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Given Y a non-compact manifold or orbifold, we define a natural subspace of the cohomology of Y called the narrow cohomology. We show that despite Y being non-compact, there is a well-defined and non-degenerate pairing on this subspace. The narrow cohomology proves useful for the study of genus zero Gromov-Witten theory. When Y is a smooth complex variety or Deligne-Mumford stack, one can define a quantum D-module on the narrow cohomology of Y. This yields a new formulation of quantum Serre duality.

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.

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 Mn 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.

In this text we develop some aspects of Harder–Narasimhan theory, slopes, semistability and canonical filtration, in the setting of combinatorial lattices. Of noticeable importance is the Harder–Narasimhan structure associated to a Galois connection between two lattices. It applies, in particular, to matroids. We then specialize this to linear codes. This could be done from at least three different approaches: using the sphere-packing analogy, or the geometric view, or the Galois connection construction just introduced. A remarkable fact is that these all lead to the same notion of semistability and canonical filtration. Relations to previous propositions toward a classification of codes, and to Wei's generalized Hamming weight hierarchy, are also discussed. Last, we study the two important questions of the preservation of semistability (or more generally the behavior of slopes) under duality, and under tensor product. The former essentially follows from Wei's duality theorem for higher weights—and its matroid version—which we revisit in an appendix, developing analogues of the Riemann–Roch, Serre duality, Clifford, and gap and gonality sequence theorems. Likewise the latter is closely related to the bound on higher weights of a tensor product, conjectured by Wei and Yang, and proved by Schaathun in the geometric language, which we reformulate directly in terms of codes. From this material we then derive semistability of tensor product.

Let $k$ be an algebraically closed field of positive characteristic. For any integer $m \geq 2$, we show that the Hodge numbers of a smooth projective $k$-variety can take on any combination of values modulo $m$, subject only to Serre duality. In particular, there are no non-trivial polynomial relations between the Hodge numbers., Published version. 15 pages

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 .

Let $k$ be a field of positive characteristic. We prove that the only linear relations between the Hodge numbers $h^{i,j}(X) = \dim H^j(X,\Omega_X^i)$ that hold for every smooth proper variety $X$ over $k$ are the ones given by Serre duality. We also show that the only linear combinations of Hodge numbers that are birational invariants of $X$ are given by the span of the $h^{i,0}(X)$ and the $h^{0,j}(X)$ (and their duals $h^{i,n}(X)$ and $h^{n,j}(X)$). The corresponding statements for compact K\"ahler manifolds were proven by Kotschick and Schreieder., Comment: 16 pages

Exceptional cycles in a triangulated category $\mathcal T$ with Serre duality, introduced by N. Broomhead, D. Pauksztello, and D. Ploog, have a notable impact on the global structure of $\mathcal T$. In this paper we show that if $\mathcal T$ is homotopy-like, then any exceptional $1$-cycle is indecomposable and at the mouth; and any object in an exceptional $n$-cycle with $n\ge 3$ is at the mouth. Let $A$ be an indecomposable gentle $k$-algebra with $A\ne k$. The Hom spaces between string complexes at the mouth are explicitly determined. The main result classifies "almost all" the exceptional cycles in $K^b(A\mbox{-}{\rm proj})$, using characteristic components and their AG-invariants, except those exceptional $1$-cycles which are band complexes. Namely, the mouth of a characteristic component $C$ of $K^b(A\mbox{-}{\rm proj})$ forms a unique exceptional cycle in $C$, up to an equivalent relation $\approx$; if the quiver of $A$ is not of type $A_3$, this gives all the exceptional $n$-cycle in $K^b(A\mbox{-}{\rm proj})$ with $n\ge 2$, up to $\approx$; and a string complex is an exceptional $1$-cycle if and only if it is at the mouth of a characteristic component with {\rm AG}-invariant $(1, m)$. However, a band complex at the mouth is possibly not an exceptional $1$-cycle., 29 pages

Let 𝒟 be a Krull–Schmidt, Hom-finite triangulated category with a Serre functor and a cluster-tilting object T. We introduce the notion of an FΛ-stable support τ-tilting module, induced by the shift functor and the Auslander–Reiten translation, in the cluster-tilted algebra Λ=End𝒟op(T). We show that there exists a bijection between basic cluster-tilting objects in 𝒟 and basic FΛ-stable support τ-tilting Λ-modules. This generalizes a result of Adachi–Iyama–Reiten [1]. As a consequence, we obtain that all cluster-tilting objects in 𝒟 have the same number of nonisomorphic indecomposable direct summands.

Let A be an artin algebra. We show that the bounded homotopy category of finitely generated right A-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].

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.

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 $\mathfrak{p}$-local and $\mathfrak{p}$-torsion subcategories of the stable category, for each homogeneous prime ideal $\mathfrak{p}$ in the cohomology ring of the group scheme., Comment: 24 pages. This version corrects a mistake in the statement of Theorem 3.1; see also Theorem 1.4, and Examples 3.6 and 3.7. References have been updated

We prove that a Hom-finite additive category having determined morphisms on both sides is a dualizing variety. This complements a result by Krause. We prove that in a Hom-finite abelian category having Serre duality, a morphism is right determined by some object if and only if it is an epimorphism. We give a characterization to abelian categories having Serre duality via determined morphisms.

For a (right and left) coherent ring $A$, we show that there exists a duality between homotopy categories ${\mathbb{K}}^{{\rm{b}}}({\rm mod}{\mbox{-}}A^{{\rm op}})$ and ${\mathbb{K}}^{{\rm{b}}}({\rm mod}{\mbox{-}}A)$. If $A=\Lambda$ is an artin algebra of finite global dimension, this duality restricts to a duality between their subcategories of acyclic complexes, ${\mathbb{K}}^{{\rm{b}}}_{\rm ac}({\rm mod}{\mbox{-}}\Lambda^{\rm op})$ and ${\mathbb{K}}^{{\rm{b}}}_{\rm ac}({\rm mod}{\mbox{-}}\Lambda).$ As a result, it will be shown that, in this case, ${\mathbb{K}}_{\rm ac}^{{\rm{b}}}({\rm mod}{\mbox{-}}\Lambda)$ admits a Serre functor and hence has Auslander-Reiten triangles., Comment: arXiv admin note: text overlap with arXiv:1605.04745

Recent research has demonstrated a connection between Weil-etale cohomology and special values of zeta functions. In particular, Lichtenbaum has shown that the order of vanishing and leading coefficient of the zeta function of a smooth, projective variety over a finite field has a Weil-etale cohomological interpretation in terms of certain secondary Euler characteristics. These results rely on a duality theorem stated in terms of cup-product in Weil-etale cohomology. We define Weil-etale cohomology for varieties over p-adic fields, and prove a duality theorem for the cohomology of G m on a smooth, proper, geometrically connected curve of index 1. This duality theorem is a p-adic analogue of Lichtenbaum's Weil-etale duality theorem for curves over finite fields, as well as a Weil-etale analogue of his classical duality theorem for curves over p-adic fields. Finally, we show that our duality theorem implies this latter classical duality theorem for index 1 curves.

We establish duality and vanishing results for the cohomology of the Weil group of a p-adic field. Among them is a duality theorem for finitely generated modules, which implies Tate–Nakayama Duality. We prove comparison results with Galois cohomology, which imply that the cohomology of the Weil group determines that of the Galois group. When the module is defined by an abelian variety, we use these comparison results to establish a duality theorem analogous to Tateʼs duality theorem for abelian varieties over p-adic fields.

In this note we recall the history of Serre's problem and its beautiful solution given by D. Quillen in 1976

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., Comment: 32 pages, 12 pictures

application/pdf, We introduce a notion of Gorenstein algebras of codimension c and demonstrate that Serre duality theory plays an essential role in the theory of derived equivalences for Gorenstein algebras.

The membership of the local cohomology modules Han(M) of a module M in certain Serre subcategories of the category of modules is studied from below (in). Generalizations of depth and regular sequences are defined. The relation of these notions to local cohomology are found. It is shown that the membership of the local cohomology modules of a finite module in a Serre subcategory in the upper range just depends on the support of the module.

We will prove a Riemann–Roch like theorem for triangulated categories satisfying Serre duality. As an application, we will prove Riemann–Roch Theorem and Adjunction Formula for noncommutative Cohen–Macaulay surfaces in terms of sheaf cohomology. We will also show that the formulas hold for stable categories over Koszul connected graded algebras in terms of Tate–Vogel cohomology by extending the BGG correspondence.

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.

Convolution structures are group-like objects that were extensively studied by harmonic analysts. We use them to define H0 and H1 for Arakelov divisors over number fields. We prove the analogs of the Riemann–Roch and Serre duality theorems. This brings more structure to the works of Tate and van der Geer and Schoof. The H1 is defined by a procedure very similar to the usual Ĉech cohomology. Serre′s duality becomes Pontryagin duality of convolution structures. The whole theory is parallel to the geometric case.

We prove that for an algebraic curvature tensor on a pseudo-Euclidean space, the Jordan-Osserman condition implies the Raki\'c duality principle, and that the Osserman condition and the duality principle are equivalent in the diagonalisable case., Comment: 5 pages

Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated N n-graded module. The functors associated with Alexander duality provide a duality on the level of free and injective resolutions, and numerous Bass and Betti number relations result as corollaries. A minimal injective resolution of a module M is equivalent to the injective resolution of its Alexander dual and contains all of the maps in the minimal free resolution of M over every Z n-graded localization. Results are obtained on the interaction of duality for resolutions with cellular resolutions and lcm-lattices. Using injective resolutions, theorems of Eagon, Reiner, and Terai are generalized to all N n-graded modules: the projective dimension of M equals the support-regularity of its Alexander dual, and M is Cohen–Macaulay if and only if its Alexander dual has a support-linear free resolution. Alexander duality is applied in the context of the Z n-graded local cohomology functors HiI(−) for squarefree monomial ideals I in the polynomial ring S, proving a duality directly generalizing local duality, which is the case when I = m is maximal. In the process, a new flat complex for calculating local cohomology at monomial ideals is introduced, showing, as a consequence, that Terai's formula for the Hilbert series of HiI(S) is equivalent to Hochster's for Hn − i m (S/I).

A global algebraic description of dualizing and residual complexes for projective varieties in terms of generalized fractions is given and applied to the study of zero-dimensional schemes.

Cohomology rings of finite groups have strong duality properties, as shown by 4 and 16 . We prove here that cohomology rings of virtual duality groups have a ring theoretic duality property, which combines the duality properties of finite groups with the cohomological duality of the subgroup of finite index. The formal behaviour of the local cohomology theorem is precisely analogous to that for a compact Lie group 6 , except that the dimension appears to be negative.

We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a duality statement between the cone of pseudo-effective divisors and the cone of ``movable curves'', which is obtained from a general theory of movable intersections and approximate Zariski decomposition for closed positive (1,1)-currents. As a corollary, a projective manifold has a pseudo-effective canonical bundle if and only if it is is not uniruled. We also prove that a 4-fold with a canonical bundle which is pseudo-effective and of numerical class zero in restriction to curves of a covering family, has non negative Kodaira dimension., Comment: 39 pages

We study the duality between corings and ring extensions. We construct a new category with a self-dual functor acting on it, which extends that duality. This construction can be seen as the non-commutative case of another duality extension: the duality between finite dimensional algebras and coalgebra. Both these duality extensions have some similarities with the Pontryagin-van Kampen duality theorem.

This is a report on recent work of Chalupnik and Touze. 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., 23 pages. Version 2: Added correct assumptions for Serre duality and included further references. Version 3: Added dedication and made minor simplifications and corrections. Version 4: Final version (slightly updated) accepted for publication in Compositio Mathematica

A topological duality is developed for a wide class of lattice ordered algebraic structures by introducing in an ordered Stone space a natural binary and continuous function. In particular, duality theorems are obtained for l-groups and for abelian l-groups.

In this paper, we consider a primal–dual infinite linear programming problem-pair, i.e. LPs on infinite dimensional spaces with infinitely many constraints. We present two duality theorems for the problem-pair: a weak and a strong duality theorem. We do not assume any topology on the vector spaces, therefore our results are algebraic duality theorems. As an application, we consider transferable utility cooperative games with arbitrarily many players.

In this paper, we formulate and prove a derived category version of Poitou-Tate duality on Galois cohomology of compact modules (with a continuous Galois action) over a pro-p ring, where p is a prime., Comment: 45 pages: final version

We study Serre duality in the singularity category of an isolated Gorenstein singularity and find an explicit formula for the duality pairing in terms of generalised fractions and residues. For hypersurfaces we recover the residue formula of the string theorists Kapustin and Li. These results are obtained from an explicit construction of complete injective resolutions of maximal Cohen-Macaulay modules., Comment: 28 pages, the exposition has been simplified and shortened

Let k be an algebraically closed field and A a k-linear hereditary category satisfying Serre duality with no infinite radicals between the preprojective objects. If A is generated by the preprojective objects, then we show that A is derived equivalent to rep_k Q for a so called strongly locally finite quiver Q. To this end, we introduce light cone distances and round trip distances on quivers which will be used to investigate sections in stable translation quivers of the form \mathbb{Z} Q., Comment: 16 pages, as accepted by Journal of Pure and Applied Algebra

Using that finite topological spaces are just finite orders, we develop a duality theory for sheaves of Abelian groups over finite spaces following closely Grothendieck's duality theory for coherent sheaves over proper schemes. Since the geometric realization of a finite space is a polyhedron, we relate this duality with the duality theory for Abelian sheaves over polyhedra.

We introduce a notion of generalized Serre duality on a Hom-finite Krull-Schmidt triangulated category $\mathcal{T}$. This duality induces the generalized Serre functor on $\mathcal{T}$, which is a linear triangle equivalence between two thick triangulated subcategories of $\mathcal{T}$. Moreover, the domain of the generalized Serre functor is the smallest additive subcategory of $\mathcal{T}$ containing all the indecomposable objects which appear as the third term of an Auslander-Reiten triangle in $\mathcal{T}$; dually, the range of the generalized Serre functor is the smallest additive subcategory of $\mathcal{T}$ containing all the indecomposable objects which appear as the first term of an Auslander-Reiten triangle in $\mathcal{T}$. 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.

A duality theorem is formulated for noncommutative association schemes. This duality theorem contains as special cases (1) the Delsarte-Tamaschke duality theorem (which was essentially obtained by Kawala in 1942) for commutative association schemes, and (2) the Tannaka-Krein duality theorem for arbitrary finite groups.

We use the anti-equivalence between Cohen-Macaulay complexes and coherent sheaves on formal schemes to shed light on some older results and prove new results. We bring out the relations between a coherent sheaf M satisfying an S_2 condition and the lowest cohomology N of its "dual" complex. We show that if a scheme has a Gorenstein complex satisfying certain coherence conditions, then in a finite \'etale neighborhood of each point, it has a dualizing complex. If the scheme already has a dualizing complex, then we show that the Gorenstein complex must be a tensor product of a dualizing complex and a vector bundle of finite rank. We relate the various results in [S] on Cousin complexes to dual results on coherent sheaves on formal schemes., Comment: 40 pages. Substantially different from earlier version(s). To appear in Journal of Algebra

A possible generalization of the Serre problem (Quillen–Suslin theorem) on the freeness of projective modules over polynomial rings, which seems to be of interest in systems theory applications, is shown to follow from the solution to the original Serre problem.

Motivated by constructions in the representation theory of finite dimensional algebras we generalize the notion of Artin–Schelter regular algebras of dimension n to algebras and categories to include Auslander algebras and a graded analogue for infinite representation type. A generalized Artin–Schelter regular algebra or a category of dimension n is shown to have common properties with the classical Artin–Schelter regular algebras. In particular, when they admit a duality, then they satisfy Serre duality formulas and the Ext-category of nice sets of simple objects of maximal projective dimension n is a finite length Frobenius category.

Every duality between a full subcategory B of topological R-modules and a full subcategory D of Mod-S which contains S and is closed under isomorphisms is determined in a natural way by a bimodule RUS. This article examines the situation when D=Mod-S, in which case one gets a duality that may be considered as being one-half of a Morita duality.

We show that from the representation theory of a locally non-trivial compact groupoid whose irreducible representations separate its points, one can reconstruct the groupoid using a procedure similar to the Tannaka-Krein duality for compact groups. We study the Fourier and Fourier-Plancherel transforms and prove the Plancherel theorem for compact groupoids. Mathematics subject classification (2010): Primary 43A65, Secondary 43A40.