Then we prove homotopy invariance of the introduced cohomology theory for undirected graphs

COHOMOLOGY OF DIGRAPHS AND (UNDIRECTED) GRAPHS^∗

ALEXANDER GRIGOR’YAN^†, YONG LIN^‡, YURI MURANOV^§, AND SHING-TUNG YAU^¶ Abstract. We construct a cohomology theory on a category of finite digraphs (directed graphs), which is based on the universal calculus on the algebra of functions on the vertices of the digraph.

We develop necessary algebraic technique and apply it for investigation of functorial properties of this theory. We introduce categories of digraphs and (undirected) graphs, and using natural isomor- phism between the introduced category of graphs and the full subcategory of symmetric digraphs we transfer our cohomology theory to the category of graphs. Then we prove homotopy invariance of the introduced cohomology theory for undirected graphs. Thus we answer the question of Babson, Barcelo, Longueville, and Laubenbacher about existence of homotopy invariant homology theory for graphs. We establish connections with cohomology of simplicial complexes that arise naturally for some special classes of digraphs. For example, the cohomologies of posets coincide with the cohomologies of a simplicial complex associated with the poset. However, in general the digraph cohomology theory can not be reduced to simplicial cohomology. We describe the behavior of di- graph cohomology groups for several topological constructions on the digraph level and prove that any given finite sequence of non-negative integers can be realized as the sequence of ranks of digraph cohomology groups. We present also sufficiently many examples that illustrate the theory.

Key words. (co)homology of digraphs, (co)homology of graphs, differential graded algebras, path complex of a digraph, simplicial homology, differential calculi on algebras.

AMS subject classifications.05C25, 05C38, 16E45, 18G35, 18G60, 55N35, 55U10, 57M15.

1. Introduction. In this paper we consider finite simple digraphs (directed graphs) and (undirected) simple graphs. A simple digraphGis couple (V, E) where V is any set andE⊂ {V ×V \ diag}. Elements ofV are called the vertices and the elements of E – directed edges. Sometimes, to avoid misunderstanding, we shall use the extended notations VG and EG instead ofV and E, respectively. The fact that (a, b)∈Ewill be denoted bya→b. A (undirected) graphGis a pair (V, E) (or more precise (VG, EG)) whereV is a set of vertices andE is a set of unordered pairs (v, w) of vertices. The elements of E are called edges. In this paper we shall consider only simple graphs, which have no edges (v, v) (loops).

A digraph is a particular case of a quiver. A particular example of a digraph is a poset (partially ordered set) when E is just a partial order (that is, a →b if and only if a ≥ b). The interest to construction of some type of algebraic topology on the digraphs and graphs is motivated by physical applications of this subject (see, for example, [6], [7], [8]), discrete mathematics [24], [18], [4], and graph theory [1], [2], [18], and [19, Part III].

Dimakis and M¨uller-Hoissen suggested [7] and [8] a certain approach to con- struction of cohomologies on digraphs, which is based on the notion of a differential

∗Received March 31, 2013; accepted for publication May 28, 2014.

†Department of Mathematics, University of Bielefeld, 33501 Bielefeld, Germany (grigor@math.

uni-bielefeld.de). Partially supported by SFB 701 of German Research Council.

‡Department of Mathematics, Renmin University of China, Beijing, China. Supported by the Fundamental Research Funds for the Central Universities and the Research Funds of (11XNI004).

§Faculty of Mathematics and Computer Science, University of Warmia and Mazury, Olsztyn, Poland (muranov@matman.uwm.edu.pl). Partially supported by the CONACyT Grants 98697 and 151338, SFB 701 of German Research Council, and the travel grant of the Commission for Developing Countries of the International Mathematical Union.

¶Department of Mathematics, Harvard University, Cambridge MA 02138, USA (yau@

math.harvard.edu). Partially supported by the grant “Geometry and Topology of Complex Net- works”, no. FA-9550-13-1-0097.

887

calculus on an abstract associative unital algebraA over a commutative unital ring K. However, this approach remained on intuitive level without a precise definition of the corresponding cochain complex. An explicit and direct construction of chain and cochain complexes on arbitrary finite digraphs was given in [13] (see also [15]). The construction in [13] of n-chains is based on the naturally defined notion of a path of lengthnon a digraph.

In the present paper we provide an alternative construction of the cochain complex that is equivalent to that of [13] (see Section 4 below). We stay here on the algebraic point of view of [3], [6] to define a functor from the category of digraphs to the category of cochain complexes. We develop necessary algebraic background for this approach, which makes most of the constructions functorial and enables one to use methods of homological algebra [17], [22]. The main construction is based on the universal calculus on the algebra of functions on the vertices of the digraph.

This constructed cohomology theory happens to be closely related to other coho- mology theories but is not covered by them. For example, the cohomology groups of a poset coincide with the simplicial cohomology groups of a simplicial complex associ- ated with the poset and with the Hochschild cohomology of corresponding incidence algebra (see [5], [12], and [14]). We would like also to point out, that the digraph cohomology theory gives new geometric connections between the digraphs and cubic lattices of topological spaces (see, [9], [10], and [15]) and new algebraic connections with algebras of quiver and incidence algebras (see [11], [21], [5], [14]).

We introduce categories of digraphs and (undirected) graphs. Using natural iso- morphism between introduced category of graphs and full subcategory of symmetric digraphs (see [16, Section 1.1]), we transfer the cohomology theory to the category of graphs. In the papers [1] and [2] the homotopy theory of graphs was constructed, and the question about natural homotopy invariant homology theory of graphs was raised.

We prove homotopy invariance of introduced cohomology theory for graphs and give several examples of computations. Note, that the previously known homology theory of digraphs (see, for example, [18, Section 3]) is not homotopy invariant.

We prove functoriality of the cohomology groups for natural maps of digraphs.

In particular, for a subcategory of digraphs with the inclusion maps we obtain direct description of relative cohomology groups. We describe behavior of introduced coho- mology groups for several transformations of digraphs that are similar to standard topological constructions.

We describe relations between the digraph cohomology and the simplicial coho- mology of various simplicial complexes which arise naturally for some special classes of digraphs. Finally, we prove the following cohomology realization theorem:

for any finite collection of nonnegative integers k0, k1, . . . kn with k0 ≥ 1, there exists a finite digraph G (that is not a poset) such that the cohomology groups of its differential calculus satisfies the conditions

dimHⁱ(ΩG) =ki for all 0≤i≤n.

The paper is organized in the following way. In Section 2 we give a short survey of the classical results on abstract differential calculi on associative algebras [3] in the form that is adapted to further application to digraphs. We provide several technical theorems which are based on the standard algebraic results (see [3], [20], and [22]) which will be helpful in the next sections.

In Section 3, we define the differential calculus on the algebra of functions on a finite set following [7] and [8] and describe its basic properties.

In Section 4, we define the calculus on simple finite digraphs. We use the algebraic machinery developed in previous sections and prove that we have a functor from the category of digraphs to the category of differential calculi with morphisms of the calculi. We describe some cohomology properties of these calculi and prove among others the cited above cohomology realization theorem.

In Section 5, we construct a cohomology theory on the category of undirected graphs that is identified naturally with the full subcategory of symmetric digraphs [16] and prove the homotopy invariance of obtained cohomology theory. Note that our homology theory of graphs is new and its construction realizes the desire of Babson, Barcelo, Longueville, and Laubenbacher ” for a homology theory associated to the A-theory of a graph” (see [1, page 32]).

In Section 6 we consider a category of acyclic digraphs and transfer to this case the results of previous sections. We describe a sufficiently wide class of acyclic digraphs for which the cohomology theory admits a geometrical realization in terms of simplicial complexes.

2. Differential calculus on algebras. In this section we give a short survey of classical results on abstract differential calculi on associative algebras in the form that is adapted to further application to digraphs. Starting with a standard construction of a first order calculus from [3], we give two methods for construction of higher order universal differential calculi and prove their equivalence. We provide several technical theorems which are based on the classical algebraic results (see [3], [20], and [22]) which will be helpful in the next sections.

LetKbe a commutative unital ring andAbe an associative unital algebra over K.

Definition 2.1. A first order differential calculus on the algebra A is a pair (Γ, d) where Γ is anA-bimodule, andd:A →Γ is aK-linear map such that

(i) d(ab) = (da)·b+a·(db) for all a, b ∈ A (where · denotes multiplication between the elements ofAand Γ).

(ii) The minimal left A-module containing dA, coincides with Γ, that is, any elementγ∈Γ can be written in the form

(2.1) γ=X

i

ai·dbi

withai, bi∈ A, wherei run over any finite set of indexes.

By [3, III, §10.2], a mapping dsatisfying (i) is called a derivation of A into Γ.

The condition (i) implies

d1A=d(1A1A) = (d1A) 1A+ 1A(d1A) = 2d1A

and hence d1A= 0. TheK-linearity implies then thatd(k1A) = 0 for any k∈K.

Let us describe a construction of the first order differential calculus for a general algebraA. The algebraA can be regarded as aK-module, and the tensor product A ⊗KA is also defined as a K-module. In what follows we will always denote ⊗K

simply by⊗.

Note that Aand A ⊗KAhave also natural structures of A-bimodules. We will denote by·the product of the elements ofAby those ofA ⊗_KA. For alla, b, c∈ A, we have

(2.2) c·(a⊗b) = (ca)⊗b and (a⊗b)·c=a⊗(bc) Define the following operator

(2.3) d:A → A ⊗ A, da:= 1A⊗a−a⊗1A,

and observe that it satisfies the product rule. Hence, d is a derivation fromA into A ⊗ A. Now we reduce the A-bimodule A ⊗ A to obtain a first order differential calculus.

Definition 2.2. Define Ω¹_Aas the minimal leftA-submodule ofA⊗Acontaining dA. In other words, Ω¹_A consists of all finite sums of the elements ofA ⊗ A of the forma·db witha, b∈ A(cf. (2.1)

Proposition 2.3. Ω¹_A is a A-bimodule and, hence, Ω¹_A, d

is a first order differential calculus on A.

Proof. Let u∈Ω¹_A and c ∈ A. We need to prove that c·u and u·c belong to Ω¹_A. By definition of Ω¹_A, it suffices to verify this for u=a·db wherea, b∈ A.Then c·u= (ca)·db∈Ω¹_A and

u·c= (a·db)·c=a·(db·c) =a·(d(bc)−b·dc) =a·d(bc)−(ab)·dc∈Ω¹_A. Hence, Ω¹_Asatisfies all the requirements of Definition 2.1.

Let us give an alternative equivalent description of Ω¹_A. Define aK-linear map

(2.4) µ:A ⊗ A → A, µ X

i

ai⊗bi

!

=X

i

aibi

where i runs over a finite index set. By (2.2) the map µ is a homomorphism of A-bimodules. It follows from (2.2), (2.3) and (2.4) that, for alla, b∈ A,

µ(a·db) =µ(a⊗b)−µ(ab⊗1A) =ab−ab= 0, so thata·db∈kerµand, hence, Ω¹_A⊂kerµ.In fact, the following is true.

Theorem 2.4. [3, III,§10.10]

(i)We have the identityΩ¹_A= kerµ, whereµis defined by (2.4).

(ii) For every differential calculus of first order (Γ, d^′) over the algebra A there exists exactly one epimorphismp of A-bimodules p: Ω¹_A→Γ such that the following diagram is commutative

(2.5)

A −→^d Ω¹_A l^id ↓^p A ^d

′

−→ Γ.

Definition 2.5. The pair Ω¹_A, d

is called theuniversal first order differential calculus onA.

Example 2.6. Consider the R-algebra A = C^m(R) and the bimodule Γ = C^m−1(R) with the usual derivative of functions f from A that will be denoted by d^′f. Let us describe explicitly the epimorphism p: Ω¹_A→ Γ from Theorem 2.4(ii).

Define a mappingp:A ⊗ A →Γ by p(f⊗g) =1

2(f g^′−f^′g)

and extend it additively to all elements of A ⊗ A. It is easy exercise to prove that p|Ω¹_A is aA-bimodule epimorphism, using thatf⊗g−g⊗f ∈Ω¹_A and

(2.6) p(f⊗g−g⊗f) = 1

2(f g^′−f^′g)−1

2(gf^′−g^′f) = (f g)^′.

Finally, for anyf ∈ Aby (2.6) we have (p◦d)f =p(1⊗f−f⊗1) =f^′ so thatp◦d is the ordinary first order derivative onA.

Let us pass to construction of a higher order differential calculus onA. We start with the following two definitions.

Definition 2.7. A graded unital algebra Λ over a commutative unital ringKis an associative unitalK-algebra that can be written as a direct sum

Λ = M

p=0,1,...

Λ^p

ofK-modules Λ^pwith the following conditions: the unity 1Λ of Λ belongs to Λ⁰and u∈Λ^p, v∈Λ^q ⇒ u∗v∈Λ^p+q,

where ∗ denotes multiplication in Λ. If u∈Λ^p then pis called thedegree of uand is denoted by degu.The operation of multiplication in a graded algebra is called an exterior (or a graded) multiplication. A homomorphism f: Λ^′ →Λ^′′ of two graded unitalK-algebras Λ^′ and Λ^′′ is a homomorphism ofK-algebras that preserves degree of elements.

Definition 2.8. A differential calculus on an associative unital K-algebraAis a couple (Λ, d), where Λ is a graded algebra

Λ = M

p=0,1,...

Λ^p

overKsuch that Λ⁰=A, andd: Λ→Λ is aK-linear map, such that (i) dΛ^p⊂Λ^p+1

(ii) d²= 0

(iii) d(u∗v) = (du)∗v+ (−1)^pu∗(dv), for allu∈Λ^p, v∈Λ^q, where∗ is the exterior multiplication in Λ;

(iv) the minimal leftA-submodule of Λ^p+1 containing dΛ^p coincides with Λ^p+1, that is, anyw∈Λ^p+1 can be represented as a finite sum of the form

(2.7) w=X

k

ak∗dvk

for someak ∈ Aandvk ∈Λ^p.

The property (iii) in Definition 2.8 is called theLeibniz ruleor theproduct rule.

A classical example of a differential calculus is the calculus of exterior differential forms on a smooth manifold with the wedge product and with the exterior derivation.

This calculus is based on the algebra Aof smooth functions on the manifold.

The following property of a differential calculus will be frequently used.

Lemma 2.9. Let (Λ, d)be a differential calculus on A. Then for any p≥0 any element w∈Λ^pcan be written as a finite sum

(2.8) w=X

j

a^j₀∗da^j₁∗da^j₂∗ · · · ∗da^j_p,

where a^j_i ∈ Afor all 0≤i≤pand∗ is the exterior multiplication inΛ.

Proof. Representation (2.8) forp= 0 is true by Λ⁰=A. Let us make an inductive step from p−1 top.By part (iv) of Definition 2.8, it suffices to show the existence of the representation (2.8) forw=a∗dvwitha∈ Aandv∈Λ^p−1. By the inductive hypothesis,v admits the representation in the form

v=X

j

a^j₁∗da^j₂∗da^j₂∗ · · · ∗da^j_p

where all a^j_i ∈ A. Using the associative law, the Leibniz rule andd²= 0, we obtain dv=X

j

da^j₁∗da^j₂∗da^j₂∗ · · · ∗da^j_p,

whence (2.8) follows with aⁱ₀=a.

The first method of construction a differential calculus onAuses multiple tensor products⊗K ofAby itself as in the following definition.

Definition 2.10. Given an arbitrary associative unital K-algebra A, define a gradedK-algebraT as follows:

where

(2.9) T= M

p=0,1,...

T^p, where T^p=





A, p= 0

A ⊗ A ⊗ · · · ⊗ A

| {z }

ptimes⊗

, p≥1,

and the exterior multiplication T^p•T^q −→T^p+q is defined by

(2.10) (a0⊗a1⊗ · · · ⊗ap)•(b0⊗b1⊗ · · · ⊗bj) :=a0⊗a1⊗ · · · ⊗apb0⊗b1⊗ · · · ⊗bq, for allai, bj ∈ A.

It is a trivial exercise to check that the multiplication • is well-defined and that T is indeed a graded associative unital K-algebra with the unity 1T = 1A. The multiplication • by elements ofA=T⁰ endows eachK-moduleT^p by a structure of A-bimodule.

Note, that the original multiplication in the algebraAcoincides with the exterior multiplicationT⁰•T⁰→T⁰, and the multiplication·of the elements ofA=T0 and A ⊗ A=T1 defined in (2.2), coincides with exterior multiplicationT⁰•T¹→T¹.

Define aK-linear mapd: T^p→T^p+1(p≥0) by a formula (2.11) d(a0⊗ · · · ⊗ap) =

p+1X

i=0

(−1)ⁱa0⊗ · · · ⊗ai−1⊗1A⊗ai⊗ · · · ⊗ap, for allai ∈ A. The next result can obtained by straightforward computation.

Proposition 2.11. For the operator (2.11) we have d² = 0. In particular, d determines the following cochain complex ofK-modules

0−→T⁰−→^d T¹−→^d T²−→. . .

Remark 2.12. The homomorphismε:K→ Adefined by ε(k) =k1Aevidently satisfies the propertyd◦ε= 0. Hence we can equip the cochain complex T^∗ by the augmentationε. We shall denote this complex with the augmentation εbyTe^ε.

Proposition 2.13. The mapd defined in (2.11) satisfies the following product rule:

(2.12) d(u•v) =du•v+ (−1)^pu•dv for allu∈T^p andv∈T^q.

Proof. It suffices to prove (2.12) for u = a0⊗a1 ⊗ · · · ⊗ap ∈ T^p and v = b0⊗b1⊗ · · · ⊗bq ∈T^q. We have

d(u•v) =d(a0⊗a1⊗ · · · ⊗apb0⊗b1⊗ · · · ⊗bq)

= Xp j=0

(−1)^ja0⊗ · · · ⊗aj−1⊗1A⊗aj⊗ · · · ⊗apb0⊗b1⊗ · · · ⊗bq

+ (−1)^p+1a0⊗ · · · ⊗ap−1⊗apb0⊗1A⊗b1⊗ · · · ⊗bq

+

q+1X

i=2

(−1)^p+ia0⊗ · · · ⊗ ⊗ap−1⊗apb0⊗b1⊗ · · · ⊗bi−1⊗1A⊗bi· · · ⊗bq

On the other hand, we have du•v+ (−1)^pu•dv

= Xp j=0

(−1)^ja0⊗ · · · ⊗aj−1⊗1A⊗aj⊗ · · · ⊗ap)•(b0⊗b1⊗ · · · ⊗bq)

+(−1)^p+1(a0⊗ · · · ⊗ap⊗1A)•(b0⊗b1⊗ · · · ⊗bq) [term withj=p+ 1]

+(−1)^p(a0⊗ · · · ⊗ap)•(1A⊗b0⊗ · · · ⊗bq) [term withi= 0]

+(−1)^p(a0⊗ · · · ⊗ap)•(−1) (b0⊗1A⊗b1⊗ · · · ⊗bq) [term withi= 1]

+ (−1)^p(a0⊗ · · · ⊗ap)• Xq+1 i=2

(−1)ⁱb0⊗ · · · ⊗bi−1⊗1A⊗bi⊗ · · · ⊗bq .

Noticing that the terms withj=p+ 1 andi= 0 cancel out, we obtain the required identity.

Now we reduce the graded algebraT introduced above, to obtain a differential calculus in the sense of Definition 2.8.

Definition 2.14. Set Ω⁰_A =A=T⁰. For all integers p≥0, define inductively Ω^p+1_A as the minimal leftA-submodule ofT^p+1containingdΩ^p_A, that is, Ω^p+1_A consists of all the elements of the form (2.7) for someak ∈ Aandvk ∈Ω^p_A.

Clearly, forp= 1 Definition 2.14 is consistent with previous Definition 2.2.

Theorem 2.15. For allp, q≥0

(2.13) u∈Ω^p_A, v∈Ω^q_A ⇒ u•v∈Ω^p+q_A . Consequently, the direct sum ΩA=L

p=0,1,...Ω^p_A, with the multiplication • and with differential dis a differential calculus on A.

Applying (2.13) with q= 0, we obtain that Ω^p_Ais also a rightA-module, that is, Ω^p_A is anA-bimodule.

Proof. The proof is by induction onp. Forp= 0 the statement is trivial, as by definition Ω^q_A is a left A-module. Let us make an inductive step fromp−1 top. It suffices to prove thatu•v∈Ω^p+q_A foru=a•dbwherea∈ Aandb∈Ω^p−1_A .We have by the associative law and by the Leibniz rule

u•v= (a•db)•v=a•((db)•v)

=a•[d(b•v) + (−1)^pb•dv]

=a•d(b•v) + (−1)^p(a•b)•dv.

By the inductive hypothesis we have b•v ∈ Ω^p+q−1_A whence d(b•v) ∈ Ω^p+q_A and a•d(b•v) ∈ Ω^p+q_A . Also, we have a•b ∈ Ω^p−1_A and dv ∈ Ω^q+1_A , whence by the inductive hypothesis (a•b)•dv∈Ω^p+q_A .It follows thatu•v∈Ω^p+q_A .

Finally, (ΩA, d) satisfies all the conditions of Definition 2.8 by Propositions 2.11, 2.13, Definition 2.14 and by (2.13). Hence, (ΩA, d) is a differential calculus onA.

Now let us describe a different construction of the differential calculus onAthat is based on the first order differential calculus Ω¹_Afrom Definition 2.2. Define for each p≥0 aA-bimoduleΩe^p_Aby

(2.14) Ωe⁰_A=A and Ωe^p_A= Ω¹_A⊗AΩ¹_A⊗A...⊗AΩ¹_A

| {z }

pfactors

forp≥1.

In particular, Ωe¹_A = Ω¹_A.Clearly, each Ωe^p_A is also a K-module. Define the following multiplication⋆ between the elementsu∈Ωe^p_Aandv∈Ωe^q_A:

(2.15) u ⋆ v=

u·v, ifp= 0 orq= 0 u⊗Av, ifp, q≥1,

where· denotes the multiplication inΩe^k_A by the elements ofA that comes from the A-bimodule structure ofΩe^k_A. Clearly, multiplication⋆ is associative, has a unity 1A, and makes the direct sum

ΩeA= M

p=0,1,...

Ωe^p_A

into a graded K-algebra. It turns out that the graded algebras ΩeA and ΩA (cf.

Definition 2.14) are isomorphic as is stated below.

Theorem 2.16. (i)There exists a unique isomorphism f:ΩeA→ΩA of graded K-algebras given byA-bimodule isomorphisms

fp:Ωe^p_A→Ω^p_A, p≥0, wheref0:A → A andf1: Ω¹_A→Ωe¹_A are identical maps.

(ii)Define an operator de:Ωe^p_A→Ωe^p+1_A to make the following diagram commuta- tive:

(2.16)

Ωe^p_A −→^de Ωe^p+1_A

↓^fp ↓^fp+1 Ω^p_A −→^d Ω^p+1_A

Then (ΩeA,d)e is a differential calculus that is isomorphic to (ΩA, d).

Clearly, the operatorsdanddeonAare the same. As in the proof of Lemma 2.9 we obtain that any element of Ωe^p_A can be represented as a finite sum of the terms a0⋆dae1⋆ ... ⋆daep,and the following identity holds:

de

a0⋆dae1⋆ ... ⋆daep

=dae0⋆dae1⋆ ... ⋆daep.

Proof. We will use the following property of the tensor product: A ⊗AA ∼=A where∼= stands for aA-bimodule isomorphism given by the mapping [3], [20], [22]

(2.17) ϕ:A → A ⊗AA, ϕ(a) =a⊗1A.

In order to construct a necessary mappingf, define first aA-bimodule Te^pby Te⁰=A, Te^p= (A ⊗ A)⊗A(A ⊗ A)⊗A...⊗A(A ⊗ A)

| {z }

pfactorsA⊗A

, p≥1.

Since Ω¹_Ais a sub-module ofA ⊗ A, it follows thatΩe^p_A is a sub-module ofTe^p. Recall that Ω^p_Ais a sub-module ofT^pwhereT^pwas defined by (2.9). Let us show that, for allp≥0,

(2.18) Te^p∼=T^p.

Forp= 0 andp= 1 it is obvious as

Te⁰=A=T⁰ and Te¹=A ⊗ A=T¹.

If (2.18) is already proved for somep≥1 then the statement follows by the associative law of tensor product and the inductive hypothesis.

Denote byfp the mapping fromTe^p toT^p that provides the isomorphism (2.18).

For p = 0,1 the mappings fp are identity mappings. It follows from (2.17) and properties of the tensor product that forp≥2 and for

(2.19) u= (a1⊗b1)⊗A(a2⊗b2)⊗A...⊗A(ap⊗bp)∈Te^p

whereai, bi∈ A, we have

(2.20) fp(u) =a1⊗b1a2⊗b2a3⊗...⊗bp−1ap⊗bp∈T^p. Set

Te= Mp p=0,1,...

Te^p

and define the exterior multiplication⋆ in Te by (2.15), so thatTe becomes a graded K-algebra. Setf =⊕p≥0fpand show that the mappingf :Te→T is an isomorphism of the graded algebrasTeandT (cf. Definition 2.10). It suffices to verify that

(2.21) f(u ⋆ v) =f(u)•f(v)

for allu, v∈T .e Letu∈Te^pandv∈Te^q. Ifp= 0, that is,u∈ A, thenu ⋆ v=u·vand f(u ⋆ v) =f(u·v) =u·f(v) =f(u)•f(v).

The same argument works for q= 0. For p= 1 it suffices to prove Assume now that p≥1 andq≥1. It suffices to verify (2.21) foruas in (2.19) and for

v= (α1⊗β1)⊗A(α2⊗β2)⊗A· · · ⊗A(αq⊗βq) where αj, βj∈ A.Then by (2.15) and (2.20)

f(u ⋆ v) =a1⊗b1a2⊗...⊗bp−1ap⊗bpα1⊗β1α2⊗...⊗βq

whereas by (2.10)

f(u)•f(v) = (a1⊗b1a2⊗...⊗bp−1ap⊗bp)•(α1⊗β1α2⊗...⊗βq−1αq⊗βq)

=a1⊗b1a2⊗...⊗bp−1ap⊗bpα1⊗β1α2⊗...⊗βq, which proves (2.21).

Let us prove that the restriction offtoΩeAprovides an isomorphism of the graded algebrasΩeAand ΩA, that is,

f(eΩ^p_A) = Ω^p_A.

For p= 0,1 it is clear. Assume p≥ 2. By Lemma 2.9 any element of Ω^p_A can be written as a finite sum of the terms

w=v1•v2• · · · •vp

where vi∈Ω¹_A. For the element

v:=v1⋆ v2⋆ ... ⋆ vp∈Ωe^p_A we have by (2.21) andf|Ω¹_A = id

f(v) =f(v1)•f(v2)•...•f(vp) =v1•v2• · · · •vp=w,

which implies the inclusion

f(eΩ^p_A)⊃Ω^p_A.

Let us prove the opposite inclusion. By definition (2.14) ofΩe^p_A, any element ofΩe^p_Ais a finite sum of the termsv=v1⋆ v2⋆ ... ⋆ vp, where vi∈Ω¹_A. As above we have (2.22) f(v) =v1•v2•...•vp

that belongs to Ω^p_Aby Theorem 2.15, whencef(eΩ^p_A)⊂Ω^p_A.The last argument proves also the uniqueness of the isomorphism of the graded algebrasΩeA and ΩA. Indeed, sincef0 and f1 must be the identical maps, they are uniquely determined, and the uniqueness offp follows from (2.22).

Finally, the claim (ii) is a trivial consequence of (i).

Theorem 2.17. The differential calculus (ΩA, d) ∼= (ΩeA,d)e has the following universal property. For any other differential calculus(Λ, d^′)overA, there exists one and only one epimorphism p: ΩA→Λ of gradedA-algebras given by

p=M

k

pk, pk: Ω^k_A→Λk

withp0= idand such that, for all k≥0, the following diagram is commutative:

Ω^k_A −→^d Ω^k+1_A

↓^pk ↓^pk+1 Λ^{k d}

′

−→ Λ^k+1

Proof. Denote by∗ the exterior multiplication in Λ. By Lemma 2.9 any element w∈Λ^k withk≥1 can be written as a finite sum

(2.23) w=X

j

a^j₀∗da^j₁∗da^j₂∗ · · · ∗da^j_k

wherea^j_l ∈ Afor all 0≤l≤k. Consider a graded algebra Λ =e M

k=0,1,...

Λe^k,

whereΛe^k fork≤1 is defined by

Λe⁰=A, Λe¹= Λ¹, Λe^k =Λe¹⊗A· · · ⊗AΛe¹

| {z }

kfactors

for k≥2.

The exterior multiplication⋆inΛ is defined as in (2.15). The condition (2.23) impliese that the mapsp0 and p1 induce an epimorphismq:Λe −→Λ of the graded algebras, whereq=

L∞ k=0

qk andqk:Λe^k −→Λ^k are defined as follows: q0andq1 are the identity mappings, while fork≥2 the mappingqk is defined by

qk(w1⋆ w2⋆· · ·⋆ wk) =q1(w1)∗q1(w2)∗ · · · ∗q1(wk)∈Λ^k

for allwi∈Λe¹.Letf0= Id. By Theorems 2.4 and 2.16 we have a unique epimorphism f1=pofA-bimodules making the following diagram commutative:

(2.24)

A −→^de Ωe¹_A

↓^f0 ↓^f1

A ^d

′

−→ Λe¹

The diagram (2.24) induces an epimorphismf :ΩeA→Λ of graded algebras given bye f =

L∞ k=0

fk,where fork≥2 the mapping fk:Ωe¹_A⋆· · ·⋆Ωe¹_A

| {z }

kfactors

−→Λ|e¹⋆· · ·{z⋆Λe¹}

k factors

is defined by

fk(w1⋆· · ·⋆ wk) =f1(w1)⋆· · ·⋆ f1(wk)∈Λe^k

for allwi ∈ Ωe¹_A.Thus, we obtain an epimorphism p : ΩeA −→ Λ of graded algebras defined by

p= M

k=0,1,...

pk = M

k=0,1,...

qk◦fk: such thatp0= Id andp1=p.

To finish the proof of the theorem we must check the commutativity of the diagram

(2.25)

Ω^k_A −→^d Ω^k+1_A

↓^pk ↓^pk+1 Λ^{k d}

′

−→ Λ^k+1

for allk≥0. By Theorem 2.16 we can identify in the first line of (2.25) the graded algebra ΩAwithΩeAand dwithd. Let us prove by induction ine kthat this diagram is commutative. Fork = 0 this is true by Theorem 2.4. Inductive step from k−1 to k assuming k ≥ 1. It suffices to check the commutativity of (2.25) only on the elements w∈Ω^k_A of the form w=a•dv,where a∈ A and v ∈Ω^k−1_A . Since pis a homomorphism of graded algebras, the inductive hypothesis and d^′2= 0, we obtain

d^′pk(a•dv) =d^′(a ⋆ pk(dv)) =d^′(a ⋆ d^′pk−1(v)) =d^′a ⋆ d^′pk−1(v).

On the other side, using the Leibniz rule and the inductive hypothesis, we obtain pk+1d(a•dv) =pk+1(da•dv) =p1(da)⋆ pk(dv) =d^′a ⋆ d^′pk−1(v).

The comparison of the above two lines proves that the diagram (2.25) is commuta- tive.

Corollary 2.18. Under the hypotheses of Theorem 2.17, there exists a two-sided graded ideal

J = M

l=1,2,...

J^l, Jl⊂ΩA

of the graded algebra ΩAsuch that (2.26) Λ^k= Ω^kA

.J^k, ΩAJΩA⊆ J, dJ^k⊂ J^k+1 for all k≥0, andJ⁰={0}.

Furthermore, the following diagram is commutative:

(2.27)

0−→ 0 −→ 0 −→ 0 −→. . .

↓ ↓ ↓

0−→ 0 −→ J¹ −→^d J² −→^d . . .

↓ ↓ ↓

0−→ Ω⁰_A −→^d Ω¹_A −→^d Ω²_A −→^d . . .

↓^p0 ↓^p1 ↓^p2 0−→ Λ^{0 d}

′

−→ Λ^{1 d}

′

−→ Λ^{2 d}

′

−→. . .

↓ ↓ ↓

0−→ 0 −→ 0 −→ 0 −→. . .

where the mappingsJ^k →Ω^k_Aare the identical inclusions. In diagram (2.27) the rows are chain complexes ofK-modules, and the columns are exact sequences ofK-modules.

Proof. Indeed, defineJ^k = Ker{pk: Ω^k_A→Λ^k}.

Definition 2.19. The differential calculus (ΩA, d) is calledthe universal differ- ential calculus on the algebraA.

Proposition 2.20. Let (ΩA, d)be the universal differential calculus on the alge- bra AandJ ⊂ΩA be a graded ideal, that satisfies the property dJ ⊂ J. Denote by dJ the map of degree oneΩA/J →ΩA/J that is induced byd. Then(ΩA/J, dJ)is a differential calculus on the algebraA.

Proof. It is easy to check thatd²_J = 0 and dJ satisfies the Leibniz rule.

Corollary 2.21. Under assumptions of Corollary 2.18, we have the following cohomology long exact sequence:

0−→H⁰(ΩA)−→H⁰(Λ)−→H¹(J)−→H¹(ΩA)−→H¹(Λ)−→. . . Proof. This follows from the commutative diagram (2.27) by means of the stan- dard homology algebra [22].

Now we describe properties of quotient calculi that we need for constructing the functorial homology theory of digraphs.

Theorem 2.22. Let E^p ⊂Ω^p_A be a K-linear subspace for all p ≥1, such that E= L^∞

k=0

E^p is a graded ideal of the exterior algebra ΩA. Consider a subspace

J =M

p≥1

J^p⊂ΩA=M

p≥0

Ω^p_A, where J^p =

E^p, for p= 1 E^p+dE^p−1, for p≥2.

Then J ⊂ΩA is a graded ideal of algebra ΩA such that dJ ⊂ J. In particular, the inclusion J −→ΩA is a morphism of cochain complexes.

Proof. Any elementw∈ J^p can be represented in the form

(2.28) w=w1+w2

wherew1∈ E^p andw2=d(v), v∈ E^p−1. Forx∈Ωⁱ_A, y∈Ω^j_A we have xwy=xw1y+xw2y=xw1y+x(dv)y.

The element xw1y lies inE, since by our assumptionE is an ideal. Now, using the Leibniz rule, we have

d(xvy) = (dx)vy+ (−1)ⁱxd(vy) = (dx)vy+ (−1)ⁱx(dv)y+ (−1)ⁱ(−1)^p−1xv(dy), and hence

x(dv)y = (−1)ⁱ[d(xvy)−(dx)vy+ (−1)^i+pxv(dy)]

= (−1)ⁱd(xvy) + (−1)ⁱ⁺¹(dx)vy+ (−1)^pxv(dy).

In the last sum (−1)ⁱd(xvy)∈dΩ^i+j+p−1 and two others element lie in Ω^i+j+p, since Eis an ideal. Thus we proved thatJ is an ideal. For an elementwwith decomposition (2.28) we have

dw=dw1+dw2=dw1+d(dv) =dw1∈dE^p∈ J^p+1, which finishes the proof.

Corollary 2.23. Under assumptions of Theorem 2.22 we have a commutative diagram of cochain complexes

(2.29)

0−→ 0 −→ 0 −→ 0 −→. . .

↓ ↓ ↓

0−→ 0 −→ J¹ −→^d J² −→^d . . .

↓ ↓ ↓

0−→ Ω⁰_A −→^d Ω¹_A −→^d Ω²_A −→^d . . .

↓ ↓ ↓

0−→ Ω⁰_A ^d

′

−→ Ω¹_A/J^{1 d}

′

−→ Ω²_A/J^{2 d}

′

−→. . .

↓ ↓ ↓

0−→ 0 −→ 0 −→ 0 −→. . .

where the columns are exact sequences of K-modules and the differentials d^′ are in- duced by d. Commutative diagram (2.29)induces a cohomology long exact sequence

0−→H⁰(ΩA)−→H⁰(ΩA/J)−→H¹(J)−→H¹(ΩA)−→H¹(ΩA/J)−→. . . Corollary 2.24. Let for anyp≥1,E^p ⊂ F^p beK-linear subspaces of Ω^p_Asuch that E = ^∞⊕

p=1E^p andF = ^∞⊕

p=1F^p are graded ideals of the exterior algebra ΩA. Define J^p andI^p by

J^p=

E^p, for p= 1 E^p+d E^p−1

, for p≥2 , I^p =

F^p, for p= 1 F^p+d F^p−1

, for p≥2, , and set J = ^∞⊕

p=1J^p,I = ^∞⊕

p=1I^p. Then J^p ⊂ I^p ⊂Ω^p_A , which induces inclusions of cochain complexes

(2.30) J −→ I −→ΩA.

Proof. The inclusionsJ^p⊂ I^p⊂Ω^p_Acommute with differentials.

Corollary 2.25. Under assumptions of Corollary 2.24we have the following short exact sequence of cochain complexes of K-modules

(2.31) 0−→ I/J −→ΩA/J −→ΩA/I −→0

which can be written in the form of a commutative diagram ofK-modules

(2.32)

0−→ 0 −→ 0 −→ 0 −→. . .

↓ ↓ ↓

0−→ 0 −→ I¹/J¹ −→ I²/J² −→. . .

↓ ↓ ↓

0−→ Ω⁰_A −→ Ω¹_A/J¹ −→ Ω²_A/J² −→. . .

↓ ↓ ↓

0−→ Ω⁰_A −→ Ω¹_A/I¹ −→ Ω²_A/I² −→. . .

↓ ↓ ↓

0−→ 0 −→ 0 −→ 0 −→. . .

In (2.32) all columns are exact and rows are cochain complexes. All the differentials in (2.32) are induced by the differential d. The diagram (2.32) induces a cohomology long exact sequence

0−→H⁰(ΩA/J)−→H⁰(ΩA/I)−→H¹(I/J)−→H¹(ΩA/J)−→. . . Proof. The proof is standard, see [20, III,§1] and [22].

Now we discuss functorial properties of differential calculi (see, for example, [3], [20], [22]). Consider a categoryALGin which objects are associative unitalK-algebras and morphisms are homomorphisms ofK-algebras.

Definition 2.26. Define a categoryDC of differential calculi by the following way. An object ofDC is a differential calculus (ΛA, dA) on a unital associative algebra A(see Definition 2.8). A morphismλ: (ΛA, dA)−→(ΛB, dB) in the categoryDC is given by a degree preserving morphism of graded algebras

λ= M

i=0,1,...

λi: ΛA→ΛB, where λi: Λⁱ_A→Λⁱ_B, i≥0,

and λ0:A −→ B is a morphism in the categoryALG, and the maps λi (i ≥0) are homomorphisms ofK-modules which commutes with the differentials.

Let A and B be unital associative algebras over a commutative unital ring K and g: A → B be a homomorphism. Now we would like to define an induced by g morphism

λ= M

0,1,...

λi=U(g) : (ΩA, dA)−→(ΩB, dB)

of the universal differential calculus (ΩA, dA) to the universal differential calculus (ΩB, dB).

LetTA, TBbe graded algebras defined by algebrasA, andBas in Definition 2.10.

Let

φk:T_A^k →T_B^k, k≥0,

be a homomorphism ofK-modules (see [3, II,§3.2]) defined by φk(a0⊗a1⊗ · · · ⊗ak) =g(a0)⊗g(a1)⊗ · · · ⊗g(ak).

Denote by

φ= ^∞⊕

k=0φk:TA= ^∞⊕

k=0T_A^k −→TB= ^∞⊕

k=0T_B^k

a graded homomorphism of graded K-modules. The map φ is a degree preserving homomorphism of graded algebras, since

φk+l[(a0⊗a1⊗ · · · ⊗ak)•(b0⊗b1⊗ · · · ⊗bl)]

=g(a0)⊗g(a1)⊗ · · · ⊗g(akb0)⊗g(b1)⊗ · · · ⊗g(bl)

=g(a0)⊗g(a1)⊗ · · · ⊗g(ak)g(b0)⊗g(b1)⊗ · · · ⊗g(bl)

=φk(a0⊗a1⊗ · · · ⊗ak)•φl(b0⊗b1⊗ · · · ⊗bl . The mapsφk commutes with differentials, sinceg(1A) = 1B.

Letλi (i≥0) be the restrictionλi =φi|Ωⁱ_A: Ωⁱ_A−→T_Bⁱ,and setλ= ^∞⊕

i=0λi. Proposition 2.27. The homomorphism ofK-modulesλk is a morphism of dif- ferential calculi(ΩA, dA)−→(ΩB, dB).

Proof. We must check only that λk(Ω^k_A)⊂Ω^k_B. This follows from the fact that φk commutes with the differentials and from the inductive definition of Ω^k_A, Ω^k_Bas in Definition 2.14.

Theorem 2.28. We can assign to any associative unital K-algebra Aa univer- sal differential calculusU(A) = (ΩA, dA) and to homomorphism g:A → B of such algebras a morphism λ = U(g) : (ΩA, dA) −→ (ΩB, dB) of the universal differential calculi. Thus,U is a functor from the category of associative unitalK-algebras to the category of differential calculi.

Proof. Trivial checking.

Theorem 2.29. Let (Ω, d) be a differential calculus on an algebra A with an exterior multiplication•. The multiplication •induces a well-defined associative mul- tiplication

H^p(Ω)•H^q(Ω)−→H^p+q(Ω).

Proof. Letw, v∈Ω and dw= 0, dv= 0. Thend(wv) = 0 by Leibniz rule. Now, letw1=w+dx, v1=v+dy, where dw= 0 anddv= 0. Then we have

w1•v1= (w+dx)•(v+dy) =w•v+w•dy+ (dx)•v+ (dx)•(dy) =

=w•v+d(±w•y) +d(x•v) +d(x•d(y)) where we have used the Leibniz rule andd(x•dy) = (dx)•(dy).

Corollary 2.30. A homomorphism g:A → B of K-algebras induces a homo- morphism of cohomology ringsH^∗(ΩA)→H^∗(ΩB). This correspondence is functorial.

3. Differential calculus on finite sets. From now on let K be a field. We apply the general constructions of the previous sections to the algebraAof functions V →Kdefined on a finite setV ={0,1, . . . , n}. In construction of differential calculus on Awe follow [7] and [8]. The we describe functorial properties of the calculus in the form which will be helpful in the next sections.

The algebraAhas aK-basis

eⁱ:V →K, i= 0,1, . . . , n, where eⁱ(j) =δ_i^j:=

1_K, i=j

0, i6=j , 0≤i, j≤n, and the following relations are satisfied:

(3.1) eⁱe^j=δ_i^je^j, Xn i=0

eⁱ= 1A.

Denote by (Ω¹_V, d) the first order differential calculus (Ω¹_A, d) defined in Section 2 with the exterior multiplication•.

Theorem 3.1. [8] The K-module Ω¹_V has a basis

eⁱ⊗e^j where 0 ≤ i, j ≤ n, i6=j. The differential d:A →Ω¹_V on the basic elements eⁱ of A is given by the formula

(3.2) deⁱ= X

0≤j≤n, j6=i

(e^j⊗eⁱ−eⁱ⊗e^j).

Also, the following identity is satisfied:

(3.3) eⁱ•de^j=

eⁱ⊗e^j, i6=j

−P

k6=ieⁱ⊗e^k, i=j.

Proof. For 0≤i, j≤n, we have by (2.4)

µ(eⁱ⊗e^j) =eⁱe^j=δ_jⁱ.

Hence eⁱ⊗e^j ∈Ω¹_V fori6=j andeⁱ⊗eⁱ ∈/ Ω¹_V for 0≤i≤n. The finite dimensional K-moduleA ⊗ Ahas basis

eⁱ⊗e^j for 0≤i, j≤n(see [3, II§7.7 Remark]), whence the first statement follows.

The identities (3.2) and (3.3) are proved by direct computation using the definition ofdand relations (3.1).

Let Ω^k_V = Ω^k_A ⊂ T_A^k (k ≥0), and ΩV = ΩA be the graded algebra defined in Section 2 with the multiplication•. Let us introduce the following notation:

e^i0...ik:=eⁱ⁰⊗eⁱ¹⊗eⁱ²⊗ · · · ⊗e^ik

assuming thatim 6=im+1 for all 0≤m≤k−1. Clearly, e^i0...ik are the elements of Ω^k_V.

Theorem 3.2. [8] (i) The elements

e^i0...ik form aK-basis in Ω^k_V.

(ii) The exterior multiplication • of the basic elements is given by the following formula

e^i0...ik•e^j0...jl =

0, ik 6=j0

e^{i0...ikj1...jl}, ik =j0.

(iii) The differentialdis given on the basic elements by

de^i0...ik=

k+1X

m=0

X

j6=im−1,im

(−1)^me^{i0...im−1jim...ik}.

Proof. (i) The elementse^i0...ik withim6=im+1 for all 0≤m≤k−1 are linearly independent in theK-moduleT^k (see [3, II§7.7 Remark]). We must only prove that such elements lie in Ω^k_V ⊂T^k. By Theorem 2.16 we have an isomorphism of graded algebrasf:ΩeA→ΩV = ΩAwith an isomorphism ofK-modules

fk:Ωe^k_A→Ω^k_V, k≥0,

which is the identity isomorphism for k = 0,1. Hence the statement (i) is true for k = 0,1 by the definition ofA and by Theorem 3.1. Fork≥2, consider an element w = e^i0...ik ∈ T^k with im 6= im+1 for all 0 ≤ m ≤ k−1. Then the elements eⁱ⁰ⁱ¹, eⁱ¹ⁱ², . . . , e^ik−1ik lie in Ωe¹_Aand hence their⋆-product

ω=eⁱ⁰ⁱ¹⋆ eⁱ¹ⁱ²⋆· · ·⋆ e^ik−1ik

is contained in Ωe^k_A, and, hence,fk(ω)∈Ω^k_V. By the definition of fk we have fk eⁱ⁰ⁱ¹⋆ eⁱ¹ⁱ²⋆· · ·⋆ e^ik−1ik

=f1 eⁱ⁰ⁱ¹

•f1 eⁱ¹ⁱ²

• · · · •f1 e^ik−1ik

=f1 eⁱ⁰⊗eⁱ¹)•f1(eⁱ¹⊗eⁱ²)•...•f1(e^ik−1⊗e^ik

=eⁱ⁰⊗(eⁱ¹eⁱ¹)⊗... e^ik−1e^ik−1

⊗e^ik

=e^i0...ik so thatfk(ω)∈Ω^k_V.

(ii) This follows from the definition of multiplication • in Definition 2.10 and (3.1).

(iii) We prove this by induction onk. Fork= 0 it is proved in Theorem 3.1. For k= 1, leti6=j. We have using (3.1)

d eⁱ⊗e^j

= 1A⊗eⁱ⊗e^j−eⁱ⊗1A⊗e^j+eⁱ⊗e^j⊗1A

=X

k

(e^k⊗eⁱ⊗e^j−eⁱ⊗e^k⊗e^j+eⁱ⊗e^j⊗e^k)

= (eⁱ⊗eⁱ⊗e^j−eⁱ⊗eⁱ⊗e^j−eⁱ⊗e^j⊗e^j+eⁱ⊗e^j⊗e^j)

+X

k6=i

e^k⊗eⁱ⊗e^j− X

k6=i,k6=j

eⁱ⊗e^k⊗e^j+X

k6=j

eⁱ⊗e^j⊗e^k.

The sum in the brackets is equal to zero, and we obtain the result for k = 1. For k≥2 we have, using the Leibniz rule,

d eⁱ⁰⊗eⁱ¹⊗ · · · ⊗e^ik

=d (eⁱ⁰⊗eⁱ¹)•(eⁱ¹⊗eⁱ²⊗ · · · ⊗e^ik)

= (eⁱ⁰⊗eⁱ¹)•(eⁱ¹⊗eⁱ²⊗ · · · ⊗e^ik)−(eⁱ⁰⊗eⁱ¹)•d(eⁱ¹⊗eⁱ²⊗ · · · ⊗e^ik).

The result then follows by the inductive hypotheses and elementary transformations.