Minimal projective resolutions

In this paper, we present an algorithmic method for computing a projective resolution of a module over an algebra over a field. If the algebra is finite dimensional, and the module is finitely generated, we have a computational way of obtaining a minimal projective resolution, maps included. This resolution turns out to be a graded resolution if our algebra and module are graded. We apply this resolution to the study of the -algebra of the algebra; namely, we present a new method for computing Yoneda products using the constructions of the resolutions. We also use our resolution to prove a case of the ``no loop' conjecture.

     

Exterior products of forms and the cohomology ring of a complex space

In recent publications, we have defined complexes of differential forms on analytic spaces which are resolutions of the constant sheaf. These complexes were used to prove the existence of a mixed Hodge structure on the cohomology of analytic spaces which possess kählerian hypercoverings, in particular, projective algebraic varieties. We define an exterior product on these forms, which induces the cup product on the cohomology of analytic spaces. The main difficulty is to prove that this exterior product is functorial with respect to morphisms of analytic spaces. This exterior product can be used to prove that the cup product is compatible with the mixed Hodge structure on the cohomology.     

Geometric cohomology frames on Hausmann-Holm-Puppe conjugation spaces

For certain manifolds with an involution the mod 2 cohomology ring of the set of fixed points is isomorphic to the cohomology ring of the manifold, up to dividing the degrees by two. Examples include complex projective spaces and Grassmannians with the standard antiholomorphic involution (with real projective spaces and Grassmannians as fixed point sets).

Hausmann, Holm and Puppe have put this observation in the framework of equivariant cohomology, and come up with the concept of conjugation spaces, where the ring homomorphisms arise naturally from the existence of what they call cohomology frames. Much earlier, Borel and Haefliger had studied the degree-halving isomorphism between the cohomology rings of complex and real projective spaces and Grassmannians using the theory of complex and real analytic cycles and cycle maps into cohomology.

The main result in the present note gives a (purely topological) connection between these two results and provides a geometric intuition into the concept of a cohomology frame. In particular, we see that if every cohomology class on a manifold with involution is the Thom class of an equivariant topological cycle of codimension twice the codimension of its fixed points (inside the fixed point set of ), these topological cycles will give rise to a cohomology frame.

     

Hyperplane arrangement cohomology and monomials in the exterior algebra

We show that if is the complement of a complex hyperplane arrangement, then the homology of has linear free resolution as a module over the exterior algebra on the first cohomology of . We study invariants of that can be deduced from this resolution. A key ingredient is a result of Aramova, Avramov, and Herzog (2000) on resolutions of monomial ideals in the exterior algebra. We give a new conceptual proof of this result.

     

Cohomology of projective space seen by residual complex

A subcomplex of a residual complex on projective space is constructed for computing the cohomology modules of locally free sheaves. A constructive new proof of the Bott formula is given by explicitly exhibiting bases for the cohomology modules.

     

Periodic Cyclic Homology as Sheaf Cohomology Dedicated to Daniel Quillen on his 60th Birthday

We study a noncommutative version of the infinitesimal site of Grothendieck. A theorem of Grothendieck establishes that the cohomology of the structure sheaf on the infinitesimal topology of a scheme of characteristic zero is de Rham cohomology. We prove that, for the noncommutative infinitesimal topology of an associative algebra over a field of characteristic zero, the cohomology of the structure sheaf modulo commutators is periodic cyclic cohomology. We also compute the noncommutative infinitesimal cohomology of other sheaves. For example, we show that infinitesimal hypercohomology with coefficients in K-theory gives the fiber of the Jones–Goodwillie character which goes from K-theory to negative cyclic homology.     

Quantum cohomology of projective bundles over

In this paper we study the quantum cohomology ring of certain projective bundles over the complex projective space . Using excessive intersection theory, we compute the leading coefficients in the relations among the generators of the quantum cohomology ring structure. In particular, Batyrev's conjectural formula for quantum cohomology of projective bundles associated to direct sum of line bundles over is partially verified. Moreover, relations between the quantum cohomology ring structure and Mori's theory of extremal rays are observed. The results could shed some light on the quantum cohomology for general projective bundles.

     

Flat covers and cotorsion envelopes of sheaves

In this paper we prove that any sheaf of modules over any topological space (in fact, any -module where is a sheaf of rings on the topological space) has a flat cover and a cotorsion envelope. This result is very useful, as we shall explain later in the introduction, in order to compute cohomology, due to the fact that the category of sheaves ( -modules) does not have in general enough projectives.

     

(Bi-)Cohen–Macaulay Simplicial Complexes and Their Associated Coherent Sheaves

ABSTRACT

Via the BGG correspondence, a simplicial complex Δ on [n] is transformed into a complex of coherent sheaves on P ^n?1. We show that this complex reduces to a coherent sheaf ? exactly when the Alexander dual Δ* is Cohen–Macaulay.

We then determine when both Δ and Δ* are Cohen–Macaulay. This corresponds to ? being a locally Cohen–Macaulay sheaf.

Lastly, we conjecture for which range of invariants of such Δ's it must be a cone, and show the existence of such Δ's which are not cones outside of this range.     

Zur lokalen Hochschild-Homologie eines komplexen Raums

For the exploration of the Hochschild homology and cohomology sheaves Tor^s(0,F) and Ext_s(0,F) of a complex space (X,0), where F is a coherent 0-Module and , an S-free resolution of 0 is needed. Such a resolution M we construct starting from an imbedding of X in an open subset of. In order to show that M is acyclic we investigate analytic tensor products and prove the analytic tensor product of an analytic local algebra with an analytic module is in a sense relatively projective, while analytic module homomorphisms are relatively split.     

Slopes of vector bundles on projective curves and applications to tight closure problems

We study different notions of slope of a vector bundle over a smooth projective curve with respect to ampleness and affineness in order to apply this to tight closure problems. This method gives new degree estimates from above and from below for the tight closure of a homogeneous -primary ideal in a two-dimensional normal standard-graded algebra in terms of the minimal and the maximal slope of the sheaf of relations for some ideal generators. If moreover this sheaf of relations is semistable, then both degree estimates coincide and we get a vanishing type theorem.

     

Ample filters and Frobenius amplitude

Let X be a projective scheme over a field. We show that the vanishing cohomology of any sequence of coherent sheaves is closely related to vanishing under pullbacks by the Frobenius morphism. We also compare various definitions of ample locally free sheaf and show that the vanishing given by the Frobenius morphism is, in a certain sense, the strongest possible. Our work can be viewed as various generalizations of the Serre Vanishing Theorem.     

Rational versus real cohomology algebras of low-dimensional toric varieties

We show that the real cohomology algebra of a compact toric variety of complex dimension  is determined, up to isomorphism, by the combinatorial data of its defining fan. Surprisingly enough, this is no longer the case when taking rational coefficients. Moreover, we show that neither the rational nor the real or complex cohomology algebras of compact quasi-smooth toric varieties are combinatorial invariants in general.

     

BOTT–BOREL–WEIL THEORY AND BERNSTEIN–GEL’FAND–GEL’FAND RECIPROCITY FOR LIE SUPERALGEBRAS

The main focus of this paper is Bott–Borel–Weil (BBW) theory for basic classical Lie superalgebras. We take a purely algebraic self-contained approach to the problem. A new element in this study is twisting functors, which we use in particular to prove that the top of the cohomology groups of BBW theory for generic weights is described by the recently introduced star action. We also study the algebra of regular functions, related to BBW theory. Then we introduce a weaker form of genericness, relative to the Borel subalgebra and show that the virtual BGG reciprocity of Gruson and Serganova becomes an actual reciprocity in the relatively generic region. We also obtain a complete solution of BBW theory for \( \mathfrak{o}\mathfrak{s}\mathfrak{p} \)(m|2), D(2, 1; α), F(4) and G(3) with distinguished Borel subalgebra. Furthermore, we derive information about the category of finite-dimensional \( \mathfrak{o}\mathfrak{s}\mathfrak{p} \)(m|2)-modules, such as BGG-type resolutions and Kostant homology of Kac modules and the structure of projective modules.     

Grothendieck topologies and deformation Theory II

Starting from a sheaf of associative algebras over a scheme we show thatits deformation theory is described by cohomologies of a canonical object,called the cotangent complex, in the derived category of sheaves ofbi-modules over this sheaf of algebras. The passage from deformations tocohomology is based on considering a site which is naturally constructed outof our sheaf of algebras. It turns out that on the one hand, cohomology ofcertain sheaves on this site control deformations, and on the other hand,they can be rewritten in terms of the category of sheaves of bi-modules.     

On cohomology algebras of complex subspace arrangements

The integer cohomology algebra of the complement of a complex subspace arrangement with geometric intersection lattice is completely determined by the combinatorial data of the arrangement. We give a combinatorial presentation of the cohomology algebra in the spirit of the Orlik-Solomon result on the cohomology algebras of complex hyperplane arrangements. Our methods are elementary: we work with simplicial models for the complements that are induced by combinatorial stratifications of complex space. We describe simplicial cochains that generate the cohomology. Among them we distinguish a linear basis, study cup product multiplication, and derive an algebra presentation in terms of generators and relations.

     

Polyanalytic forms on compact Riemann surfaces

A sheaf of differentials on a compact Riemann surface supplied with a projective structure is said to be n-analytic if, in local projective coordinates, sections of the sheaf satisfy the differential equation { } For the projective structure induced by a covering mapping from the disk, an explicit characterization of the space of cross sections and of the space of first cohomologies of an n-analytic sheaf is given in terms of known spaces of sections of certain holomorphic sheaves. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 15–25. Translated by S. V. Kislyakov.     

The cohomology of the cotangent bundle of Heisenberg groups

Given a parabolic subalgebra g₁×n of a semisimple Lie algebra, Kostant (Ann. Math. 1963) and Griffiths (Acta Math. 1963) independently computed the g₁ invariants in the cohomology group of n with exterior adjoint coefficients. By a theorem of Bott (Ann. Math. 1957), this is the cohomology of the associated compact homogeneous space with coefficients in the sheaf of local holomorphic forms. In this paper we determine explicitly the full module structure, over the symplectic group, of the cohomology group of the Heisenberg Lie algebra with exterior adjoint coefficients. This is the cohomology of the cotangent bundle of the Heisenberg group.     

Sheaf coalgebras and duality

In this paper we construct an adjoint pair of functors between the category of sheaves on a smooth manifold M and the category of coalgebras over the ring of smooth functions with compact support on M. We show that the sheaf coalgebra associated to a sheaf E on M determines the sheaf E uniquely up to an isomorphism, and that the adjoint functors restrict to an equivalence between sheaves on M and sheaf coalgebras over .     

Invariants de Von Neumann des faisceaux analytiques cohérents

In order to study the group of holomorphic sections of the pull-back to the universal covering space of an holomorphic vector bundle on a compact complex manifold, it would be convenient to have a cohomological formalism, generalizing Atiyah's index theorem. In [Eys99], such a formalism is proposed in a restricted context. To each coherent analytic sheaf on a n-dimensionnal smooth projective variety and each Galois infinite unramified covering , whose Galois group is denoted by , cohomology groups denoted by are attached, such that: 1. The underly a cohomological functor on the abelian category of coherent analytic sheaves on X. 2. If is locally free, is the group of holomorphic sections of the pull-back to of the holomorphic vector bundle underlying . 3. belongs to a category of -modules on which a dimension function with real values is defined. 4. Atiyah's index theorem holds [Ati76]: The present work constructs such a formalism in the natural context of complex analytic spaces. Here is a sketch of the main ideas of this construction, which is a Cartan-Serre version of [Ati76]. A major ingredient will be the construction [Farb96] of an abelian category containing every closed -submodule of the left regular representation. In topology, this device enables one to use standard sheaf theoretic methods to study Betti numbers [Ati76] and Novikov-Shubin invariants [NovShu87]. It will play a similar r?le here. We first construct a -cohomology theory () for coherent analytic sheaves on a complex space endowed with a proper action of a group such that conditions 1-2 are fulfilled. The -cohomology on the Galois covering of a coherent analytic sheaf onX is the ordinary cohomology of a sheaf on X obtained by an adequate completion of the tensor product of by the locally constant sheaf on X associated to the left regular representation of the discrete group in the space of functions on . Then, we introduce an homological algebra device, montelian modules, which can be used to calculate the derived category of and are a good model of the Čech complex calculating -cohomology. Using this we prove that , if X is compact. This is stronger than condition 3, since this also yields Novikov-Shubin type invariants. To explain the title of the article, Betti numbers and Novikov-Shubin invariants of are the Von Neumann invariants of the coherent analytic sheaf . We also make the connection with Atiyah's -index theorem [Ati76] thanks to a Leray-Serre spectral sequence. From this, condition 4 is easily deduced.

---

---

Received: 30 October 1998 / Published online: 8 May 2000