Distributive lattices and cohomology

DISTRIBUTIV E LA TTICES AND COHOMOLOGY. TOMASZ MASZCZYK † Abstract. A reso lutio n of the intersection of a finite num ber o f subgroups of an abelian group b y means of their sums is c on- structed, provided the lattice generated by these subgroups is dis- tributive. This is used fo r de tec ting singularities of mo dules over Dedekind rings. A generalized Chines e remainder theorem is de- rived as a consequence of the abov e resolutio n. The Gelfand- Naimark dualit y betw een finite closed cov erings of compact Ha us- dorff spaces and the generalized Chinese remainder theor em is clar - ified. 1. Intro duction The Gelfand-Naimark duality iden tifies lattices of closed subsets in compact Hausdorff spaces with lattices opp osite to surjectiv e systems of quotien ts of unital comm utat iv e C*-algebras. Therefore, giv en a finite collection I 0 , . . . , I n of closed *-ideals in a C*-algebra A = C ( X ) of con tinuous functions o n a compact H ausdorff space X , it identifies co equalizers in the catego ry of compact Hausdorff spaces ( V ( I ) ⊂ X is the zero lo cus of the ideal I ⊂ A = C ( X )) n [ α =0 V ( I α ) ← n a α =0 V ( I α ) ⇔ n a α,β =0 V ( I α ) ∩ V ( I β ) (1) with equalizers in the category of unital comm utative C*- algebras A/ n \ α =0 I α → n Y α =0 A/I α ⇒ n Y α,β =0 A/I α + I β . (2) In particular, finite families of closed *-ideals intersec ting to zero cor- resp ond to finite families of closed subsets co v ering X . In general, lat- tices of closed *-ideals in comm utativ e unital C*-algebras are alw ay s distributiv e, since they are isomorphic b y the Gelfand-Naimark dualit y to lattices opp osite to sublattices of subsets. Therefore one can think ab out finite families o f closed subsets in a compact Hausdorff space as † The author was partially suppo rted by KBN grants N201 1770 33 and 1 1 5/E- 343/SP B/6.PR UE/DIE 50/20 05-2 0 08. Mathematics Subje ct Classific ation (2000): 06 D99, 46L52 , 1 3 D07, 13F05 , 16E60. 1 2 TOMASZ M ASZCZYK of finite subsets in a distributive lattice of ideals. By Hilb ert’s Null- stellensatz one can replace a compact Hausdorff space and its closed subsets by an affine alg ebraic set X o v er an algebraically closed field and its algebraic subsets on one hand, and closed ideals in a C*-algebra b y radical ideals in the algebra O [ X ] of p olynomial functions on X , on the other hand. One can tak e also a finite set of monomial ideals in a ring of p olynomials ov er a field [5 ] as we ll as a finite set of congru- ences in the ring of in tegers and the family of corresp onding ideals. In all a b o v e cases the fact that the diag ram (2 ) is an equalizer is a conse- quence of distributivit y of a corresp onding lattice of ideals, and in view of the la st example can b e regarded as a generalized Chinese remain- der theorem. More examples can b e obtained from the fact that ev ery algebra o f finite represen tation t yp e ha s distributiv e lattice of ideals [12] and the pro p ert y of having distributiv e lattice o f ideals is Morita in v ariant and op en under deformations of finite dimensional algebras [14]. The aim of the prese n t pap er is to sho w that the generalized Chinese theorem is a consequenc e of v anishing of first cohomolo gy of a canon- ical complex asso ciated with a finite num b er o f mem b ers I 0 , . . . , I n of a distributiv e lattice L of subgroups of a n ab elian group A . The re- sp ectiv e v anishing theorem (Theorem 1) dep ends only on that lattice. Since it is indep enden t of t he am bien t ab elian gr o up A , Theorem 1 is prior to the generalized Chinese remainder theorem. It is also more general, since it claims v anishing of the whole higher cohomology . This can b e used for computing some higher Ext’s detecting singularity of mo dules o ver D edekind rings (Corollary 1). 2. Distributive la t tices and homological algebra In this section w e consider lattices of subgroups of a giv en ab elian group with the in tersection and the sum as the me et and the join op er- ations, respectiv ely . As fo r a general lattice the dis tributivit y conditio n can b e written in t w o equiv a len t forms: • The sum distributes o v er the in tersection P 0 ∩ ( P 1 + P 2 ) = ( P 0 ∩ P 1 ) + ( P 0 ∩ P 2 ) . (3) • The in tersection distributes o v er the sum I 0 + I 1 ∩ I 2 = ( I 0 + I 1 ) ∩ ( I 0 + I 2 ) . (4) The aim of this section is to explain homolo gic al na t ur e of the first and c ohomolo gic al nature o f the second form of distributivit y . Homolo g ical c hara cterization of distributiv it y w as use d b y Zharino v in [18] to gener- alize famous edge-of - the-w edge theorem of Bogolyub ov. In the presen t DISTRIBUTIVE LA TTICES AND COHOMOLOGY. 3 pap er w e prov e a cohomological c ha r acterization of distributivit y a nd deriv e from it a generalized Chinese remainder theorem. In this sec- tion w e show also t ha t b oth characterizations hav e consequences f or arithmetic. 2.1. Homology. Let P 0 , . . . , P n b e a finite f a mily of mem b ers of some fixed lattice L of subgroups in an ab elian g r o up A . W e define a group of q -c hains C q ( P 0 , . . . , P n ) as a quotien t of the direct sum M 0 ≤ α 0 ,...,α q ≤ n P α 0 ∩ . . . ∩ P α q (5) b y a subgroup generated b y elemen ts p α 0 ,...,α ′ ,...,α ′′ ,...,α q + p α 0 ,...,α ′′ ,...,α ′ ,...,α q , p α 0 ,...,α,...,α,...,α q , (6) and b oundary op erato rs (in terms of represen tativ es of elemen ts of quotien t groups) ∂ : C q ( P 0 , . . . , P n ) → C q − 1 ( P 0 , . . . , P n ) , ( ∂ p ) α 0 ...α q − 1 = X α q p α 0 ...α q − 1 α q . (7) By a standar d argumen t from homological algebra ∂ ◦ ∂ = 0. W e de note b y H • ( P 0 , . . . , P n ) the ho mology of the complex (C • ( P 0 , . . . , P n ) , ∂ ). Theorem 1. [18 ] 1) H 0 ( P 0 , . . . , P n ) = P 0 + · · · + P n , 2) If the lattic e L is distributive then H q ( P 0 , . . . , P n ) = 0 for q > 0 , 3) If H 1 ( P 0 , P 1 , P 2 ) = 0 for al l P 0 , P 1 , P 2 ∈ L then the lattic e L is distributive. The follo wing corollary pro vides a homo lo gic al char acterization of distributivit y of a lattice L . Corollary 1. T h e fol lowing c onditions ar e e quivale nt. 1) L is distributive, 2) F or al l P 0 , . . . , P n ∈ L the c anonic al morph isms of c omplexes C • ( P 0 , . . . , P n ) → P 0 + . . . + P n (8) ar e quasiisomo rp hisms. In particular, since C q ( P 0 , . . . , P n ) can b e iden tified with the direct sum L 0 ≤ α 0 <...<α q ≤ n P α 0 ∩ . . . ∩ P α q , the ab ov e corollary pro vides a ho- molo gic al resolution of the sum of subgroups P 0 + . . . + P n b y means of their in tersections P α 0 ∩ . . . ∩ P α q , 0 ≤ α 0 < . . . < α q ≤ n , prov ided the lattice L is distributiv e. 4 TOMASZ M ASZCZYK In [1] authors in tro duce a class o f so called G *GCD rings, defined as suc h for whic h gcd( P 1 , P 2 ) exists for all finitely generated pro jectiv e ideals P 1 , P 2 . This c lass includes GG CD rings, se mihereditary rings , f.f. rings (and hence flat rings), v on Neumann regular rings, arithmetical rings, Pr ¨ ufer domains and GGCD domains. F o r ev ery such a ring the class of finitely generated pro jectiv e ideals is closed under intersec tion [1]. Therefore, for an arithmetical ring R ev ery sum P 0 + . . . + P n of finitely g enerated pro jectiv e ideals in A admits a canonical resolution (8) by finitely g enerated pro jectiv e mo dules, whic h implies the f ollo w- ing corolla ry . Corollary 2. L et P 0 , . . . , P n b e finitely g ener ate d pr oje ctive ide a l s in an arithmetic al ring R . Then Ext q R ( P 0 + . . . + P n , − ) = H q (Hom R (C • ( P 0 , . . . , P n ) , − )) , (9) T or R q ( P 0 + . . . + P n , − ) = H q (C • ( P 0 , . . . , P n ) ⊗ R − ) . (10) 2.2. Cohomology. Let I 0 , . . . , I n b e a finite family of mem b ers of some fixed lattice L of subgroups in an ab elian group A . W e define a group o f q -co c hains C q ( I 0 , . . . , I n ) as a subgroup of the pro duc t Y 0 ≤ α 0 ,...,α q ≤ n I α 0 + . . . + I α q (11) consisting of sequences ( i α 0 ...α q ) whic h are completely alternating with resp ect to indices α 0 , . . . , α q , i.e. i α 0 ,...,α ′ ,...,α ′′ ,...,α q + i α 0 ,...,α ′′ ,...,α ′ ,...,α q = 0 , i α 0 ,...,α,...,α,...,α q = 0 , (12) and cob o undary op erators d : C q ( I 0 , . . . , I n ) → C q +1 ( I 0 , . . . , I n ) , ( di ) α 0 ...α q +1 = q +1 X p =0 ( − 1) p i α 0 ... c α p ...α q +1 . (13) By a standard argumen t fro m homological a lgebra d ◦ d = 0. W e denote b y H • ( I 0 , . . . , I n ) the cohomology of the complex (C • ( I 0 , . . . , I n ) , d ). Theorem 2. 1) H 0 ( I 0 , . . . , I n ) = I 0 ∩ . . . ∩ I n , 2) If the lattic e L is distributive then H q ( I 0 , . . . , I n ) = 0 for q > 0 , 3) I f H 1 ( I 0 , I 1 , I 2 ) = 0 for al l I 0 , I 1 , I 2 ∈ L then the lattic e L is distributive. Pr o of. Let us no te first that C q ( I 0 , . . . , I n ) can b e iden tified with the pro duct Q 0 ≤ α 0 <...<α q ≤ n I α 0 + · · · + I α q . DISTRIBUTIVE LA TTICES AND COHOMOLOGY. 5 1) Since the difference i β − i α is a lternating with respect t o t he indices α, β w e ha ve H 0 ( I 0 , . . . , I n ) = k er( Y 0 ≤ α ≤ n I α → Y 0 ≤ α<β ≤ n I α + I β ) , ( i α ) 7→ ( i β − i α ) whic h consists of constan t sequenc es ( i α = i | i ∈ I 0 ∩ . . . ∩ I n ). 2) F or q > 0 induction on n . F or n = 0 o b vious. Inductiv e s tep: Con- sider ( I 0 , . . . , I n ) for n > 0. Then ev ery q -co c hain ( i α 0 ...α q , i α 0 ...α q − 1 n ), for q > 0, can b e iden t ified with a sequence consisting of elemen ts i α 0 ...α q ∈ I α 0 + . . . + I α q , for 0 ≤ α 0 < . . . < α q ≤ n − 1 , (14) i α 0 ...α q − 1 n ∈ I α 0 + . . . + I α q − 1 + I n , for 0 ≤ α 0 < . . . < α q − 1 ≤ n − 1 . This is a co cycle iff q +1 X p =0 ( − 1) p i α 0 ... c α p ...α q +1 = 0 , (15) for a ll 0 ≤ α 0 < . . . < α q +1 ≤ n − 1, and q X p =0 ( − 1) p i α 0 ... c α p ...α q n + ( − 1) q +1 i α 0 ...α q = 0 . (16) for a ll 0 ≤ α 0 < . . . < α q ≤ n − 1. By t he inductiv e h yp othesis H q ( I 0 , . . . , I n − 1 ) = 0 for q > 0. Then (15) implies that for all 0 ≤ α 0 < . . . < α q − 1 ≤ n − 1 there exist i α 0 ...α q − 1 ∈ I α 0 + . . . + I α q − 1 , suc h that for all 0 ≤ α 0 < . . . < α q ≤ n − 1 i α 0 ...α q = q X p =0 ( − 1) p i α 0 ... c α p ...α q , (17) hence b y (16) q X p =0 ( − 1) p i α 0 ... c α p ...α q n + q X p =0 ( − 1) p + q +1 i α 0 ... c α p ...α q = 0 , (18) whic h can b e rewritten as q X p =0 ( − 1) p ( i α 0 ... c α p ...α q n − ( − 1) q i α 0 ... c α p ...α q ) = 0 . (19) F or q = 1 w e know b y already pro v en point 2) of the theorem that H q − 1 ( I 0 + I n , . . . , I n − 1 + I n ) = ( I 0 + I n ) ∩ . . . ∩ ( I n − 1 + I n ) whic h is equal to I 0 ∩ . . . ∩ I n − 1 + I n , since the lattice L is distributiv e. Therefore b y 6 TOMASZ M ASZCZYK (19) fo r q = 1 there exist i ∈ I 0 ∩ . . . ∩ I n − 1 and i n ∈ I n suc h that for all 0 ≤ α ≤ n − 1 i αn + i α = i + i n , (20) hence i αn = i n − ( i α − i ) . (21) Equations (1 7 ) for q = 1, whic h reads as i αβ = i β − i α = ( i β − i ) − ( i α − i ) , (22) and (21) together mean that i = ( i αβ , i αn | 0 ≤ α < β ≤ n − 1) ∈ C 1 ( I 0 , . . . , I n ) is cob oundary of ( i α − i, i n | 0 ≤ α ≤ n − 1) ∈ C 0 ( I 0 , . . . , I n ) whic h prov es that H 1 ( I 0 , . . . , I n ) = 0. F or q > 1 by the inductiv e h yp o thesis H q − 1 ( I 0 + I n , . . . , I n − 1 + I n ) = 0, hence (19) implies that for all 0 ≤ α 0 < . . . < α q − 2 ≤ n − 1 there exist i α 0 ...α q − 2 n ∈ ( I α 0 + I n ) + . . . + ( I α q − 2 + I n ) = I α 0 + . . . + I α q − 2 + I n , suc h that fo r all 0 ≤ α 0 < . . . < α q − 1 ≤ n − 1 i α 0 ...α q − 1 n − ( − 1) q i α 0 ...α q − 1 = q − 1 X p =0 ( − 1) p i α 0 ... c α p ...α q − 1 n , (23) whic h can b e rewritten as i α 0 ...α q − 1 n = q − 1 X p =0 ( − 1) p i α 0 ... c α p ...α q − 1 n + ( − 1) q i α 0 ...α q − 1 . (24) Equations (17) and (24) to gether mean that ( i α 0 ...α q , i α 0 ...α q − 1 n ) is cob ound- ary of ( i α 0 ...α q − 1 , i α 0 ...α q − 2 n ), hence H q ( I 0 , . . . , I n ) = 0 for q > 1. 3) W e ha v e to pro ve that for all I 0 , I 1 , I 2 ∈ L ( I 0 + I 1 ) ∩ ( I 0 + I 2 ) = I 0 + I 1 ∩ I 2 . (25) The inclusion ( I 0 + I 1 ) ∩ ( I 0 + I 2 ) ⊃ I 0 + I 1 ∩ I 2 is obvious . T o prov e the opp osite inclusion ta k e i ∈ ( I 0 + I 1 ) ∩ ( I 0 + I 2 ). It can b e written in tw o w a ys a s i = i 01 + i ′ 12 , where i 01 ∈ I 0 ⊂ I 0 + I 1 , i ′ 12 ∈ I 1 ⊂ I 1 + I 2 , (26) i = i 02 + i ′′ 12 , where i 02 ∈ I 0 ⊂ I 0 + I 2 , i ′′ 12 ∈ I 2 ⊂ I 1 + I 2 . (27) Define i 12 := i ′ 12 − i ′′ 12 . Subtra cting (27) from (26 ) w e get the co cycle condition i 12 − i 02 + i 01 = 0 . (28) DISTRIBUTIVE LA TTICES AND COHOMOLOGY. 7 Since H 1 ( I 0 , I 1 , I 2 ) = 0 (28 ) implies that there ex ist i α ∈ I α , α = 0 , 1 , 2, suc h that i αβ = i β − i α , (29) in particular i ′ 12 − i ′′ 12 = i 12 = i 2 − i 1 , (30) hence i 1 + i ′ 12 = i 2 + i ′′ 12 . (31) Since i 0 ∈ I 0 and by (26) i 01 ∈ I 0 (29) implies tha t i 1 = i 0 + i 01 ∈ I 0 . (32) By (26) (resp. (27)) the left (resp. right) hand side of (31) b elong s to I 1 (resp. I 2 ), hence i 1 + i ′ 12 ∈ I 1 ∩ I 2 . (33) Finally , b y (2 6), (3 2) and (33) i = i 01 + i ′ 12 = ( i 01 − i 1 ) + ( i 1 + i ′ 12 ) ∈ I 0 + I 1 ∩ I 2 .  (34) The following corollary pro vides a c ohomolo gic al char acterization of distributivit y of a lattice L . Corollary 3. T h e fol lowing c onditions ar e e quivale nt. 1) L is distributive, 2) F or al l I 0 , . . . , I n ∈ L the c anonic al morphisms of c omplexe s I 0 ∩ . . . ∩ I n → C • ( I 0 , . . . , I n ) , (35) ar e quasiisomo rp hisms. In particular, since C q ( I 0 , . . . , I n ) can b e iden tified with the pro duct Q 0 ≤ α 0 <...<α q ≤ n I α 0 + · · · + I α q , the ab ov e corollary provides a c ohomo- lo gic al resolution of the in t ersection of subgroups I 0 ∩ . . . ∩ I n b y means of t heir sums I α 0 + · · · + I α q , 0 ≤ α 0 < . . . < α q ≤ n , provid ed the lattice L is distributiv e. This fact together with the fact that eac h finitely generated mo d- ule ov er a Dedekind ring is a direct sum of distributive mo dules (i.e. mo dules whose lattice of submo dules is distributiv e) [15] can b e used for detecting singularities of mo dules o v er Dedekind rings. Fir st of all, in a non-singular left mo dule (i.e. left mo dule without nonzero elemen ts annihilated b y all essen tia l left ideals) the in tersection of in- jectiv e submo dule s is again injectiv e [17 ]. T herefore, giv en injectiv e submo dules I 0 , . . . , I n in a left mo dule A ov er a ring R , the functors 8 TOMASZ M ASZCZYK Ext q R ( − , I 0 ∩ . . . ∩ I n ) for q > 0 detect singularity of A . These func- tors can b e computed with use o f our resolution whenev er ev ery sum of injectiv e submo dules of a left R -mo dule A is injectiv e. The la t t er prop erty c haracterizes left No ethe rian left hereditary rings [9], hence it holds for Dedekind rings. Therefore w e can apply Theorem 2 to obtain the follo wing corollary . Corollary 4. L et I 0 , . . . , I n b e in je ctive submo dules in a distributive left R -mo dule A over a left No etherian and left her e ditary ring R . Then Ext q R ( − , I 0 ∩ . . . ∩ I n ) = H q (Hom R ( − , C • ( I 0 , . . . , I n ))) . (36) Ther efor e, if A is a fin itely gener ate d and non-singular mo dule over a De dekind ring R H q (Hom R ( − , C • ( I 0 , . . . , I n ))) = 0 (37) for q > 0 . 3. Generalized Chinese Remainder Theorem As next application we will prov e t he follow ing generalized Chinese remainder theorem. Corollary 5. F or any finite family I 0 , . . . , I n of memb ers of some fixe d distributive lattic e L of sub gr oups in an ab elian gr oup A the c ano n ic al diagr am A/ n \ α =0 I α → n Y α =0 A/I α ⇒ n Y α,β =0 A/I α + I β . (38) is an e qualizer. Pr o of. Injectivit y of the first arrow is ob vious. Exactness of (3 8) in the middle term is equiv alen t to exactness in the middle term of the canonical complex A π → Y 0 ≤ α ≤ n A/I α δ → Y 0 ≤ α<β ≤ n A/I α + I β , (39) where π ( a ) = ( a + I α | 0 ≤ α ≤ n ), δ ( a α + I α | 0 ≤ α ≤ n ) = ( a β − a α + I α + I β | 0 ≤ α < β ≤ n ). W e ha v e k er δ = ( a α + I α | a β − a α ∈ I α + I β ) , (40) hence ( i αβ := a β − a α | 0 ≤ α < β ≤ n ) is a co cycle in C 1 ( I 0 , . . . , I n ). By Theorem 1 there exist i α ∈ I α suc h that a β − a α = i β − i α . Let a := a α − i α = a β − i β . Then ( a α + I α | 0 ≤ α ≤ n ) = π ( a ), whic h pro v es exactness of (39) in the middle term.  DISTRIBUTIVE LA TTICES AND COHOMOLOGY. 9 Remark. It is w ell kno wn t ha t if all I α ’s are pa ir wise coprime ideals in a unital asso ciative ring A , i.e. I α + I β = A for α 6 = β , then the dia- gram (38) is an eq ualizer and I 1 ∩ . . . ∩ I n = P σ I σ (1) . . . I σ ( n ) , where σ ’s are sufficien t ly many p erm utations of { 1 , . . . , n } . These facts are inde- p enden t of distributivity of the lat tice of ideals. Therefore Corollary 5 (essen tialy presen t already in [10], next redisco v ered and generalized man y t imes, e.g. [4, 3, 16, 5]) should b e unde rsto o d as a generalization of the Chinese r emainder theorem to the non-coprime case, for whic h distributivit y of the la t t ice of ideals is a sufficien t condition. In fact, the lattice of left ideals in a (unital asso ciativ e) ring is distributive iff the ab o v e generalized Chinese remainder theorem holds for suc h ideals [3]. Therefore in t he commutativ e case there is “one ne c ess a ry an d sufficient c ondition that plac es the the or em in pr op er p ersp e ctive. It states that the C h inese r emaind er the or em holds in a c om mutative ring if and only if the lattic e of ide als of the ring is distributive” [13 ]. The aim of this section w as to sho w ho w lattice theory communic ates with mo dular arithmetic through homology theory . 4. Noncommut a tive Topology Finite families of closed subsets co vering a top ological space are im- p ortant for the Ma y er-Vietoris principle in sheaf cohomology with sup- p orts and to p ological K-theory . Since b y the Gelfand-Naimark dualit y gluing of a compact Hausdorff space X from finite num b er of com- pact Hausdorff pieces is equiv alen t to a generalized Chinese remainder theorem (2) fo r closed *-ideals in a comm utativ e unital C*-algebra C ( X ), one is tempted to define a “nonc ommutative close d c overing of a nonc om m utative sp ac e d ual to an asso ciative C*-algebr a A ” as a fi- nite collection o f closed * -ideals in tersecting to zero and generating a distributiv e latt ice [6], [8]. In [8] authors f o cus on the combinatorial side o f suc h gluing in terms of the p oset structure on X induced by suc h a co vering. This p oset structure has its own top ology (Alexandro v top olo gy), drastically dif- feren t from the original compact H ausdorff one. After fi xing an or der of the finite closed cov ering, they represen t the distributive lattice gener- ated b y these originally closed (now Alex andro v o p en) subsets as a ho - momorphic image of the free distributive lattice generated b y the same finite set of generators. Next, authors pull-ba c k quotient C*-algebras A/I to that free la ttice and view the resulting surjectiv e system of quotien t algebras as flabby she a f of C*- algebras on the Alexandro v top ology corresp onding to that free lattice. Fina lly , they form ulate the 10 TOMASZ M ASZCZYK Gelfand-Naimark duality b etw een ordered co v erings of compact Haus- dorff spaces b y N closed sets and flabb y shea v es of comm utativ e unital C*-algebras o n the Alexandrov top ology corresp onding to the free dis- tributiv e lattice with N generators. The aim of the presen t c hapter is to av oid the auxiliary Alexandro v top ology . In fact, creating new to p ology by declaring old clos ed subsets to be new o p ens is not necess ary . The reason is that there is no need to see the generalized Chinese remainder theorem a s the sheaf condition. The following definitions in tro duce a not io n, whic h replaces shea v es when finite closed co v erings replace op en cov erings. Definition. F or any top olog ical space X w e define a categor y of func- tors P (w e call them p atterns ) from the lattice of close d subsets of X to the category of sets (ab elian g roups, rings, algebras etc) satisfying t he follo wing unique gluing pr op erty with resp ect to finite closed co verings ( C 0 , . . . , C n ) of closed subsets C = C 0 ∪ . . . ∪ C n ⊂ X , whic h demands that a ll canonical diagr ams P ( C ) → n Y α =0 P ( C α ) ⇒ n Y α,β =0 P ( C α ∩ C β ) (41) are equalizers. Definition. W e call a pat t ern P on X glob al if for a n y closed subset C ⊂ X the restriction morphism P ( X ) → P ( C ) is surjectiv e. Definition. F or a con tinuous map f : X → Y t he preimage f − 1 ( D ) of an y closed subset D ⊂ Y is closed in X and f defines the dir e ct image functor f ∗ of pat terns: ( f ∗ P )( D ) := P ( f − 1 ( D )). W e call ( glob al ly ) algebr aize d sp a c e a pair consisting o f a top ological space X equipp ed with a (global) pattern A X of a lg ebras. Definition. A morphism of (globally) a lgebraized spaces consists of a con tinuous map of top ological spaces f : X → Y and a mor phism o f patterns of algebras A Y → f ∗ A X . In this framew ork the afo remen tioned Gelfand-Naimark dualit y b e- t wee n gluing of compact Hausdorff spaces a nd the generalized Chinese remainder theorem for C*-alg ebras reads now as follo ws. Theorem 3. T h e Gelfand-Naimark duality in d uc es a ful l emb e dding of the c ate gory opp o s i te to unital c ommutative C *-algebr as e quipp e d with lattic es of close d *-ide als into the c a te gory of c omp act Hausdorff glob al ly algebr aize d sp ac es. DISTRIBUTIVE LA TTICES AND COHOMOLOGY. 11 Note that in the ab o v e theorem the Gelfand-Naimark duality b e- t wee n pairs ( A, L ) and ( X , A X ) dualizes a unital commutativ e C*- algebra A to a compact Hausdorff space X and the lattice L of closed *-ideals in A to a glo bal pattern of a lg ebras A X . Note that ev ery lattice of closed *- ideals in a C*- a lgebra is distribu- tiv e. Therefore, according to the general ideology of no ncomm utative top ology , a pair consisting of a unital asso ciative C*-algebra A and a lattice L of closed *-ideals in A should b e regarded as a “ nonc ommu- tative c om p a ct Hausdorff glob a l ly algebr aize d sp ac e” . 4.1. C*-algebras and patterns. 4.1.1. Continuous fields of C*-algebr as. In functional analysis of func- tion C*-algebras, in opp osite to algebraic geometry , the notion of sheaf pla ys no a significant role. The appropriate replacemen t is then the notion of sections of c ontinuous fields of C*-alg e b r as [7]. They are con trav arian t functors on the categor y of closed subsets transforming closed em b eddings in t o surjectiv e restriction homomorphisms. It is easy t o observ e that they satisfy the unique gluing prop ert y with re- sp ect to finite closed co v erings of closed subsets , i.e. they a re patterns in our terminology . This prop erty w a s used in computation of K -theory of an imp ortan t class of T o eplitz algebras on Lie gr o ups, with use of the Ma y er-Vietoris sequence [11]. 4.1.2. Continuous functions vanis h ing at infinity. Another example of patterns arising in theory of function C*-algebras comes from contin u- ous functions v anishing a t infinity . F o r an y lo cally compact space o ne has a non-unital C*-alg ebra C 0 ( X ) of con tin uous functions v anishing at infinit y . It is widely accepted that C 0 ( X ) is an appropriate C*- algebraic r eplacemen t of the lo cally compact space X , mostly in view of the G elfand-Naimark duality in t he unital-v ersus-compact case. A b eautiful part of functional analysis w a s created to ex tend the Gelfand- Naimark duality in this wa y . Ho w ev er, an idea that relaxing compact- ness to lo cal compactness can b e dualized to forgetting ab o ut the unit of the C*-algebra is not true if one wan ts to preserv e the usual relation b et w een contin uous functions and top olo gy . First, a b out C*-algebras and lo calit y . Although con tinuous f unctions form a sheaf under restriction to op en subse ts, the v anishing at infin- it y prop erty do es not surviv e the r estriction. This means that giv en t wo op e n subsets U ⊂ V ⊂ X there is no a r estriction homomorphism from C 0 ( V ) to C 0 ( U ). Strange enough, there is a w ell define d injec- tiv e homomorphism of non-unital algebras in t he opp o site direction 12 TOMASZ M ASZCZYK C 0 ( U ) → C 0 ( V ), giv en by the extension b y zero. Moreov er, giv en op en subsets U 0 , . . . , U n one has a canonical equalizer diagra m C 0 ( U ) → n Y α =0 C 0 ( U α ) ⇒ n Y α,β =0 C 0 ( U α ∪ U β ) , (42) whose arro ws are defined as collections of extensions by zero with re- sp ect to inclusions U = U 0 ∩ . . . ∩ U n ⊂ U α , U α ⊂ U α ∪ U β and U β ⊂ U α ∪ U β . The ˇ Cec h-Stone compactification X ֒ → β X and the Gelfand-Na ima r k dualit y put the problem of geometric description of the extension b y zero in to the righ t p ers p ectiv e. The extension b y zero C 0 ( U ) → C 0 ( V ) is equiv alen t to a surjectiv e restriction homo mo r phism of unital quotient algebras C ( β X \ U ) = C ( β X ) / C 0 ( U ) → C ( β X ) / C 0 ( V ) = C ( β X \ V ) , (43) when w e rega rd C 0 ( U ) as a closed *-ideal in the C*-algebra C ( β X ) ∼ = C b ( X ) of contin uous functions on β X isomorphic to the C*-algebra of b ounded functions on X . This restriction homomorphism is Gelfand- Naimark dual to the closed inclusion C := β X \ U ⊂ β X \ V = : D , and mak es (42) the equalizer diagram v erifying the pattern prop ert y of the assignmen t C 7→ I ( C ) := C 0 ( β X \ C ) on the finite closed cov ering β X \ U = n [ α =0 ( β X \ U α ) . (44) This means that the pattern I is a (sub)pattern of ideals in the constan t pattern C ( β X ) of algebras. The pattern C 7→ C ( C ) is then the pattern of quotien t algebras. 4.1.3. Pattern c ohomolo gy on top olo g ic al sp ac es. Patterns admit a n analog of the ˇ Cec h cohomolog y with resp ect to finite closed cov erings. Assume that there is giv en suc h a cov ering X = C 0 ∪ . . . ∪ C n of a top ological space X and a pattern P . Mimic king the ˇ Cec h complex construction we define the p attern c ohomolo gy H p ( C 0 , . . . , C n ; P ) := H p ( Y i 0 <... 0 H p ( C 0 , . . . , C n ; P ) = 0 . (49) In particular, for the constan t pattern A ( C ) := A H p ( C 0 , . . . , C n ; A ) =  A if p = 0 , 0 if p 6 = 0 (50) and fo r its subpattern I taking v alues in a distributiv e lat t ice of sub- groups of A , satisfying (47) and (48), and suc h that I ( X ) = 0 H p ( C 0 , . . . , C n ; I ) = 0 (51) for a ll p . The short exact seque nce of patterns 0 → I → A → A/ I → 0 (52) induces then a lo ng exact sequence of pattern cohomology , whic h im- plies that H p ( C 0 , . . . , C n ; A/ I ) =  A if p = 0 , 0 if p 6 = 0 (53) This implies that the cohomological b ehavior of t aking remainders mo dulo ideals (restrictions t o closed subsets) of an arit hmetical ring A expresse d in terms o f the globally algebraized space structure defined on the Z ariski top ology of Sp ec( A ) is similar to the cohomological b e- ha vior of lo caliz ations (restrictions to op en subsets) of A expressed in terms of the lo cally ringed space structure on Sp ec( A ). 4.1.4. She aves versus p atterns. D ue to Cartan, Leray’s “faisc e aux c on- tinus” on lo cally compact spaces are equiv alen t to shea v es. Actually , giv en a sheaf F on a lo cally compact space X one can assign to eve ry closed subset C ⊂ X the stalk of F along C . This assignmen t is differ- en t from our “pattern”. F or a sheaf of con tinuous functions the stalk at a p oint consists of germs, while the ev aluation of the pattern on a p oin t consists of v alues. If the space X is not discrete the ke rnel of the surjectiv e ev aluation map from t he stalk to the set of v alues is usually big. 14 TOMASZ M ASZCZYK Reference s [1] Ali, M.M.; Smith, D.J.: Gener alize d GCD rings. II. B eitr¨ age Algebra Geom. 44 ( 2003 ), no. 1, 75–9 8. [2] Ali, M.M.; Smith, D.J.: Gener alize d GCD mo dules. , Beitr¨ age Algebra Geom. 46 ( 2005 ), no. 2, 447– 466. [3] Achk ar, H.: Su r les anne aux arithm´ etiques , S´ eminaire d’Alg` e br e Non Commutativ e (ann´ ee 1972 /197 3 ). Pr emi` ere partie, Exp. No.1, Publ. Math. Orsay , N o. 27, Univ. Paris XI, Orsay , 1973. [4] Bala chandran, V. K.: The Chinese re mainder the or em for the distributive lattic e , J.Indian Math. Soc. (N.S.) 13 (1949), 76–80. 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[18] Zhar inov, V.V.: D istributive lattic es and the e dge of the we dge the or em o f Bo golyub ov. Gener alized functions and their applica tions in mathematical ph ysics, Pro c. In t. Conf., Moscow 1 980, 237–244 ( 1981 ). Institute of Ma thema tics, Polish Academy of Sciences, Sniadeckich 8, 00–956 W arsza w a, Pol and DISTRIBUTIVE LA TTICES AND COHOMOLOGY. 15 Institute of Ma thema tics, University of W arsa w, Banacha 2, 02–097 W arsza w a, Pol and E-mail addr ess : maszc zyk@m imuw.edu.p l