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application/pdf, It is proved that the Hodge decomposition and Serre duality hold on a non-compact weighted CR manifold with negligible boundary. A complete CR manifold has negligible boundary. Some examples of complete CR manifolds are presented., First published in Proceedings of the American Mathematical Society in volume136 and number10, 2008, published by the American Mathematical Society

We prove a duality theorem for graded algebras over a field that implies several known duality results : graded local duality, versions of Serre duality for local cohomology and of Suzuki duality for generalized local cohomology, and Herzog-Rahimi bigraded duality., 6 pages

We prove the Serre duality theorem for the noncommutative projective scheme proj A \;{\operatorname {proj }}\;A when A A is a graded noetherian PI ring or a graded noetherian AS-Gorenstein ring.

We are characterizing the categories of coherent sheaves on a weighted projective line as the small hereditary noetherian categories without projectives and admitting a tilting complex. The paper is related to recent work with de la Pena (Math. Z., to appear) characterizing finite dimensional algebras with a sincere separating tubular family, and further gives a partial answer to a question of Happel, Reiten, Smalo (Mem. Amer. Math. Soc. 120 (1996), no. 575) regarding the characterization of hereditary categories with a tilting object. A characterization of weighted projective lines We are going to characterize the categories coh(X) of coherent sheaves on a weighted projective line X [3, 4]. Theorem 1. Let k be an algebraically closed field. For a small connected abelian k-category H with finite dimensional morphism and extension spaces the following assertions are equivalent: (i) H is equivalent to the category of coherent sheaves on a weighted projective line. (ii) Each object of H is noetherian. H is hereditary, has no non-zero projectives, and admits a tilting complex. (iii) Each object of H is noetherian, moreover (a) There exists an equivalence τ : H → H (Auslander-Reiten translation) such that Serre duality DExt(A,B) ∼= Hom(B, τA) holds functorially in A,B ∈ H. (b) The Grothendieck group K0(H) is finitely generated free, and the Euler form 〈−,−〉 : K0(H) × K0(H) → Z given on classes of objects of H by 〈[X ], [Y ]〉 = dimk Hom(X,Y )−dimk Ext(X,Y ) is non-degenerate of determinant ±1. (c) H has an object without self-extensions which is not of finite length. Received by the editors February 9, 1995. 1991 Mathematics Subject Classification. Primary 14G14, 16G20; Secondary 18F20, 18E30.

A version of Serre-duality is proved for Artin’s non-commutative projective schemes.

We prove that the signature of a two dimensional compact complex spin manifold is divisible by 16. Rohlin [R] showed that the signature of a smooth closed oriented spin 4manifold is divisible by 16. We give a proof for those manifolds which carry a complex structure. Recall that a manifold is spin if and only if the StiefelWitney class w2(X) vanishes. The key observation we need is that a complex manifold is spin if and only if its canonical bundle (the determinant of the holomorphic cotangent bundle) has a square root [A, 3.2]. By the dimension of a complex manifold, we mean its complex dimension. Theorem. If X is a two dimensional compact complex spin manifold then the signature o(X) is divisible by 16. Proof. Choose a holomorphic line bundle L such that L 0 L K, where K is the canonical bundle. Denote the holomorphic euler characteristic x(Ox(L)) by A. By Serre duality [S], the cup product gives a perfect pairing II'(X, Ox(L)) x H (X, Ox(L)) --H (X, Ox(K)) = C. Consequently H (X,Ox(L)) carries a nondegenerate skew-symmetric form, hence it is even dimensional. Therefore A = 2dimH0(X,Ox(L))-dimH1(X,Ox(L)) _0 (mod 2). On the other hand, by the Riemann-Roch and Hirzebruch signature theorems [AS], A= (cl (L)2 + c1(L). cl (X))/2 + (cl(X)2 + c2(X))/12 = (C1 (X)2 2c2(X))/24 =6r(X)/8 which shows that v(X) has the required divisibility. Received by the editors March 13, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 57N 13, 32J 15. This work was done at MSRI where the author was supported by NSF grant DMS-8505550. ( 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page 1143 This content downloaded from 157.55.39.72 on Wed, 14 Sep 2016 04:19:52 UTC All use subject to http://about.jstor.org/terms

Let F -E -B be a fibration such that F, E, and B are homotopy equivalent to finite complexes. Then the following fact is proved. E is a Poincare duality space if and only if B and F are Poincare duality spaces. Let F -E -* B be a fibration with F, B and hence E being homotopy equivalent to finite complexes. We shall give a proof of the following theorem. THEOREM 1. E is a Poincare duality space if and only if B and F are Poincare duality spaces. Here Poincare duality space is as defined in [W]. This theorem was announced by F. Quinn in [Q], but up to now no proof has appeared in the literature; R. Schultz and R. Rigdon have also discovered proofs. The result has been used in [CG] and by R. Schultz. So a brief proof in the literature has become desirable. Suppose X is a finite complex. We may embed X in Rn so that there exists a regular neighborhood N of X which has X as a deformation retract. Then the inclusion aN N X has a homotopy theoretical fibre denoted Ox. We know from work of Spivak, [S], that X satisfies Poincare duality for twisted coefficients over 71(X) if and only if Ox is homotopy equivalent to a sphere. We shall show that OE is homotopy equivalent to the join OB * OF (We may embed F, E and B in large enough Euclidean spaces to insure that OE9 OB and OF are simply connected.) Then by the Kunneth formula OE is a sphere if and only if OF and OB are spheres. In view of Spivak's result, this will establish the theorem. We shall prove the theorem for the special case of a locally trivial fibre bundle in which the fibre, total space and base are manifolds with boundary. This will establish the general case for Hurewicz fibrations since by the closed fibre smoothing theorem of [CG] we know that for any fibration F -, E -, B where B is finite dimensional and F is a finite complex, there is a torus Tn and a locally trivial smooth bundle N -* M -> B such that N -* M -B is fibre homotopy equivalent to F x T7" E X T7 -n B. Thus if (DM oN * (B then (ExTn FxTn * 0 B' But 0ExTn n 'E * Tn E * S m, SO Received by the editors September 29, 1978. AMS (MOS) subject classifications (1970). Primary 55F05, 57B 10. 'This work was partially supported by a grant from the National Science Foundation. ? 1979 American Mathematical Society 0002-9939/79/0000-0380/$01 .75 148 This content downloaded from 40.77.167.18 on Sun, 11 Sep 2016 06:13:37 UTC All use subject to http://about.jstor.org/terms POINCARi DUALITY AND FIBRATIONS 149 E * Sk F * OB * S'. Hence the theorem for fibrations will follow from the special case of fibre bundles. So assume that F -* E -* B is a locally trivial fibre bundle for which F, E and B are manifolds with boundaries. Now if N is a manifold with boundary aN, we convert the inclusion AN C N into a fibration by letting E be the space of paths a in N such that o(O) E aN. The projection p: E > N is given by p(a) = a(l) and the homotopy theoretic fibre 1N over the base point nO E Nis (N = {a E N1Ia(O) E aN and a(l) = no). Now let eO E E be a base point and let P(E), P(F), and P(B) be the space of paths in E, F, and B, respectively, which end at eo, eo, and 7i(eo), respectively, where E B is the projection. Observe that OF C P(F) and oB c P(B) and oE C P(E). We shall prove that P(E) is homeomorphic to P(F) X P(B). Thus OE = (?B X P(F)) U (P(B) X OF) LEMMA. P(E) = P(B) X P(F) and (E = (OB X P(F)) U (P(B) X OA PROOF. For any fibration F -E T B there is a lifting function X: B > El where B is the subspace of E X B' given by {(e, a)17(e) = a(0)) and A(e, a) is a path 6 E E' such that S o 6 = a and a(0) = e. So let A be the lifting function for the principle fibre bundle G -E(F) B associated to F -E -> B. Here G is the group of homeomorphisms of F and E(F) is the space of maps of the distinguished fibre F into any other fibre which is a homeomorphism onto that fibre. If we let i: F -E be the obvious inclusion we see that every path a E P(B) gives rise to an isotopy of homeomorphisms a,: F -> E such that S o d,(eo) = a,(t). Note t F-* 6,(eo) defines a lifting 6 of a. Now we define a continuous map y: P(F) X P(B) -P(E) by (, a) F-* p where p(t) = a,(T(t)). Also define 8: P(E) -> P(F) X P(B) by p F-+ (T, a) where a = 7 o p and T(t) = d,-i (p(t)). Now p and 8 are inverse to each other so P(E) is homeomorphic to P(B) X P(F). To see that OE = ( B X P(F)) U (P(B) X OF) we need only apply the construction to the boundary aE which is the union of p '(aB) and the subbundle of E given by all the points which lie in the boundary of some fibre of E. Now the pairs (P(E), OE)' (P(F), OF) and (P(B), OB) are homotopy equivalent to the pairs (COE, OE), (COF, OF) and (COB, OB), respectively, where CIE is the cone on OE9 etc. This follows since the space P(E) is contractible and the pair (P(E), (E) has the homotopy extension property since aE has a collar neighborhood in E. Thus (P(E), OE) is homotopy equivalent to (C(QF) X C(OB), OF * OB) So OE is homotopy equivalent to OF * OB

In this note we wish to point out a relationship between two problems in the theory of Poincare Duality groups, which, for dialectical purposes, we state as conjectures. They are Homology Equivalence Conjecture. For any finite Poincare complex X, not homotopy equivalent to S2 or RP2, there is a finitely presented Poincare Duality group G and a homology equivalencef: K(G, 1) -X. Realisation Conjecture. If G is a finitely presented Poincare Duality group then there is a closed manifold XG of homotopy type K(G, 1). By a homology equivalence f: Y -->X we mean a map which induces homology/cohomology isomorphisms with respect to all local coefficient systems on X. The Homology Equivalence Conjecture is the natural extension to Poincare complexes of a problem first raised for smooth closed manifolds by Kan and Thurston in [5]. We shall prove

Here 77* (X) denotes the integral homology of X; Zp denotes the integers mod p. Similar results to these have been obtained by M. Rueff [4] using Seifert's "Linking invariant" [5].2 C. T. C. Wall [7] has also studied a similar situation. Our methods have similarity to theirs, but use of the Bockstein spectral sequence instead of more conventional homology theories allows us to refine the results somewhat, bringing in Steenrod squares. I am endebted to Wall for his comments, and to Emery Thomas who pointed out Lemma 7, enabling me to sharpen my original version of Theorem 2.