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Let (, ,) be an Ext -finite, Krull–Schmidt and k -linear n -exangulated category with k a commutative artinian ring. In this note, we prove that has Auslander–Reiten–Serre duality if and only if has Auslander–Reiten n -exangles. Moreover, we also give an equivalent condition for the existence of Serre duality (which is a special type of Auslander–Reiten–Serre duality). Finally, assume further that has Auslander–Reiten–Serre duality. We exploit a bijection triangle, which involves the restricted Auslander bijection and the Auslander–Reiten–Serre duality. [ABSTRACT FROM AUTHOR]

It is often stated that Frobenius quantales are necessarily unital. By taking negation as a primitive operation, we can define Frobenius quantales that may not have a unit. We develop the elementary theory of these structures and show, in particular, how to define nuclei whose quotients are Frobenius quantales. This yields a phase semantics and a representation theorem via phase quantales. Important examples of these structures arise from Raney’s notion of tight Galois connection: tight endomaps of a complete lattice always form a Girard quantale which is unital if and only if the lattice is completely distributive. We give a characterisation and an enumeration of tight endomaps of the diamond lattices M n and exemplify the Frobenius structure on these maps. By means of phase semantics, we exhibit analogous examples built up from trace class operators on an infinite dimensional Hilbert space. Finally, we argue that units cannot be properly added to Frobenius quantales: every possible extention to a unital quantale fails to preserve negations. [ABSTRACT FROM AUTHOR]

Using the exceptional inverse image functor for quasi-finite proper morphisms of separated tame Deligne–Mumford stacks of finite type over a field k, Serre duality is obtained in varying degrees of generality for tame Deligne–Mumford stacks. The approach follows that for schemes. [ABSTRACT FROM AUTHOR]

We adapt ideas from Ekedahl to prove a Serre-type duality for Witt-divisorial sheaves of Q–Cartier divisors on a smooth projective variety over a perfect field of finite characteristic. We also explain its relationship to Tanaka’s vanishing theorems. [ABSTRACT FROM AUTHOR]

We study basic Dolbeault cohomology and find new Weitzenböck formulas on a transversely Kähler foliation. We investigate conditions on mean curvature and Ricci curvature that impose restrictions on basic Dolbeault cohomology. For example, we prove that on a transversely Kähler foliation with positive transversal Ricci curvature, there are no nonzero basic-harmonic forms of type (r , 0) , among other results. [ABSTRACT FROM AUTHOR]

Extriangulated categories were introduced by Nakaoka and Palu, which unify exact categories and extension-closed subcategories of triangulated categories, and recently Iyama, Nakaoka and Palu investigated Auslander-Reiten theory in terms of Auslander-Reiten-Serre duality in extriangulated categories. In this paper, we introduce the notion of Serre duality, as a special type of Auslander-Reiten-Serre duality, and then give an equivalent condition for the existence of Serre duality. On the other hand, we study a bijection triangle in extriangulated categories which involves the restricted Auslander bijection and Auslander-Reiten-Serre duality. Let R be a commutative artinian ring. We show that the restricted Auslander bijection holds true in Hom-finite R -linear Krull-Schmidt extriangulated categories having Auslander-Reiten-Serre duality, and especially obtain the Auslander bijection in Hom-finite R -linear Krull-Schmidt extriangulated categories having Serre duality. We also give some applications, and in particular, we show that a conjecture given by Ringel holds true in this setting. [ABSTRACT FROM AUTHOR]

We solve the ∂ ¯ -equation for (p, q)-forms locally on any reduced pure-dimensional complex space and we prove an explicit version of Serre duality by introducing suitable concrete fine sheaves of certain (p, q)-currents. In particular this gives a condition for the ∂ ¯ -equation to be globally solvable. Our results also give information about holomorphic p-forms on singular spaces. [ABSTRACT FROM AUTHOR]

Let n ≥ 3 and let Ω be a bounded domain in Cn with a smooth negative plurisubharmonic exhaustion function φ. 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 in Ω can be extended to the whole domainΩ . To prove it, we combine an L2 version of Serre duality and Donnelly-Fefferman type estimates on (n,n−1)- and (n,n)-forms. [ABSTRACT FROM AUTHOR]

We characterize directs sums of twists of symmetric powers of the universal quotient bundle over the Grassmannian of lines. We use a method that could be used for analogue results on any arbitrary variety, and that should give stronger results than using the standard technique of Beilinson's spectral sequence. [ABSTRACT FROM AUTHOR]

A duality theorem for the stable module category of representations of a finite group scheme is proved. One of its consequences is an analogue of Serre duality, and the existence of Auslander–Reiten triangles for the p-local and p-torsion subcategories of the stable category, for each homogeneous prime ideal p in the cohomology ring of the group scheme. [ABSTRACT FROM AUTHOR]

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 $$n=2$$ . [ABSTRACT FROM AUTHOR]

Text We develop geometric methods to study the generating weights of free modules of vector-valued modular forms of half-integral weight, taking values in a complex representation of the metaplectic group. We then compute the generating weights for modular forms taking values in the Weil representation attached to cyclic quadratic modules of order 2 p r , where p ≥ 5 is a prime. We also show that the generating weights approach a simple limiting distribution as p grows, or as r grows and p remains fixed. Video For a video summary of this paper, please visit https://youtu.be/QNbPSXXKot4 . [ABSTRACT FROM AUTHOR]

For an additive category with a Serre duality and a finite group action, we compute explicitly the Serre duality on the category of equivariant objects. We prove that under certain conditions, the equivarianzation of an additive category with a periodic Serre duality still has a periodic Serre duality. A similar result is proved for fractionally Calabi-Yau triangulated categories. [ABSTRACT FROM AUTHOR]

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 certain fine sheaves B X n , q of currents on X of bidegree ( n , q ) , such that the Dolbeault complex ( B X n , • , ∂ ¯ ) becomes, in a certain sense, a dualizing complex. In particular, if X is Cohen–Macaulay then ( B X n , • , ∂ ¯ ) is an explicit fine resolution of the Grothendieck dualizing sheaf. [ABSTRACT FROM AUTHOR]

We study the role of the Serre functor in the theory of derived equivalences. Let A be an abelian category and let ( U , V ) be a t -structure on the bounded derived category D b A with heart H . We investigate when the natural embedding H → D b A can be extended to a triangle equivalence D b H → D b A . Our focus of study is the case where 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 U of the t -structure is closed under the Serre functor. [ABSTRACT FROM AUTHOR]

LetAbe an artin algebra. We show that the bounded homotopy category of finitely generated rightA-modules has Auslander–Reiten triangles. Two applications are given: (1) we provide an alternative proof of a theorem of Happel in [14]; (2) we prove that over a Gorenstein algebra, the bounded homotopy category of finitely generated Gorenstein projective (resp. injective) modules, admits Auslander–Reiten triangles, which improve a main result in [12]. [ABSTRACT FROM PUBLISHER]

We use duality in the manner of Serre to generalize a theorem of Hedenmalm on solution of the ∂ ¯ equation with inverse of the weight in Hörmander L 2 estimates. [ABSTRACT FROM AUTHOR]

The principal aim of this paper is to extend Abel’s theorem to the setting of complex supermanifolds of dimension 1 | q over a finite-dimensional local supercommutative C -algebra. The theorem is proved by establishing a compatibility of Serre duality for the supercurve with Poincaré duality on the reduced curve. We include an elementary algebraic proof of the requisite form of Serre duality, closely based on the account of the reduced case given by Serre in Algebraic groups and class fields , combined with an invariance result for the topology on the dual of the space of répartitions. Our Abel map, taking Cartier divisors of degree zero to the dual of the space of sections of the Berezinian sheaf, modulo periods, is defined via Penkov’s characterization of the Berezinian sheaf as the cohomology of the de Rham complex of the sheaf D of differential operators. We discuss the Jacobi inversion problem for the Abel map and give an example demonstrating that if n is an integer sufficiently large that the generic divisor of degree n is linearly equivalent to an effective divisor, this need not be the case for all divisors of degree n . [ABSTRACT FROM AUTHOR]

This is a report on recent work of Chałupnik and Touzé. We explain the Koszul duality for the category of strict polynomial functors and make explicit the underlying monoidal structure which seems to be of independent interest. Then we connect this to Ringel duality for Schur algebras and describe Serre duality for strict polynomial functors. [ABSTRACT FROM PUBLISHER]

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 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. [ABSTRACT FROM AUTHOR]

An L2 version of the Serre duality on domains in complex manifolds involving duality of Hilbert space realizations of the ∂-operator is established. This duality is used to study the solution of the ∂-equation with prescribed support. Applications are given to ∂-closed extension of forms, as well as to Bochner-Hartogs type extension of CR functions. [ABSTRACT FROM AUTHOR]

Abstract: We define holomorphic structures on canonical line bundles of the quantum projective space and identify their space of holomorphic sections. This determines the quantum homogeneous coordinate ring of the quantum projective space. We show that the fundamental class of is naturally presented by a twisted positive Hochschild cocycle. Finally, we verify the main statements of the Riemann–Roch formula and the Serre duality for and . [Copyright &y& Elsevier]

Abstract: We show that every k-linear abelian Ext-finite hereditary category with Serre duality which is generated by preprojective objects is derived equivalent to the category of representations of a strongly locally finite thread quiver. [Copyright &y& Elsevier]

Abstract: We introduce a notion of generalized Serre duality on a Hom-finite Krull–Schmidt triangulated category . This duality induces the generalized Serre functor on , which is a linear triangle equivalence between two thick triangulated subcategories of . Moreover, the domain of the generalized Serre functor is the smallest additive subcategory of containing all the indecomposable objects which appear as the third term of an Auslander–Reiten triangle in ; dually, the range of the generalized Serre functor is the smallest additive subcategory of containing all the indecomposable objects which appear as the first term of an Auslander–Reiten triangle in . We compute explicitly the generalized Serre duality on the bounded derived categories of artin algebras and of certain noncommutative projective schemes in the sense of Artin and Zhang. We obtain a characterization of Gorenstein algebras: an artin algebra A is Gorenstein if and only if the bounded homotopy category of finitely generated projective A-modules has Serre duality in the sense of Bondal and Kapranov. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

Let X be strongly q-convex domain of an n-dimensional Kähler manifold M and E be a holomorphic vector bundle over M. Then, if E satisfies certain positivity conditions, we prove vanishing theorems for the $\bar\partial$-cohomology groups of X with values in E. [ABSTRACT FROM AUTHOR]

The main result of the paper shows that, under Koszul duality between quiver algebras, cohomology of tails is identified with graded Vogel cohomology. As an application, a new proof of the noncommutative Serre duality over generalized Artin–Schelter regular Koszul quiver algebras is given. It is deduced from a similar formula over an arbitrary (i.e., not necessarily Koszul) Frobenius algebra, which turns out to be equivalent to the Auslander–Reiten formula. As another application, it is shown that, over a generalized Artin–Schelter regular Koszul quiver algebra, any algebra automorphism appearing in the noncommutative Serre duality formula is closely related, under Koszul duality, to the Nakayama automorphism of the Koszul-dual algebra. [Copyright &y& Elsevier]