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For every positive integer r, we introduce two new cohomologies, that we call E r {E_{r}} -Bott–Chern and E r {E_{r}} -Aeppli, on compact complex manifolds. When r = 1 {r\kern-1.0pt=\kern-1.0pt1} , they coincide with the usual Bott–Chern and Aeppli cohomologies, but they are coarser, respectively finer, than these when r ≥ 2 {r\geq 2} . They provide analogues in the Bott–Chern–Aeppli context of the E r {E_{r}} -cohomologies featuring in the Frölicher spectral sequence of the manifold. We apply these new cohomologies in several ways to characterise the notion of page- ( r - 1 ) {(r-1)} - ∂ ⁡ ∂ ¯ {\partial\bar{\partial}} -manifolds that we introduced very recently. We also prove analogues of the Serre duality for these higher-page Bott–Chern and Aeppli cohomologies and for the spaces featuring in the Frölicher spectral sequence. We obtain a further group of applications of our cohomologies to the study of Hermitian-symplectic and strongly Gauduchon metrics for which we show that they provide the natural cohomological framework.

Let n ≥ 3 n \geq 3 and let Ω \Omega be a bounded domain in C n \mathbb {C}^n with a smooth negative plurisubharmonic exhaustion function φ \varphi . As a generalization of Y. Tiba’s result, we prove that any holomorphic function on a connected open neighborhood of the support of ( i ∂ ∂ ¯ φ ) n − 2 (i\partial \bar \partial \varphi )^{n-2} in Ω \Omega can be extended to the whole domain Ω \Omega . To prove it, we combine an L 2 L^2 version of Serre duality and Donnelly-Fefferman type estimates on ( n , n − 1 ) (n,n-1) - and ( n , n ) (n,n) -forms.

In the present paper, we devise a version of topological $L^2$-Serre duality for singular complex spaces with arbitrary singularities. This duality is used to deduce various new $L^2$-vanishing theorems for the $\bar{\partial}$-equation on singular spaces. It is shown that complex spaces with rational singularities behave quite tame with respect to the $\bar{\partial}$-equation in the $L^2$-sense. More precisely: a singular point is rational if and only if the $L^2$-$\bar{\partial}$-complex is exact in this point. So, we obtain an $L^2$-$\bar{\partial}$-resolution of the structure sheaf in rational singular points., Comment: 30 pages; completely revised version with more details on the topological difficulties

Let \(\Omega ={\widetilde{\Omega }}{\setminus } \overline{D}\) where \({\widetilde{\Omega }}\) is a bounded domain with connected complement in \({\mathbb {C}}^n\) (or more generally in a Stein manifold) and D is relatively compact open subset of \({\widetilde{\Omega }}\) with connected complement in \(\widetilde{\Omega }\). We obtain characterizations of pseudoconvexity of \({\widetilde{\Omega }}\) and D through the vanishing or Hausdorff property of the Dolbeault cohomology groups of \(\Omega \) on various function spaces. In particular, we show that if the boundaries of \({\widetilde{\Omega }}\) and D are Lipschitz and \(C^2\)-smooth respectively, then both \({\widetilde{\Omega }}\) and D are pseudoconvex if and only if 0 is not in the spectrum of the \(\overline{\partial }\)-Neumann Laplacian of \(\Omega \) on (0, q)-forms for \(1\le q\le n-2\) when \(n\ge 3\); or 0 is not a limit point of the spectrum of the \(\overline{\partial }\)-Neumann Laplacian on (0, 1)-forms when \(n=2\).

We prove that the full twist is a Serre functor in the homotopy category of type A Soergel bimodules. As a consequence, we relate the top and bottom Hochschild degrees in Khovanov–Rozansky homology, categorifying a theorem of Kalman.

A duality theorem for the singularity category of a finite dimensional Gorenstein algebra is proved. It complements a duality on the category of perfect complexes, discovered by Happel. One of its consequences is an analogue of Serre duality, and the existence of Auslander-Reiten triangles for the $\mathfrak{p}$-local and $\mathfrak{p}$-torsion subcategories of the derived category, for each homogeneous prime ideal $\mathfrak{p}$ arising from the action of a commutative ring via Hochschild cohomology., 19 pages

We study the role of the Serre functor in the theory of derived equivalences. Let $\mathcal{A}$ be an abelian category and let $(\mathcal{U}, \mathcal{V})$ be a $t$-structure on the bounded derived category $D^b \mathcal{A}$ with heart $\mathcal{H}$. We investigate when the natural embedding $\mathcal{H} \to D^b \mathcal{A}$ can be extended to a triangle equivalence $D^b \mathcal{H} \to D^b \mathcal{A}$. Our focus of study is the case where $\mathcal{A}$ is the category of finite-dimensional modules over a finite-dimensional hereditary algebra. In this case, we prove that such an extension exists if and only if the $t$-structure is bounded and the aisle $\mathcal{U}$ of the $t$-structure is closed under the Serre functor., As accepted by Advances in Mathematics; some differences in label and page numbering may occur between this version and the published version

We use duality in the manner of Serre to generalize a theorem of Hedenmalm on solution of the $\bar \partial $ equation with inverse of the weight in Hormander $\displaystyle L^{2}$ estimates.\

Schur-Weyl duality theory has many applications in quantum information and quantum computation theory. In this paper, we first give an equivalent relationship about the subgroup of unitary group U(d) and give several examples of subgroup of U(d) which satisfy the equivalent relationship. Next, we establish a new generalized Schur-Weyl which is an improvement of the classical Schur-Weyl duality and a generalized Schur-Weyl duality proved recently.

The purpose of this paper is to construct a duality theory for finite commutative multivalued groups and to demonstrate its connection with the classical duality in the theory of ordinary groups and the Kawada–Delsarte duality in algebraic combinatorics. We study in detail the case of multivalued groups of order three, construct a parameterization of the set of these groups, and obtain explicit formulas for the duality. In future, we plan to use this duality in the study of the coset problem. Bibliography: 26 titles.

We reformulate the construction of Kontsevich's completion and use Lawson homology to define many new motivic invariants. We show that the dimensions of subspaces generated by algebraic cycles of the cohomology groups of two $K$-equivalent varieties are the same, which implies that several conjectures of algebraic cycles are $K$-statements. We define stringy functions which enable us to ask stringy Grothendieck standard conjecture and stringy Hodge conjecture. We prove a projective bundle theorem in morphic cohomology for trivial bundles over any normal quasi-projective varieties., Comment: 24 pages

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 make explicitPoincaré dualityfor the equivariantK-theory of equivariant complex projective spaces. The case of the trivial group provides a new approach to theK-theory orientation [3].

Auslander-Reiten duality for module categories is generalised to Grothendieck abelian categories that have a sufficient supply of finitely presented objects. It is shown that Auslander-Reiten duality amounts to the fact that the functor Ext^1(C,-) into modules over the endomorphism ring of C admits a partially defined right adjoint when C is a finitely presented object. This result seems to be new even for module categories. For appropriate schemes over a field, the connection with Serre duality is discussed., Comment: 16 pages

Let k be an algebraically closed field of positive characteristic. The goal of this paper is to characterize the proper smooth curves X/k of positive genus g equipped with a k-rational point P such that X\P can be realized as an etale cover of the affine line. A first elementary analysis shows that such a cover exists if and only if there exists an exact regular differential form on X having a unique zero at P (of order 2g-2). The main inconvenient of this approach is that it doesn't give any control on the degree of the cover. A finer investigation, essentially based on the duality between the Cartier operator and the Frobenius map allows to overcome this problem., Comment: 19 pages

The first goal of this paper is to provide an abstract framework in which to formulate and study local duality in various algebraic and topological contexts. For any stable $\infty$-category $\mathcal{C}$ together with a collection of compact objects $\mathcal{K} \subset \mathcal{C}$ we construct local cohomology and local homology functors satisfying an abstract version of local duality. When specialized to the derived category of a commutative ring $A$ and a suitable ideal in $A$, we recover the classical local duality due to Grothendieck as well as generalizations by Greenlees and May. More generally, applying our result to the derived category of quasi-coherent sheaves on a quasi-compact and separated scheme $X$ implies the local duality theorem of Alonso Tarr\'io, Jerem\'ias L\'opez, and Lipman. As a second objective, we establish local duality for quasi-coherent sheaves over many algebraic stacks, in particular those arising naturally in stable homotopy theory. After constructing an appropriate model of the derived category in terms of comodules over a Hopf algebroid, we show that, in familiar cases, the resulting local cohomology and local homology theories coincide with functors previously studied by Hovey and Strickland. Furthermore, our framework applies to global and local stable homotopy theory, in a way which is compatible with the algebraic avatars of these theories. In order to aid computability, we provide spectral sequences relating the algebraic and topological local duality contexts., Comment: Significantly revised version to appear in Advances in Mathematics. Section 6 has been removed, and an expanded version will appear in a separate paper

We study a one-parameter family of gauged linear sigma models (GLSMs) naturally associated to a complete intersection in weighted projective space. In the positive phase of the family we recover Gromov-Witten theory of the complete intersection, while in the negative phase we obtain a Landau--Ginzburg-type theory. Focusing on the negative phase, we develop foundational properties which allow us to state and prove a genus-zero comparison theorem that generalizes the multiple log-canonical correspondence and should be viewed as analogous to quantum Serre duality in the positive phase. Using this comparison result, along with the crepant transformation conjecture and quantum Serre duality, we prove a genus-zero correspondence between the GLSMs which arise at the two phases, thereby generalizing the Landau-Ginzburg/Calabi-Yau correspondence to complete intersections., 47 pages, comments welcome!

An Auslander–Reiten formula for complexes of modules is presented. This formula contains as a special case the classical Auslander–Reiten formula. The Auslander–Reiten translate of a complex is described explicitly, and various applications are discussed.

We develop the fundamentals of hereditary noetherian categories with Serre duality over an arbitrary field k, where the category of coherent sheaves over a smooth projective curve over k serves as the prime example and others are coming from the representation theory of finite dimensional algebras. The proper way to view such a category is to think of coherent sheaves on a possibly non-commutative smooth projective curve. We define for each such category notions like function field and Euler characteristic, determine its Auslander-Reiten components and study stable and semistable bundles for an appropriate notion of degree. We provide a complete classification of hereditary noetherian categories for the case of positive Euler characteristic by relating these to finite dimensional representations of (locally bounded) hereditary k-algebras whose underlying valued quiver admits a positive additive function.

In this paper we take some classical ideas from commutative algebra, mostly ideas involving duality, and apply them in algebraic topology. To accomplish this we interpret properties of ordinary commutative rings in such a way that they can be extended to the more general rings that come up in homotopy theory. Amongst the rings we work with are the differential graded ring of cochains on a space, the differential graded ring of chains on the loop space, and various ring spectra, e.g., the Spanier-Whitehead duals of finite spectra or chromatic localizations of the sphere spectrum. Maybe the most important contribution of this paper is the conceptual framework, which allows us to view all of the following dualities: Poincare duality for manifolds, Gorenstein duality for commutative rings, Benson-Carlson duality for cohomology rings of finite groups, Poincare duality for groups, Gross-Hopkins duality in chromatic stable homotopy theory, as examples of a single phenomenon. Beyond setting up this framework, though, we prove some new results, both in algebra and topology, and give new proofs of a number of old results., Comment: 49 pages. To appear in the Advances in Mathematics

Let X be a smooth scheme of finite type over a field K, let e be a locally free O X -bimodule of rank n, and let A be the non-commutative symmetric algebra generated by e. We construct an internal Hom functor, H omGrA (-, -), on the category of graded right A-modules. When e has rank 2, we prove that A is Gorenstein by computing the right derived functors of Hom Gr.A (O X , -). When X is a smooth projective variety, we prove a version of Serre Duality for ProjA using the right derived functors of lim Hom GrA (A/A ≥n ,-).

We describe in detail Serre's application of spectral sequence theory to the study of the relations between the homology of total space, base space and fibre in a Serre fibration; and we apply the results to establish that a 1-connected space X has homology groups (in positive dimension) in a Serre class C if and only if its homotopy groups are in C . We include in this paper some personal reflections on the contact the author had with Serre during the decade of the 1950's when Serre's revolutionary work in homotopy theory was completely changing the face of algebraic topology.

We establish a duality in the cohomology of arbitrary tori over smooth but not necessarily projective curves over a p-adic field. This generalises Lichtenbaum–Tate duality between the Picard group and the Brauer group of a smooth projective curve.

In this paper, we study the geometry and topology on the oriented Grassmann manifolds. In particular, we use characteristic classes and the Poincare duality to study the homology groups of Grassmann manifolds. We show that for k=2 or n \leq 8, the cohomology groups H*(G(k,n), R) are generated by the first Pontrjagin class, the Euler classes of the canonical vector bundles. In these cases, the Poincare duality: Hq(G(k,n), R) \to Hk(n-k)-q(G(k,n), R) can be expressed explicitly.

Borel and Serre calculated the cohomology of the building associated to a reductive group and used the result to deduce that torsion-free S S -arithmetic groups are duality groups. By replacing their group-theoretic arguments with proofs relying only upon the geometry of buildings, we show that Borel and Serre’s approach can be modified to calculate the cohomology of any locally finite affine building. As an application we show that any finitely presented A ~ n \widetilde {A}_n -group is a virtual duality group. A number of other finiteness conditions for A ~ n \widetilde {A}_n -groups are also established.

In the present paper we study higher order embeddings in the context of cyclic coverings. Analyzing the positivity of the line bundle downstairs and its relationship with the branch divisor, we provide criteria for its pull-back to define an embedding of given order. We show that the obtained criteria are sharp. Finally, we apply them to various – sometimes seemingly unrelated–problems in algebraic geometry.

We study the essential dimension of exceptional connected simply connected algebraic groups over algebraically closed fields. In the present paper, we find upper estimates for the essential dimensions of the groupsF4,E6, andE7. For the groupF4, the upper estimate, thus obtained coincides with the known lower estimate. We also prove the second Serre conjecture for the groupE6 and for a function field.

We prove a separation criterion for the compactly supported Dolbeault cohomology. As an application we give a simple proof of the following result: If X is an ( n − q )-convex complex manifold of dimension n, 1≤q≤n−1 , and K is a compact subset of X which admits a basis of q-convex neighborhoods, then H p , n − q ( X \ K , E ) is separated for all p and each holomorphic vector bundle E over X.

Let p, q be terms of the same similarity type. An identity p = q is called normal if it is either of the form x = x or none of p, q is equal to a single variable. For a variety Y, denote by Id V or IdN V the set of all identities or the set of all normal identities of Y, respectively. Of course, IdN V Q Id V and hence V is a subvariety of the variety N ( V } defined by IdN V. V is called normally presented, [1] if V = N ( Y } . If f ^ N ( Y ) then N(V] covers Y in the lattice of all varieties of a given type, see [3], [5]. An algebra stf = (A, F) is called a constant algebra if there exists an element 0 6 A (the so called constant of jtf) such that

Applying the methods of Serre duality in the setting of CR manifolds we prove approximation theorems and we study the Hartogs-Bochner phenomenon in 1-concave CR generic manifolds.

Two theorems are established which properly extend the ckss of artinian rings with Morita duality. A question of Anh about QF-rings is answered in the negative.

In this paper we use recently developed calculus of residue currents together with integral formulas to give a new explicit analytic realization, as well as a new analytic proof of Serre duality on any reduced pure $n$-dimensional paracompact complex space $X$. At the core of the paper is the introduction of concrete fine sheaves $\mathscr{B}_X^{n,q}$ of certain currents on $X$ of bidegree $(n,q)$, such that the Dolbeault complex $(\mathscr{B}_X^{n,\bullet},\,\bar{\partial})$ becomes, in a certain sense, a dualizing complex. In particular, if $X$ is Cohen-Macaulay (e.g., Gorenstein or a complete intersection) then $(\mathscr{B}_X^{n,\bullet},\,\bar{\partial})$ is an explicit fine resolution of the Grothendieck dualizing sheaf., Final version

We introduce thread quivers as an (infinite) generalization of quivers, and show that every k-linear (k algebraically closed) hereditary category with Serre duality and enough projectives is equivalent to the category of finitely presented representations of a thread quiver. In this way, we obtain an explicit construction of a new class of hereditary categories with Serre duality., Comment: As accepted by the Proceedings of the London Mathematical Society; some differences in label and page numbering may occur between this version and the published version

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 prove a sheaf-theoretic derived-category generalization of Greenlees-May duality (a far-reaching generalization of Grothendieck's local duality theorem): for a quasi-compact separated scheme X and a "proregular" subscheme Z---for example, any separated noetherian scheme and any closed subscheme---there is a sort of sheafified adjointness between local cohomology supported in Z and left-derived completion along Z. In particular, the i-th left-derived completion functor is the "local homology" sheaf $Ext^i(\R\Gamma_Z\O_X, -)$. Sheafified generalizations of a number of duality theorems scattered about the literature result, e.g., the Peskine-Szpiro duality sequence (generalizing local duality), the Warwick Duality theorem of Greenlees, the Affine Duality theorem of Hartshorne. Using Grothendieck Duality, we also get a generalization of a Formal Duality theorem of Hartshorne, and of a related local-global duality theorem. In a sequel we will develop the latter results further, to study Grothendieck duality and residues on formal schemes., Comment: DVI file pub/lipman/homology.dvi (214776K, 38 pages) available via anonymous ftp (binary) at ftp.math.purdue.edu, AMSLaTeX v 1.2

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.

Let A be an abelian hereditary category with Serre duality. We provide a classification of such categories up to derived equivalence under the additional condition that the Grothendieck group modulo the radical of the Euler form is a free abelian group of finite rank. Such categories are called numerically finite, and this condition is satisfied by the category of coherent sheaves on a smooth projective variety., 41 pages, as accepted by the Transactions of the American Mathematical Society

We study abelian localizations of triangulated categories induced by rigid contravariantly finite subcategories, and also triangulated structures on subfactor categories of triangulated categories. In this context we generalize recent results of Buan-Marsh and Iyama-Yoshino. We also extend basic results of Keller-Reiten concerning the Gorenstein and the Calabi-Yau property for categories arising from certain rigid, not necessarily cluster tilting, subcategories, as well as several results of the literature concerning the connections between 2-cluster tilting subcategories of triangulated categories and tilting subcategories of the associated abelian category of coherent functors. Finally we characterize 2-cluster tilting subcategories along these lines., Comment: 37 pages, a version, with minor changes, of a paper accepted for publication in Mathematische Zeitschrift

We show a surprising link between singularity theory and the invariant subspace problem of nilpotent operators as recently studied by C. M. Ringel and M. Schmidmeier, a problem with a longstanding history going back to G. Birkhoff. The link is established via weighted projective lines and (stable) categories of vector bundles on those. The setup yields a new approach to attack the subspace problem. In particular, we deduce the main results of Ringel and Schmidmeier for nilpotency degree p from properties of the category of vector bundles on the weighted projective line of weight type (2,3,p), obtained by Serre construction from the triangle singularity x^2+y^3+z^p. For p=6 the Ringel-Schmidmeier classification is thus covered by the classification of vector bundles for tubular type (2,3,6), and then is closely related to Atiyah's classification of vector bundles on a smooth elliptic curve. Returning to the general case, we establish that the stable categories associated to vector bundles or invariant subspaces of nilpotent operators may be naturally identified as triangulated categories. They satisfy Serre duality and also have tilting objects whose endomorphism rings play a role in singularity theory. In fact, we thus obtain a whole sequence of triangulated (fractional) Calabi-Yau categories, indexed by p, which naturally form an ADE-chain., Comment: More details added. 33 pages

Given a smooth proper dg-algebra $A$, a perfect dg $A$-module $M$, and an endomorphism $f$ of $M$, we define the Hochschild class of the pair $(M,f)$ with values in the Hochschild homology of $A$. Our main result is a Riemann-Roch type formula involving the convolution of two such Hochschild classes., Comment: 26 pages. Many changes

We prove the existence of regular foliations with a prescribed tangency divisor in neighborhoods of negatively embedded holomorphic curves; this is related to a linearization theorem due to Grauert. We give also examples of neighborhoods which can not be linearized., Pub. Mat. 55, 2011

An abelian Krull-Schmidt category is said to be uniserial if the isomorphism classes of subobjects of a given indecomposable object form a linearly ordered poset. In this paper, we classify the hereditary uniserial categories with Serre duality. They fall into two types: the first type is given by the representations of the quiver A_n with linear orientation (and infinite variants thereof), the second type by tubes (and an infinite variant). These last categories give a new class of hereditary categories with Serre duality, called big tubes., 24 pages, 2 figures

In this paper a duality theory for morphism X→Y of rigid analytic spaces over some non-archimedean valued complete field K is developed. In order to keep the statements and the proofs reasonable we make the restrictions Y is affinoid and X→Y is smooth and proper. The successful approach of B. Chiarellotto [Ch] for Stein spaces is the basis for this paper. An essential step is to show a strong form of unicity for the residue maps Res. One uses further a suitable definition for cohomology with compact support; the work of R. Kiehl [Kl] on proper mappings; geometric points as introduced in [P]; the strong G-topology of [K1,K2] in the version of [BGR]. As one expects the paper is rather technical in nature and the results are not surprising. However for the investigation of proper rigid analytic manifolds the Serre-duality can not be missed. We thank B. Chiarellotto for his careful reading of the manuscript, which has led to improvements of the paper.

The purpose of this note is to give a proof of a theorem of Serre, which states that if G is a p -group which is not elementary abelian, then there exist an integer m and non-zero elements x 1 , … x m ∈ H 1 ( G , Z / p ) such that formula here with β the Bockstein homomorphism. Denote by m G the smallest integer m satisfying the above property. The theorem was originally proved by Serre [ 5 ], without any bound on m G . Later, in [ 2 ], Kroll showed that m G [les ] p k −1, with k =dim Z / p H 1 ( G , Z / p ). Serre, in [ 6 ], also showed that m G [les ]( p k −1)/ ( p −1). In [ 3 ], using the Evens norm map, Okuyama and Sasaki gave a proof with a slight improvement on Serre's bound; it follows from their proof (see, for example, [ 1 , Theorem 4.7.3]) that m G [les ]( p +1) p k −2 . However, m G can be sharpened further, as we see below. For convenience, write H *ast;( G , Z / p )= H *( G ). For every x i ∈ H 1 ( G ), set formula here