Cohomology Rings of Precubical Sets

COHOMOLOGY RINGS OF PRECUBICAL SETS Lopatkin V.E. Abstract The aim of this pap er is to define the structure of a ring on a graded cohomology group of a precubical set in co efficien ts in a ring with unit. Keyw ords: precubical cohomology rings, cohomology of small categories, precubical sets. In tro duction Let G b e the homologous system Ab elian groups ov er a precubical set X [1], then for an y integral n > 0, H n ( X ; G ) are defined by v alues of satellites of the colimit functor lim ← − n : Ab (  + /X ) op → Ab, here  + /X is a category of singular cub es of a precubical set X , Ab is the category of Ab elian groups and homomorphisms, further for any small category C w e denote by C op the opp osite category and finally Ab (  + /X ) op is the category of functors from (  + /X ) op to Ab. This observ ation is generalizing the Serre’s sp ectral sequence for precubical sets [1]. F or the cohomology groups there exist a opp osite statement. A cohomologous system o v er a precubical set we define as a functor on a category of singular cub es. In general, v alues of this functor on morphisms are not isomorphisms. Supp ose that the cohomologous system take constan t v alues which are any ring R then w e can to define a structure of a ring ov er a graded cohomology group with co efficien ts in this system. The aim of this paper is to define the structure of a graded ring ov er a graded cohomology group of precubical sets with co efficients in the cohomologous sys- tem witch is taken a constant v alue. The basic result of this pap er is Theorem 4.4. W e use following notations. The category of sets and maps we denote by Ens, Ab is the category of Ab elian groups and homomorphisms and Ring is the category of rings and ring’s homomorphisms whic h are sa v e the unit. 1 Precubical Sets Definition 1.1 A pr e cubic al set X = ( X n , ∂ n,ε i ) is a se quenc e of sets ( X n ) n ∈ N with a fimile of maps ∂ n,ε i : X n → X n − 1 , define d for i 6 i 6 n, ε ∈ { 0 , 1 } , for which the fol lowing diagr ams is c ommutative for al l α, β ∈ { 0 , 1 } , n > 2 , 1 6 i < j 6 n : Q n ∂ n,β j / / ∂ n,α i   Q n − 1 ∂ n − 1 ,α i   Q n − 1 ∂ n − 1 ,β j − 1 / / Q n − 2 1 Let  + b e a category consisting of finite sets I n = { 0 , 1 } ordered as the Cartesian p ow er of I . Any morphism of the  + is defined as an ascending map whic h admits a decomp osition of the form V k,ε i : I k − 1 → I k where V k,ε i ( u 1 , . . . , u k − 1 ) = ( u 1 , . . . , u i − 1 , ε, u i , . . . , u k − 1 ) , ε ∈ { 0 , 1 } , 0 6 i 6 k . here ε ∈ { 0 , 1 } , 1 6 i 6 k . Also we’ll denote maps V n,ε i b y V ε i . It w ell kno w [1] that any precubical set X is a functor X :  op + → Ens. Let H b e a ordered subset { h 1 , . . . , h p } of the set { 1 , 2 , . . . , n } . Let us define a map λ ε H : I p → I n b y the follo wing formula λ ε h ( u 1 , . . . , u p ) = ( v 1 , . . . , v n ) , where v i = ε , if i / ∈ H , and v h r = u r , r = 1 , . . . , p . Prop osition 1.1 Supp ose that we have a subset H = { h 1 , . . . , h p } of the set { 1 , 2 , . . . , n } . L et us define fol lowing sets; b H µ = { h 1 , . . . , h µ − 1 , h µ +1 , . . . , h p } , e H µ = { h 1 , . . . , h µ − 1 , h µ +1 − 1 , . . . , h p − 1 } . F urther, let H j b e a { h 1 , . . . , h r , h r +1 − 1 , . . . , h p − 1 } if j / ∈ H and h r < j < h r +1 . Ther e ar e fol lowing formulas for ε, η ∈ { 0 , 1 } λ η H ◦ V ε µ = V ε h µ ◦ λ η e H µ ; λ ε H ◦ V ε µ = λ ε b H µ ; λ ε H = V ε j ◦ λ ε H j . In this c ase, H j and e H µ ar e subsets of set { 1 , 2 , . . . , n − 1 } . Pro of This prop osition was pro ved in [2, Prop osition 9.3.4] 2 A Diagonal Inclusion In this section w e’ll introduce a diagonoal inclusion and show that this inclusion is the c hain map. First let us in truduce some notices from [1]. Let X = ( X n , ∂ n,ε i ) b e the presubical set, let  + [ X p ] = L ( X p ) for p > 0 b e free Ab elian group and  + [ X p ] = 0 for p < 0. Assume that D ε i = L ( ∂ ε i ) :  + [ X p ] →  + [ X p − 1 ]. F urther let us define homomorphisms D :  + [ X p ] →  + [ X p − 1 ] , p > 1 , b y the form ula D = p X i =1 ( − 1) i  D 1 i − D 0 i  . Let us assume that  + [ X ] = L p > 0  + [ X p ] be the direct sum of groups  + [ X p ]. 2 F ollo wing [1], identify cub es f ∈ X p with corresp onding natural transforma- tions e f : h I p → X which are called singular cub es . Th us, singular p -cub es are elemen ts of the group  + [ X p ]. Let us consider functor morphisms h λ ε H : h I p → h I n , it’s hard to see that the homomorphism D can define by the following corresp onding D ε : f 7→ f ◦ h V ε i . It is clear that the f ◦ h V ε i is define any face of the singular p -cub e. There are rules of commutation functor morphisms h λ ε H with the homomorphism D in the follo wing prop osition whic h is a mo dification of prop osition 1.1 Prop osition 2.1 L et us assume that we have a or der e d subset G = { g 1 , . . . , g p } of set { 1 , 2 , . . . , n } . Supp ose that b G µ = { g 1 , . . . , g µ − 1 , g µ +1 , . . . , g p } and e G µ = { g 1 , . . . , g µ − 1 , g µ +1 − 1 , . . . , g p − 1 } . F urhter, supp ose that G j = { g 1 , . . . , g r , g r +1 − 1 , . . . , g p − 1 } if j / ∈ G and g r < j < g r +1 . L et us assume that we have a pr e- cubic al set X = ( X n , ∂ n,ε i ) , let f : h I n → X b e a singular n -cub e. Ther e ar e fol lowing formulas for ε, η ∈ { 0 , 1 } : D ε µ  f ◦ h λ η G  = D ε g µ ( f ) ◦ h λ η ˜ G µ (1) D ε µ  f ◦ h λ ε G  = f ◦ h λ ε b G µ (2) f ◦ h λ ε G = D ε j ( f ) ◦ h λ ε G j (3) We assume d that G j and e G µ ar e or der e d subsets of set { 1 , 2 , . . . , n − 1 } . Pro of. F rom proposition 1.1 it follo ws that there are follo wing form ulas h λ η G ◦ h V ε µ = h V ε g µ ◦ h λ η e G µ ; h λ ε G ◦ h V ε µ = h λ ε b G µ ; h λ ε G = h λ G ε j ◦ h V ε j . Multiplying b oth sides by f , we complete the pro of (see the commutativ e 3 diagramm). X h I p − 1 h λ η e G µ " " D D D D D D D D D D D D D D D D D D ε g µ ( f ) ◦ h λ η e G µ = D ε µ ( f ) ◦ h λ η G = = h V ε µ / / h I p h λ η G A A A A A A A A A A A A A A A A f ◦ h λ η G [ [ h I n − 1 D ε g µ ( f ) Q Q h V ε g µ / / h I n f g g It well know (see [3]) that the tensor pro duct  + [ X ] ⊗  + [ X ] of the chain complex  + [ X ] with itself is the c hain complex  + [ X ⊗ X ], where  + [( X ⊗ X ) n ] = M p + q = n  + [ X p ] ⊗  + [ X q ] , (4) and b ound op erators is defined ov er generators x ⊗ x 0 b y the form ula ∂ ( x ⊗ x 0 ) = ∂ x ⊗ x 0 + ( − 1) dim x x ⊗ ∂ x 0 . (5) Prop osition 2.2 L et X ∈  op + Ens b e a pr e cubic al set, let us assume that  + [ X ] b e afor esaid chain c omplex. F urther let  + [ X ⊗ X ] b e the tensor pr o duct of the chain c omplex  + [ X ] with itself which define d by the formulas (4), (5). A map ∆ (diagonal inclusion) which define d by the formula for any singular cub f : h I n → X : ∆( f ) = X G % GK  f ◦ h λ 0 G  ⊗  f ◦ h λ 1 K  , is the chain map. Her e K is the c omplement of a set G = { g 1 , . . . , g p } ⊆ { 1 , 2 , . . . , n } , v ar r ho GK is a signatur e of a p ermutation GK of inte gr al numb ers 1 , 2 , . . . , n . The summation is taken over al l or der e d subsets G of set { 1 , 2 , . . . n } . Pro of This p oropsition was pro ved in [2, Prop osition 9.3.5] 4 3 Cohomology of Precubical Sets with Co effi- cien ts in a Cohomologous System of Rings Definition 3. 1 A c ohomolo gous system of rings and a c ohomolo gous system of Ab elian gr oups over a pr e cubic al set X ∈  op + Ens ar e some functors R :  + /X → Ring and G :  + /X → Ab , r esp e ctively. Let us consider Ab elian grpups n  + [ X, G ] = Q ϑ ∈ X n G ( ϑ ). Let us define differ- en tials δ n,ε i : n  + [ X, G ] → n +1  + [ X, G ] as homomorphisms making following diagrams comm utativ e Q ϑ ∈ X n G ( ϑ ) pr ϑ ◦ V n +1 ,ε i   δ n,ε i / / Q ϑ ∈ X n +1 G ( ϑ ) pr ϑ   G  ϑ ◦ V n +1 ,ε i  G ( V n +1 ,ε i : ϑV n +1 ,ε i → ϑ ) / / G ( ϑ ) Definition 3. 2 L et X b e a pr e cubic al set, let G :  + / X → Ab b e a c ohomolo- gous system of Ab elian gr oups over this pr e cubic al set X . A c ohomolo gy gr oups H n ( X ; G ) of this pr e cubic al set X with c o efficients in G ar e n -th c ohomolo gy gr oups of a chain c omplex ∗  + [ X, G ] c onsisting of ab elian gr oups n  + [ X, G ] = Y σ ∈ X n G ( σ ) and differ entials δ n = n +1 X i =1 ( − 1) i ( δ n, 1 i − δ n, 0 i ) . Supp ose that the cohomologous system of rings R :  + /X → Ring o v er a precubical set X take a constan t v alue which is a ring R with a unity . Consider- ing an additive comp onent of the ring R we can examine a cohomology groups H ∗ ( X ; R ) with co efficient in the ring R . Let ∗  + [ X ; R ] b e a co chain complex. F ollowing [2, § 5.7, 5.7.27] let us con- sider the homomorphism π : ∗  + [ X ; R ] ⊗ R ∗  + [ X ; R ] → ∗  + [ X ⊗ X ; R ] , whic h defined b y the form ula ( π ( u ⊗ u 0 )) ( c ⊗ c 0 ) = η ( u ( c ) ⊗ R u 0 ( c 0 )) , here c, c 0 ∈  + [ X ], u, u 0 ∈ ∗  + [ X ; R ] and η : R ⊗ R R → R is an isomorphism of rings wic h defined b y the following form ula η ( u ( c ) ⊗ u 0 ( c 0 )) = u ( c ) · u 0 ( c 0 ) , 5 this pro do ct is the multiplication op eration in the ring R . F rom [2, Proposition 5.7.28] follow that the homorphism π is the co chain map. Th us it’s not hard to see that a map ^ = ∆ ∗ π : ∗  + [ X ; R ] ⊗ R ∗  + [ X ; R ] → ∗  + [ X ; R ] is the co c hain map b ecause from prop osition 2.2 follo ws that the map ∆ ∗ is the co c hain map. It means that the ^ generate some a pro duct in H ∗ ( X ; R ). Thus w e ha ve the follo wing Theorem 3.1 The gr ade d gr oup H ∗ ( X ; R ) with afor e–mentione d ^ -pr o duct is a ring. Let us describ e the ^ -pro duct ov er co chains. Let ϕ ∈ p  + [ X ; R ] and ψ ∈ q  + [ X ; R ] are co chains. Let u ∈ X p + q b e a p + q -cub e. W e hav e a formula ( ϕ ^ ψ )( u ) = X G % GK ϕ  u ◦ h λ 0 G  · ψ  u ◦ h λ 1 K  , (6) Here G = { g 1 , . . . , g p } ⊆ { 1 , 2 , . . . , n } , % GK is a signature of a permutation GK of integral n umbers 1 , 2 , . . . , n . The summation is tak en ov er all ordered subsets G of set { 1 , 2 , . . . n } . The notices of form u ◦ h λ 0 G w e also denote b y uh λ 0 G . 4 Prop erties of the Precubical Cohomology Ring Here we will en umerate and we’ll pro of algebraic prop erties of the ^ -pro duct in the ring H ∗ ( X ; R ). Theorem 4.1 The ^ -pr o duct of c o chains in the ring ∗  + [ X ; R ] is asso ciative and distributive with r esp e ct to the addition. If the ring R has left (right, two– side d) unit then the ring ∗  + [ X ; R ] has same unit. Pro of. F rom asso ciativ e and distributive of the pro duct in the ring R follows asso ciativ e and distributive of pro duct in the ring ∗  + [ X ; R ]. F urther, let 1 — b e a left unit of the ring R and let ι b e a co chain which tak e each the 0-cub e of the  + [ X ] to 1. It’s not hard to see that for any cochain ξ there is the following equalit y ι ^ ξ = ξ . In the same wa y we’ll get the pro of of this theorem if 1 is righ t o r t wo–sided unit. The co c hain complex ∗  + [ X ; R ] with ^ -pro duct is a graded ring. Theorem 4.2 F or ϕ ∈ p  + [ X ; R ] ψ ∈ q  + [ X ; R ] ther e is the fol lowing for- mula δ ( ϕ ^ ψ ) = δ ϕ ^ ψ + ( − 1) p ϕ ^ δψ 6 Pro of. W e ha ve ( δ ϕ ^ ψ ) ( f ) = X G % GK ( δ ϕ )  f h λ 0 G  · ψ  f h λ 1 K  = = X G % GK p +1 X µ =1 ( − 1) µ   δ 1 µ ϕ   f h λ 0 G  −  δ 0 µ ϕ   f h λ 0 G  ! · ψ  f h λ 1 K  , ( ϕ ^ δψ ) ( f ) = X G % GK ( ϕ )  f h λ 0 G  · ( δ ψ )  f h λ 1 K  = = X G % GK ϕ  f h λ 0 G  · p + q +1 X η = p ( − 1) η   δ 1 η ψ   f h λ 1 K  −  δ 0 η ψ   f h λ 1 K  ! , here G ⊂ { 1 , 2 , . . . , p + q + 1 } , G = ( h 1 , . . . , h p +1 ) and K is the complement of the set G . F rom the diagram h I p + q +1 f   h I p +1 h λ ξ G o o f h λ ξ G y y y y y y y | | y y y y y y y y X h I p D ε µ „ f h λ ξ G « o o h V ε µ O O and prop osition 2.1 it follo ws that, w e ha v e ( δ ϕ ^ ψ ) ( f ) = X G % GK p +1 X µ =1 ( − 1) µ  ϕ  D 1 h µ  f h λ 0 e G µ  − ϕ  f h λ 0 b G µ  ! · ψ h f h λ 1 K i , ( ϕ ^ δψ ) ( f ) = X G % GK ϕ h f h λ 0 G i · p + q +1 X η = p ( − 1) η  ψ h f λ 1 b K η i − ψ  D 0 k η  f h λ 1 f K η  ! . Let ˇ K µ b e a complemen t of the set b G µ . Let us consider a sum ( δ ϕ ^ ψ ) ( f ) + ( − 1) p ( ϕ ^ δψ ) ( f ). It’s not hard to see that ϕ  f h λ 0 b G µ  · ψ h f h λ 1 K i will app ear t wice; in the first place it will appear as a result of a deletion the g µ from the G in the comp onent ( G, K ) and in the second place it will app ear as a result of a deletion the g µ from the ˇ K µ in the comp onen t ( b G µ , ˇ K µ ). In the first place ϕ  f h λ 0 b G µ  · ψ h f h λ 1 K i hase a sign % GK ( − 1) µ +1 , , further, in the second place it hase a sign % b G µ ˇ K µ ( − 1) p ( − 1) α , here k α < g µ < k α +1 . But we hav e % b G µ ˇ K µ = ( − 1) p − µ + α % GK , 7 it means that the ϕ  f h λ 0 b G µ  · ψ h f h λ 1 K i will app ear twice with different signs. So that w e ha ve ( δ ϕ ^ ψ ) ( f ) + ( − 1) p ( ϕ ^ δψ ) ( f ) = = X G % GK p +1 X µ =1 ( − 1) µ ϕ  D 1 g µ  f h λ 0 e G µ  · ψ h f h λ 1 K i + +( − 1) p p + q +1 X η = p ( − 1) η +1 ϕ h f h λ 0 G i · ψ  D 0 k η  f h λ 1 f K η  ! . (7) F rom other side w e ha ve ( δ ( ϕ ^ ψ )) ( f ) = p + q +1 X i =1 ( − 1) i  ( ϕ ^ ψ )  D 1 i f  − ( ϕ ^ ψ )  D 0 i f  = = p + q +1 X i =1 ( − 1) i X F % F T  ϕ h D 1 i ( f ) h λ 0 F i · ψ h D 1 i ( f ) h λ 1 T i − − ϕ h D 0 i ( f ) h λ 0 F i · ψ h D 0 i ( f ) h λ 1 T i , (8) here F is an ordered subset of the set { 1 , 2 , . . . , p + q } and T is its complement. Using (1) – (3) of prop osition 2.2, and assume that F = ( e G j ; j ∈ G G j ; j / ∈ G T = ( K j ; j ∈ G e K j ; j / ∈ G w e get a bijection b etw een triples ( F, T , i ) and ( G, K , j ) here i = j . It means that we hav e a bijection b etw een (7) and (8) up to the sign. Let us prov e that this signs are equal. W e must chec k the following equation ( − 1) µ % GK = ( − 1) h µ % e G µ K µ , ( − 1) η % GK = ( − 1) k η % G µ e K µ . Let us compare follo wings p erm utations GK : g 1 , . . . , g µ − 1 , g µ , . . . , g p , k 1 , . . . , k α , k α +1 , . . . , k q and e G µ K h µ : g 1 , . . . , g µ − 1 , g µ +1 − 1 , . . . , g p − 1 , k 1 , . . . , k α , k α +1 − 1 , . . . , k q − 1 , n. It’s not hard to see that follo wing p erm utations g µ , . . . , g p , k α +1 , . . . , k q and g µ +1 − 1 , . . . , g p − 1 , k α +1 − 1 , . . . , k q − 1 , n ha ve same signs, b ecause we can get from first to second p erm utation by tw o steps: in the first step, w e add 1 to all num b ers, so w e get g µ +1 , . . . , g p , k α +1 , . . . , k q , g µ , 8 and in the second step we transfer g µ in the b eginning. Eac h of this steps m ul- tiply the sing by ( − 1) n − g µ . It means that sings of the last p ermutation are differen ts with resp ect to the ( − 1) α . Here α is a num b er of k which are smaller than g µ , so that α = g µ − µ and we complete to pro of the first equation. In just the same w a y we can to pro of the second equation. Q.E.D. Let a co c hain complex is a graded ring with respect to any pro duct, then this co c hain complex is said [2] to b e a c o chain ring , if this pro duct satisfy theorem 4.2. F rom theorem 4.2 we get the following Corollary 4. 3 If ϕ and ψ ar e c o cycles, then ϕ ^ ψ is a c o cycle. Mor e over if ξ is a c ob oundary and ζ is a c o cycle then ξ ^ ζ is a c ob oundary. Pro of. Indeed, using theorem 4.2, w e get δ ( ϕ ^ ψ ) = δ ( ϕ ) ^ ψ + ( − 1) dim ϕ ϕ ^ ( δψ ) = 0 + 0 = 0 . Let us supp ose that ξ = δ ϑ and let ξ b e a cob oundary , further let ζ b e a co cycle, then δ ( ϑ ^ ζ ) = ( δ ϑ ) ^ ζ + ( − 1) dim ϑ ϑ ^ ( δ ζ ) = ξ ^ ζ . This completes the pro of of this Corollary . No w w e formulate the basic result of this pap er. Theorem 4.4 A set Z ( X ; R ) of c o cycles is a subring of the ring ∗  + [ X ; R ] ; a set B ( X ; R ) of c ob oundaries is a two–side d ide al in the ring Z ( X ; R ) . The c ohomolo gy ring H ∗ ( X ; R ) of the a pr e cubic al set X ∈  op + Ens is isomorphic to the quotient–ring Z ( X ; R ) /B ( X ; R ) . The ring H ∗ ( X ; R ) is a gr ade d ring. If the ring R has left (right, two–side d) unity, then the ring H ∗ ( X ; R ) has the same unity. Pro of. F rom Corollary 4.3 it follows that a set Z ( X ; R ) is a subring of the ring ∗  + [ X ; R ] and a set B ( X ; R ) is a tw o–sided ideal in the ring Z ( X ; R ). F urther, from Definition 3.2 we get a additive isomorphism H ∗ ( X ; R ) ∼ = Z ( X ; R ) /B ( X ; R ). Supp ose that f , g ∈ H ∗ ( X ; R ), let us consider their represen tatives [ f ] and [ g ] in Z ( X ; R ), resp ectiv ely . It’s not hard to see that using (6), w e hav e that a represen tative of f ^ g b e [ f ^ g ]. It’s evident that the ab ov e–cited co chain ι is a co cycle, this completes the pro of of this Theorem. Let us sho w that there is the follo wing Theorem 4.5 If the ring R is a c ommutative then the ring H ∗ ( X ; R ) is an antic ommutative. 9 Pro of. Since for any permutation GK of integral n umbers 1 , 2 , . . . , n there is the following equation % GK = % K G then we get for any ϕ ∈ p  + [ X ; R ], ψ ∈ q  + [ X ; R ] the following equation ϕ ^ ψ = ( − 1) pq ψ ^ ϕ. Q.E.D Example 4.1 L et us to c alculate the c ohomolo gy ring of the torus T 2 . We pr esent the torus T 2 as a pr e cubic al set T 2 =  Q n T 2 ; ∂ n,ε i  , se e the figur e 1. t A t B t C t D - ?  6 p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p α p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p β Figure 1: Here is shown the expanding of the torus; D A is identified with C B and AB is identified with D C . So we have Q 0 T 2 = { o = A = B = C = D } , Q 1 T 2 = { t 1 = D A = C B , t 2 = AB = D C } , Q 2 T 2 = { ϑ = AB C D } . In the figur e 2 ar e shown values which ar e taken b ound differ entials on the one and the two–dimension cub es. We have the fol lowing c o chain c omplex 0 → Z 1 δ 0 − → Z 2 δ 1 − → Z 1 δ 2 − → 0 L et us to assign k –dimension c o chain ϑ ∗ to e ach k -cub e ϑ of the pr e cubic al torus. This c o chain ϑ ∗ is taken 1 on the cub e ϑ and it is taken 0 on others cub es. We’l l c onsider c o chains which ar e the sum of c o chains of form ϑ ∗ . Sinc e the fol lowing diagr am is c ommutative f ∈ Q ϑ ∈ Q n T 2 Z pr ϑ ◦ V n +1 ,ε i   δ n,ε i / / ( δ n,ε i f ) ( ϑ )   f ( ϑV n +1 ,ε i ) . . Q ϑ ∈ Q n +1 T 2 Z pr ϑ   Z Z ( V n +1 ,ε i : ϑ ◦ V n +1 ,ε i → ϑ ) / / Z 10 then ther e exist the fol lowing e quation ( δ n,ε i f ) ( ϑ ) = f  ϑV n +1 ,ε i  . F r om this e quation it’s not har d to se e that the one–dimension c o chain f is taken differ ent sign values on two e dges of the b ound of the 2 -cub e (ac c or ding to the sign of the orientation of this 2 -sub e) then f is the c o cycle.(se e fig. ?? ). - ?  6 + ϑV 2 , 0 1 − ϑV 2 , 1 1 + ϑV 2 , 1 2 − ϑV 2 , 0 2 ϑ - q q t − tV 1 , 0 1 + tV 1 , 1 1 Figure 2: Here are shown the orientation of 2-cub e and v alues of b ound differ- en tials ϑV n,ε i = ∂ n,ε i ϑ . In figur e 1, we have sketchy shown b asic c o cycles on the torus: if the dotte d line is cr osse d any e dge of the cub e then the c o cycle take 1 on this e dge, and this c o cyle take 0 on others e dges. L et us c onsider the ^ -pr o duct of b asic c o cycles. Sinc e ϑ ◦ V 2 ,ε i = h λ ε { i } ◦ ϑ , we get (se e (6) and figur e 2.) ( α ^ β )( ϑ ) = α  ϑV 2 , 0 1  · β  ϑV 2 , 1 2  − α  ϑV 2 , 0 2  · β  ϑV 2 , 1 1  = 0 · 0 − ( − 1) · ( − 1) = − 1 . Thus, β ^ α — is a b asic c o cycle of H 2 ( T 2 ; Z ) . F urther ( β ^ α )( ϑ ) = β  ϑV 2 , 0 1  · α  ϑV 2 , 1 2  − β  ϑV 2 , 0 2  · α  ϑV 2 , 1 1  = 1 · 1 − 0 · 0 = 1 So, we se e that the c ohomolo gy ring H ∗ ( T 2 , Z ) c an b e identifie d with the exterior algebr a over the Z -mo dule Z whose gener ators ar e α and β . Concluding Remark So, let us to sum up. F or any precubical set X ∈  op + Ens and for any ring R we get a graded cohomomology ring H ∗ ( X ; R ). If the ring R has the unit then the ring H ∗ ( X ; R ) has the same unit. F urther, if the ring R is commutativ e then the ring H ∗ ( X ; R ) is anticomm utative. References [1] Husainov A. On the Cubical Homology Groups of F ree Partially Com- m utative Monoids // New Y ork: Cornell Univ, Preprin t, 2006. 47 pp. h 11 [2] P .J. Hilton, S. Wylie, ”Homology theory . An in tro duction to algebraic top ology” , Cam bridge Univ. Press (1960) [3] S. MacLane. ”Homology”, New Y ork, Academic Press, 1963 12