Connected sequences of stable derived functors and their applications

Daniel Simson; Andrzej Tyc

Displaying similar documents to “Connected sequences of stable derived functors and their applications”

Taylor towers of symmetric and exterior powers

Brenda Johnson, Randy McCarthy (2008)

Fundamenta Mathematicae

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We study the Taylor towers of the nth symmetric and exterior power functors, Spⁿ and Λⁿ. We obtain a description of the layers of the Taylor towers, $D_{k} S p ⁿ$ and $D_{k} Λ ⁿ$ , in terms of the first terms in the Taylor towers of $S p^{t}$ and $Λ^{t}$ for t < n. The homology of these first terms is related to the stable derived functors (in the sense of Dold and Puppe) of $S p^{t}$ and $Λ^{t}$ . We use stable derived functor calculations of Dold and Puppe to determine the lowest nontrivial homology groups for $D_{k} S p ⁿ$ and $D_{k} Λ ⁿ$ .

Bipartite coalgebras and a reduction functor for coradical square complete coalgebras

Justyna Kosakowska, Daniel Simson (2008)

Colloquium Mathematicae

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Let C be a coalgebra over an arbitrary field K. We show that the study of the category C-Comod of left C-comodules reduces to the study of the category of (co)representations of a certain bicomodule, in case C is a bipartite coalgebra or a coradical square complete coalgebra, that is, C = C₁, the second term of the coradical filtration of C. If C = C₁, we associate with C a K-linear functor $ℍ_{C} : C - C o m o d \to H_{C} - C o m o d$ that restricts to a representation equivalence $ℍ_{C} : C - c o m o d \to H_{C} - c o m o d_{s p}^{•}$ , where $H_{C}$ is a coradical square complete hereditary...

Compatibility of the theta correspondence with the Whittaker functors

Vincent Lafforgue, Sergey Lysenko (2011)

Bulletin de la Société Mathématique de France

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We prove that the global geometric theta-lifting functor for the dual pair $(H, G)$ is compatible with the Whittaker functors, where $(H, G)$ is one of the pairs $({S 𝕆}_{2 n}, {𝕊 p}_{2 n})$ , $({𝕊 p}_{2 n}, {S 𝕆}_{2 n + 2})$ or $({𝔾 L}_{n}, {𝔾 L}_{n + 1})$ . That is, the composition of the theta-lifting functor from $H$ to $G$ with the Whittaker functor for $G$ is isomorphic to the Whittaker functor for $H$ .

Troesch complexes and extensions of strict polynomial functors

Antoine Touzé (2012)

Annales scientifiques de l'École Normale Supérieure

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We develop a new approach of extension calculus in the category of strict polynomial functors, based on Troesch complexes. We obtain new short elementary proofs of numerous classical $Ext$ -computations as well as new results. In particular, we get a cohomological version of the “fundamental theorems” from classical invariant theory for $G L_{n}$ for $n$ big enough (and we give a conjecture for smaller values of $n$ ). We also study the “twisting spectral sequence” $E^{s, t} (F, G, r)$ converging to the extension groups...

Limits and colimits in certain categories of spaces of continuous functions

Marvin W. Grossman

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CONTENTSIntroduction................................................................................................................................................................................5§ 1. Notation and preliminaries.............................................................................................................................................6§ 2. Epimorphisms and monomorphisms.........................................................................................................................7§...

Two results of $n$ -exangulated categories

Jian He, Jing He, Panyue Zhou (2024)

Czechoslovak Mathematical Journal

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M. Herschend, Y. Liu, H. Nakaoka introduced $n$ -exangulated categories, which are a simultaneous generalization of $n$ -exact categories and $(n + 2)$ -angulated categories. This paper consists of two results on $n$ -exangulated categories: (1) we give an equivalent characterization of axiom (EA2); (2) we provide a new way to construct a closed subfunctor of an $n$ -exangulated category.

The Roquette category of finite $p$ -groups

Serge Bouc (2015)

Journal of the European Mathematical Society

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Let $p$ be a prime number. This paper introduces the Roquette category $ℛ_{p}$ of finite $p$ -groups, which is an additive tensor category containing all finite $p$ -groups among its objects. In $ℛ_{p}$ , every finite $p$ -group $P$ admits a canonical direct summand $\partial P$ , called the edge of $P$ . Moreover $P$ splits uniquely as a direct sum of edges of Roquette $p$ -groups, and the tensor structure of $ℛ_{p}$ can be described in terms of such edges. The main motivation for considering this category is that the additive functors...

Stable solutions of $- Δ u = f (u)$ in $ℝ^{N}$

Louis Dupaigne, Alberto Farina (2010)

Journal of the European Mathematical Society

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Several Liouville-type theorems are presented for stable solutions of the equation $- Δ u = f (u)$ in $ℝ^{N}$ , where $f > 0$ is a general convex, nondecreasing function. Extensions to solutions which are merely stable outside a compact set are discussed.

On almost complex structures from classical linear connections

Jan Kurek, Włodzimierz M. Mikulski (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $ℳ f_{m}$ be the category of $m$ -dimensional manifolds and local diffeomorphisms and let $T$ be the tangent functor on $ℳ f_{m}$ . Let $𝒱$ be the category of real vector spaces and linear maps and let $𝒱_{m}$ be the category of $m$ -dimensional real vector spaces and linear isomorphisms. We characterize all regular covariant functors $F : 𝒱_{m} \to 𝒱$ admitting $ℳ f_{m}$ -natural operators $\tilde{J}$ transforming classical linear connections $\nabla$ on $m$ -dimensional manifolds $M$ into almost complex structures $\tilde{J} (\nabla)$ on $F (T) M = ⋃_{x \in M} F (T_{x} M)$ .

$A F$ -algebras and topology of mapping tori

Igor Nikolaev (2015)

Czechoslovak Mathematical Journal

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The paper studies applications of $C^{*}$ -algebras in geometric topology. Namely, a covariant functor from the category of mapping tori to a category of $A F$ -algebras is constructed; the functor takes continuous maps between such manifolds to stable homomorphisms between the corresponding $A F$ -algebras. We use this functor to develop an obstruction theory for the torus bundles of dimension $2$ , $3$ and $4$ . In conclusion, we consider two numerical examples illustrating our main results.

On the structure of triangulated categories with finitely many indecomposables

Claire Amiot (2007)

Bulletin de la Société Mathématique de France

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We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field $k$ . We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category $𝒯$ is of the form $ℤ Δ / G$ where $Δ$ is a disjoint union of simply-laced Dynkin diagrams and $G$ a weakly admissible group of automorphisms of $ℤ Δ$ . Then we prove that for ‘most’ groups $G$ , the category $𝒯$ is standard, ...

Generic representations of orthogonal groups: projective functors in the category $ℱ_{q u a d}$

Christine Vespa (2008)

Fundamenta Mathematicae

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We continue the study of the category of functors $ℱ_{q u a d}$ , associated to ₂-vector spaces equipped with a nondegenerate quadratic form, initiated in J. Pure Appl. Algebra 212 (2008) and Algebr. Geom. Topology 7 (2007). We define a filtration of the standard projective objects in $ℱ_{q u a d}$ ; this refines to give a decomposition into indecomposable factors of the first two standard projective objects in $ℱ_{q u a d}$ : $P_{H ₀}$ and $P_{H ₁}$ . As an application of these two decompositions, we give a complete description of the polynomial...

Hall algebras of two equivalent extriangulated categories

Shiquan Ruan, Li Wang, Haicheng Zhang (2024)

Czechoslovak Mathematical Journal

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For any positive integer $n$ , let $A_{n}$ be a linearly oriented quiver of type $A$ with $n$ vertices. It is well-known that the quotient of an exact category by projective-injectives is an extriangulated category. We show that there exists an extriangulated equivalence between the extriangulated categories $ℳ_{n + 1}$ and $ℱ_{n}$ , where $ℳ_{n + 1}$ and $ℱ_{n}$ are the two extriangulated categories corresponding to the representation category of $A_{n + 1}$ and the morphism category of projective representations of $A_{n}$ , respectively. As a...

On category $𝒪$ for cyclotomic rational Cherednik algebras

Iain G. Gordon, Ivan Losev (2014)

Journal of the European Mathematical Society

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We study equivalences for category $𝒪_{p}$ of the rational Cherednik algebras $𝐇_{p}$ of type $G_{ℓ} (n) = {(μ_{ℓ})}^{n} ⋊ 𝔖_{n}$ : a highest weight equivalence between $𝒪_{p}$ and $𝒪_{σ (p)}$ for $σ \in 𝔖_{ℓ}$ and an action of $𝔖_{ℓ}$ on an explicit non-empty Zariski open set of parameters $p$ ; a derived equivalence between $𝒪_{p}$ and $𝒪_{p^{'}}$ whenever $p$ and $p^{'}$ have integral difference; a highest weight equivalence between $𝒪_{p}$ and a parabolic category $𝒪$ for the general linear group, under a non-rationality assumption on the parameter $p$ . As a consequence, we confirm special cases...

Stabilization of monomial maps in higher codimension

Jan-Li Lin, Elizabeth Wulcan (2014)

Annales de l’institut Fourier

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A monomial self-map $f$ on a complex toric variety is said to be $k$ -stable if the action induced on the $2 k$ -cohomology is compatible with iteration. We show that under suitable conditions on the eigenvalues of the matrix of exponents of $f$ , we can find a toric model with at worst quotient singularities where $f$ is $k$ -stable. If $f$ is replaced by an iterate one can find a $k$ -stable model as soon as the dynamical degrees $λ_{k}$ of $f$ satisfy $λ_{k}^{2} > λ_{k - 1} λ_{k + 1}$ . On the other hand, we give examples of monomial maps $f$ , where...

The general rigidity result for bundles of $A$ -covelocities and $A$ -jets

Jiří M. Tomáš (2017)

Czechoslovak Mathematical Journal

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Let $M$ be an $m$ -dimensional manifold and $A = 𝔻_{k}^{r} / I = ℝ \oplus N_{A}$ a Weil algebra of height $r$ . We prove that any $A$ -covelocity $T_{x}^{A} f \in T_{x}^{A *} M$ , $x \in M$ is determined by its values over arbitrary $max {width A, m}$ regular and under the first jet projection linearly independent elements of $T_{x}^{A} M$ . Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result $T^{A *} M ≃ T^{r *} M$ without coordinate computations, which improves and generalizes the partial...

Definable orthogonality classes in accessible categories are small

Joan Bagaria, Carles Casacuberta, A. R. D. Mathias, Jiří Rosický (2015)

Journal of the European Mathematical Society

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We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopěnka’s principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Lévy hierarchy. For example, the statement that, for a class $𝒮$ of morphisms in a locally presentable category $𝒞$ of structures, the orthogonal class of objects...

The correspondence between Barsotti-Tate groups and Kisin modules when $p = 2$

Tong Liu (2013)

Journal de Théorie des Nombres de Bordeaux

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Let $K$ be a finite extension over $ℚ_{2}$ and $𝒪_{K}$ the ring of integers. We prove the equivalence of categories between the category of Kisin modules of height 1 and the category of Barsotti-Tate groups over $𝒪_{K}$ .

Coleff-Herrera currents, duality, and noetherian operators

Mats Andersson (2011)

Bulletin de la Société Mathématique de France

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Let $ℐ$ be a coherent subsheaf of a locally free sheaf $𝒪 (E_{0})$ and suppose that $ℱ = 𝒪 (E_{0}) / ℐ$ has pure codimension. Starting with a residue current $R$ obtained from a locally free resolution of $ℱ$ we construct a vector-valued Coleff-Herrera current $μ$ with support on the variety associated to $ℱ$ such that $φ$ is in $ℐ$ if and only if $μ φ = 0$ . Such a current $μ$ can also be derived algebraically from a fundamental theorem of Roos about the bidualizing functor, and the relation between these two approaches is discussed....

Hardness of embedding simplicial complexes in $ℝ^{d}$

Jiří Matoušek, Martin Tancer, Uli Wagner (2011)

Journal of the European Mathematical Society

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Let ${𝙴𝙼𝙱𝙴𝙳}_{k \to d}$ be the following algorithmic problem: Given a ﬁnite simplicial complex $K$ of dimension at most $k$ , does there exist a (piecewise linear) embedding of $K$ into $ℝ^{d}$ ? Known results easily imply polynomiality of ${𝙴𝙼𝙱𝙴𝙳}_{k \to 2}$ ( $k = 1, 2$ ; the case $k = 1, d = 2$ is graph planarity) and of ${𝙴𝙼𝙱𝙴𝙳}_{k \to 2 k}$ for all $k \geq 3$ . We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that ${𝙴𝙼𝙱𝙴𝙳}_{d \to d}$ and ${𝙴𝙼𝙱𝙴𝙳}_{(d - 1) \to d}$ are undecidable for each $_{d} \geq 5$ . Our main result is NP-hardness of ${𝙴𝙼𝙱𝙴𝙳}_{2 \to 4}$ and, more generally, of ${𝙴𝙼𝙱𝙴𝙳}_{k \to d}$ for all...

Simultaneous stabilization in $A_{ℝ} ()$

Raymond Mortini, Brett D. Wick (2009)

Studia Mathematica

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We study the problem of simultaneous stabilization for the algebra $A_{ℝ} ()$ . Invertible pairs $(f_{j}, g_{j})$ , j = 1,..., n, in a commutative unital algebra are called simultaneously stabilizable if there exists a pair (α,β) of elements such that $α f_{j} + β g_{j}$ is invertible in this algebra for j = 1,..., n. For n = 2, the simultaneous stabilization problem admits a positive solution for any data if and only if the Bass stable rank of the algebra is one. Since $A_{ℝ} ()$ has stable rank two, we are faced here with a different...

Geometric theta-lifting for the dual pair ${𝕊𝕆}_{2 m}, 𝕊 p_{2 n}$

Sergey Lysenko (2011)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be a smooth projective curve over an algebraically closed field of characteristic $> 2$ . Consider the dual pair $H = {SO}_{2 m}, G = {Sp}_{2 n}$ over $X$ with $H$ split. Write ${Bun}_{G}$ and ${Bun}_{H}$ for the stacks of $G$ -torsors and $H$ -torsors on $X$ . The theta-kernel ${Aut}_{G, H}$ on ${Bun}_{G} \times {Bun}_{H}$ yields theta-lifting functors $F_{G} : D ({Bun}_{H}) \to D ({Bun}_{G})$ and $F_{H} : D ({Bun}_{G}) \to D ({Bun}_{H})$ between the corresponding derived categories. We describe the relation of these functors with Hecke operators. In two particular cases these functors realize the geometric Langlands functoriality for the above pair (in the non...