somehow cumbersome and unnatural at this point, the use of this condition will become evident throughout the proof of the next result. Moreover, let us mention that ( †) is a purely combinatorial condition, depending only on the structure of the poset Q and the set F.
Theorem 5.8. Let P be a graded poset and let t ∈ P denote a maximal element. Let A and B denote the incidence algebras of P and Q := P \ {t}, respectively. If the set F of all predecessors of t in P satisfies the condition ( †), then A is a Koszul R-ring if and only if B is a Koszul S-ring.
Proof. As we have already noticed, in view of Theorem 5.3, we have to check that Tor B n−1,m (S, M ) = 0, for all m = n. We proceed by induction on the cardinality of F, which we denote by f .
For f = 1, we have M = Be u,t , where u is the unique element of F. Hence we conclude the proof of the basic case by applying Lemma 5.6. Note that in this case we do not need the condition ( †).
Assume now that the result is true for any set F ′ of cardinality f and take F a set with f + 1 elements which satisfies the condition ( †). Let s denote the common predecessor of the elements of F. We pick u ∈ F and we take F ′ := F \ {u}. It is clear that F ′ satisfies the condition ( †), with respect to the same s. Therefore, if M ′ := v∈F ′ Be v,t , then Tor For this, note that we can make the following identifications: The last isomorphism holds true by an immediate generalization of the relation (22). Indeed, if we take an element w in Be u,t ∩ x∈F ′ Be x,t , then w = y≤u α y e y,t = x∈F ′ z≤x β z,x e z,t . For every y such that α y = 0, the element e y,t must also appear in the double sum with a nonzero coefficient. In particular, there must exist x ∈ F ′ such that y ≤ x. Since y ≤ u as well, we get that e y,t ∈ Be u,t ∩ Be x,t = Be s,t . Thus w ∈ Be s,t . In conclusion, Be u,t ∩ x∈F ′ Be x,t is a submodule of Be s,t . The other inclusion is obvious, so the above intersection and Be s,t coincide, as we claimed. Let us consider the following exact sequence: · · · → Tor B n−1,m (S, Be u,t ) → Tor B n−1,m (S, Be u,t Be s,t ) → Tor B n−2,m (S, Be s,t ) → . . .
We already know that Tor B n−1,m (S, Be u,t ) = 0, cf. Lemma 5.6. To conclude the proof of the Theorem it remains to show that Tor B n−2,m (S, Be s,t ) = 0. Proceeding as in the proof of Lemma 5.6, one show that there is an isomorphism of degree 2 between the graded B-modules Be s,s and Be s,t , which is given by e x,s → e x,t for all x ≤ s. Thus: Tor B n−2,m (S, Be s,t ) ≃ Tor B n−2,m−2 (S, Be s,s ). Hence, by Lemma 5.4, we get Tor B n−2,m (S, Be s,t ) = 0. Recall that the dual poset of P coincides, as a set, with P. On the other hand, x is less than y in the dual poset of P if and only if x is greater than y in P.
For a subset F ⊆ P, the condition ( †) with respect to the dual partial order relation, can be stated as follows. We shall say that F satisfies the condition ( ‡) if and only if either F is a singleton or there is a common successor s of all elements in F such that s = sup{u, v}, for all u = v ∈ F.
Since an R-ring is Koszul if and only if its opposite R op -ring is so, working with the dual poset of P, we get the theorem below.
Theorem 5.9. Let P be a graded poset and let t ∈ P denote a minimal element. Let A and B denote the incidence algebras of P and Q := P \ {t}, respectively. If the set F of all successors of t in P satisfies the condition ( ‡), then A is a Koszul R-ring if and only if B is a Koszul S-ring.
We are now able to describe an algorithm, based on Theorem 5.8 and Theorem 5.9, which will help us to produce examples Koszul posets.
We shall consider here four types of sufficient conditions that guarantee the Koszulity of P = Q ∪ T . (a) The case Q ∩ T = ∅. Since the intersection of Q and T is empty, we have k a [P] = k a [Q] × k a [T ]. Thus, by Proposition 5.1, k a [P] is Koszul.
(b) The case |Q ∩ T | = 1. Let us assume that Q ∩ T = {u T }. Using the algorithm, one adds successively the missing elements of T , so that at every step one obtains a new Koszul poset. More precisely, one first applies the construction (1) of the algorithm to add t T to Q. Then, one adds v T using the construction (3). Finally, one takes F := {u T , v T } and uses (4) to patch completely the tile T , by adjoining s T . Note that the elements in F have a unique successor, namely t T . Henceforth, the condition ( †) is trivially fulfilled for this particular choice of F.
If Q ∩ T := {t T }, then we first add u T and v T to Q, using in both cases (3), and then we apply (4) as above. To handle the cases when Q ∩ T is either {v T } or {s T }, one proceeds in a similar way.
(c) The case Q ∩ T = {x, y}, where x is a predecessor of y in T . Such a set uniquely determine an edge of the Hasse diagram of T . The fact that P is Koszul can be proved using the same method, independently of the choice of {x, y}. So we only discuss the case when x = u T and y = t T . First, using (3), we adjoin the element v T to Q. Then we can repeat the last step from the preceding case to add the remaining element s T .
(d) The case Q ∩ T = {u T , v T , x}, where x is either t T or s T . If x = t T then s T can be added to Q taking F := {u T , v T } and using once again (4). We have to show that F satisfies the condition ( ‡). To simplify the notation, we shall denote the elements of Q ∩ T by u, v and t. We consider the maximal sequences of tiles {T 1 , T 2 , . . . , T n } and {T ′ 1 , T ′ 2 , . . . , T ′ m }, as in the figure below.
Keeping the notation from this picture, and assuming that x in an element in Q great than or equal to v, then either x ≥ t or there exists i ∈ {0, . . . , m} so that x = v i . The elements x ≥ u can be characterized in a similar way. Thus an upper bound x of {u, v} must be great than or equal to t, so t = sup{u, v}. In particular, a set F as above must satisfy the condition ( ‡). The dual case can be managed analogously.

5.
13. An example of planar tiling. To illustrate the construction of planar tilings we consider the following picture.
One of the possible ways to show that it represents the Hasse diagram of a Koszul poset is to patch step by step the tiles 1 through 8, as in the case (c). Similarly, we can add the tiles 9 up to 19. The adjoining of tiles 20 and 21 follows the case (b), since they intersect the poset previously constructed in one element. On the other hand, the tile 22 can be patched as having empty intersection with the other component of the poset, while for the tiles 23 and 24 we use the case (c) once again. Finally we add the tile 25 as in (d).
5.14. Nested Diamonds. In a similar way we get a new Koszul poset, the 'nested vertical diamonds', whose Hasse diagram is depicted in the first picture of the figure below. Start with the trivial poset {s} and adjoin the elements u 1 , . . . , u n using the construction (1) from the algorithm. Thus s is the infimum of each couple u i and u j and these elements are not comparable in the resulting poset. Hence we can apply (3), taking F = {u 1 , u 2 , . . . , u n }. As a last example we consider the poset P i,j , the 'nested horizontal diamonds', represented in the second picture of the next figure. If i, j > 1, then P i,j cannot be constructed using our algorithm, as the infimum and the supremum of F = {u, v} do not exist, so the conditions ( †) and ( ‡) do not hold.
However, if k is a field of characteristic zero, then P i,j is a Koszul poset. To see this, we shall prove that k a [P i,j ] is a braided symmetric R-bialgebra. For the definition of braided R-bialgebras in general and of symmetric braided bialgebras in particular, the reader is referred to [JPS,§6.1].
Let V be the R-bimodule generated by all elements e x,y of the canonical basis of k a [P i,j ], such that [x, y] is an interval of length one. Hence, the set {e sp,u ⊗ R e u,tq , e sp,v ⊗ R e v,tq | p ≤ i and q ≤ j} is a basis of V ⊗ R V as a linear space. Moreover, since there are no maximal chains of length greater than 2, it follows that V ⊗Rl = 0, for all l > 2. Further, we define the R-bilinear braiding c : V ⊗ R V → V ⊗ R V such that it 'interchanges' the chains of length two that correspond to the same p and q. More precisely, for all p ≤ i < q, we have: c(e sp,u ⊗ R e u,tq ) = e sp,v ⊗ R e v,tq and c(e sp,v ⊗ R e v,tq ) = e sp,u ⊗ R e u,tq .
The linear map c satisfies trivially the braid equation, because V ⊗R3 = 0. Note that c 2 = I V ⊗RV . Let I denote the ideal generated by the image of I V ⊗RV − c. By definition, S R (V, c) := T a R (V )/I. Now we can prove the isomorphism k a [P i,j ] ∼ = S R (V, c) remarking that the R-ring k a [P i,j ] coincides with the quotient of T a R (V ) modulo the ideal generated by the differences e sp,u ⊗ R e u,tq − e sp,v ⊗ R e v,tq , where p ≤ i and q ≤ j. By [JPS,Theorem 6.2 ], it follows that k a [P i,j ] is a Koszul braided R-bialgebra.
Remark 5.15. There is another class of Koszul rings that can be associated to 'nested vertical diamonds'. Let P be such a poset and let us denote the homogeneous component of degree 1 of k a [P] by V . Then, by Theorem 4.11, the following ring is Koszul: where ζ s,t = n i=1 e s,ui ⊗ e ui,t is the element associated to the unique interval [s, t] of length 2 in P. Remark 5.16. In [Wo,Definition 4.6], the author defines a graded poset Ω as being exactly thin whenever x < y and l(x, y) = 2 imply that the interval (x, y) consists of precisely two elements. Note that what we termed 'planar tilings' are examples of such posets, which we proved to be Koszul. Furthermore, our 'nested horizontal posets' are also exactly thin and Koszul.