Commutator theory, action groupoids, and an intrinsic Schreier-Mac Lane extension theorem

Beyond groups of automorphisms in the category Gp of groups and Lie-algebras of derivations in the category K-Lie of Lie algebras, there are structures of internal groupoids (called action groupoids) in both categories. They allow a synthesis of the notion of obstruction to extensions. This leads, in any pointed protomodular category C with split extension classifiers, to a general treatment of non-abelian extensions which can be understood as morphisms in a certain groupoid TorsC.     

The bar derived category of a curved dg algebra

Curved A_∞-algebras appear in nature as deformations of dg algebras. We develop the basic theory of curved A_∞-algebras and, in particular, curved dg algebras. We investigate their link with a suitable class of dg coalgebras via the bar construction and produce Quillen model structures on their module categories. We define the analogue of the relative derived category for a curved dg algebra.     

Mould calculus,polyhedral cones,and characters of combinatorial Hopf algebras

We describe a method for constructing characters of combinatorial Hopf algebras by means of integrals over certain polyhedral cones. This is based on ideas from resurgence theory, in particular on the construction of well-behaved averages induced by diffusion processes on the real line. We give several interpretations and proofs of the main result in terms of noncommutative symmetric and quasi-symmetric functions, as well as generalizations involving matrix quasi-symmetric functions. The interpretation of noncommutative symmetric functions as alien operators in resurgence theory is also discussed, and a new family of Lie idempotents of descent algebras is derived from this interpretation.     

Homology of generalized partition posets

We define a family of posets of partitions associated to an operad. We prove that the operad is Koszul if and only if the posets are Cohen-Macaulay. On the one hand, this characterization allows us to compute completely the homology of the posets. The homology groups are isomorphic to the Koszul dual cooperad. On the other hand, we get new methods for proving that an operad is Koszul.     

The phantom cover of a module

A morphism of left R-modules is a phantom morphism if for any morphism , with A finitely presented, the composition fg factors through a projective module. Equivalently, Tor₁(X,f)=0 for every right R-module X. It is proved that every R-module possesses a phantom cover, whose kernel is pure injective.If is the category of finitely presented right R-modules modulo projectives, then the association M?Tor₁(−,M) is a functor from the category of left R-modules to that of the flat functors on . The phantom cover is used to characterize when this functor is faithful or full. It is faithful if and only if the flat cover of every module has a pure injective kernel; this is equivalent to the flat cover being the phantom cover. The question of fullness is only reasonable when the functor is restricted to the subcategory of cotorsion modules. This restriction is full if and only if every phantom cover of a cotorsion module is pure injective.     

Cocycle deformations and Galois objects for semisimple Hopf algebras of dimension p3 and pq2

Let p and q be distinct prime numbers. We study the Galois objects and cocycle deformations of the noncommutative, noncocommutative, semisimple Hopf algebras of odd dimension $p^3$ and of dimension $p{q}^2$. We obtain that the $p+1$ non-isomorphic self-dual semisimple Hopf algebras of dimension $p^3$ classified by Masuoka have no non-trivial cocycle deformations, extending his previous results for the 8-dimensional Kac–Paljutkin Hopf algebra. This is done as a consequence of the classification of categorical Morita equivalence classes among semisimple Hopf algebras of odd dimension $p^3$, established by the third-named author in an appendix.     

Recipe theorem for the Tutte polynomial for matroids,renormalization group-like approach

Using a quantum field theory renormalization group-like differential equation, we give a new proof of the recipe theorem for the Tutte polynomial for matroids. The solution of such an equation is in fact given by some appropriate characters of the Hopf algebra of isomorphic classes of matroids, characters which are then related to the Tutte polynomial for matroids. This Hopf algebraic approach also allows to prove, in a new way, a matroid Tutte polynomial convolution formula appearing in [W. Kook, V. Reiner, D. Stanton, A convolution formula for the Tutte polynomial, J. Combin. Theory Ser. B 76 (1999) 297–300] and [G. Etienne, M. Las Vergnas, External and internal elements of a matroid basis, Discrete Math. 179 (1998) 111–119].     

Cluster-tilted algebras are Gorenstein and stably Calabi-Yau

We prove that in a 2-Calabi-Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its Cohen-Macaulay modules is 3-Calabi-Yau. We deduce in particular that cluster-tilted algebras are Gorenstein of dimension at most one, and hereditary if they are of finite global dimension. Our results also apply to the stable (!) endomorphism rings of maximal rigid modules of [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: math.RT/0503324, Invent. Math., in press]. In addition, we prove a general result about relative 3-Calabi-Yau duality over non-stable endomorphism rings. This strengthens and generalizes the Ext-group symmetries obtained in [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: math.RT/0503324, Invent. Math., in press] for simple modules. Finally, we generalize the results on relative Calabi-Yau duality from 2-Calabi-Yau to d-Calabi-Yau categories. We show how to produce many examples of d-cluster tilted algebras.     

Perpendicular categories of infinite dimensional partial tilting modules and transfers of tilting torsion classes

Let R be a ring and P be an (infinite dimensional) partial tilting module. We show that the perpendicular category of P is equivalent to the full module category where and ?R is the Bongartz complement of P modulo its P-trace. Moreover, there is a ring epimorphism φ:R→S. We characterize the case when φ is a perfect localization. By [Riccardo Colpi, Alberto Tonolo, Jan Trlifaj, Partial cotilting modules and the lattices induced by them, Comm. Algebra 25 (10) (1997) 3225-3237], there exist mutually inverse isomorphisms μ^′ and ν^′ between the interval in the lattice of torsion classes in , and the lattice of all torsion classes in . We provide necessary and sufficient conditions for μ^′ and ν^′ to preserve tilting torsion classes. As a consequence, we show that these conditions are always satisfied when R is a Dedekind domain, and if P is finitely presented and R is an artin algebra, then the conditions reduce to the trivial ones, namely that each value of μ^′ and ν^′ contains all injectives.     

When the heart of a faithful torsion pair is a module category

An abelian category with arbitrary coproducts and a small projective generator is equivalent to a module category (Mitchell (1964) [17]). A tilting object in an abelian category is a natural generalization of a small projective generator. Moreover, any abelian category with a tilting object admits arbitrary coproducts (Colpi et al. (2007) [8]). It naturally arises the question when an abelian category with a tilting object is equivalent to a module category. By Colpi et al. (2007) [8], the problem simplifies in understanding when, given an associative ring R and a faithful torsion pair (X,Y) in the category of right R-modules, the heartH(X,Y)of the t-structure associated with (X,Y) is equivalent to a category of modules. In this paper, we give a complete answer to this question, proving necessary and sufficient conditions on (X,Y) for H(X,Y) to be equivalent to a module category. We analyze in detail the case when R is right artinian.     

Dynamics of a thermoelastic von Kármán plate in a subsonic gas flow

We discuss the problem of non-linear oscillations of a clamped thermoelastic plate in a subsonic gas flow. The dynamics of the plate is described by von Kármán system in the presence of thermal effects. No mechanical damping is assumed. To describe the influence of the gas flow we apply the linearized theory of potential flows. Our main result states that each weak solution of the problem considered tends to the set of the stationary points of the problem. A similar problem was considered in [27], but with rotational inertia accounted for, i.e. with the additional term −αΔu_tt,α > 0, and the same result on stabilization was obtained. There was introduced the decomposition of the solution such that the one term tends to zero and the other is compact in special (“local energy”) topology. This decomposition enables us to prove the main result. But the case of rotational inertia neglected (α = 0) appears more difficult. Low a priori smoothness of u_t in the case α = 0 prevents us to construct such a decomposition. In order to prove additional smoothness of u_t we use analyticity of the corresponding thermoelastic semigroup proved in [25]. The isothermal variant of this problem with additional mechanical damping term −εΔu_t , ε > 0 was considered in [13] and stabilization to the set of stationary solutions to the problem was proved. The problem, considered in the present work can also be regarded as an extension of the result of [18] to the case when gas occupies an unbounded domain.     

Blowing up behavior for a class of nonlinear VIEs connected with parabolic PDEs

We study the blowing-up behavior of solutions of a class of nonlinear integral equations of Volterra type that is connected with parabolic partial differential equations with concentrated nonlinearities. We present some analytic results and, in the case of the kernel of Abel-kind with power nonlinearity and fixed initial data, we give a numerical approximation by using one-point collocation methods.     

Liftings of distributive lattices by locally matricial algebras with respect to the Id c functor

We study representations of distributive -lattices, considered as join-semilattices, by semilattices of finitely generated two-sided ideals of locally matricial algebras over a field k, aiming to find a functorial solution of the problem. We find simple examples of a finite subcategory of the category Ld of distributive -lattices and of a subcategory of L_d corresponding to a partially ordered class which cannot be lifted with respect to the Id_c functor. On the other hand, we prove that there is such a lifting of every diagram in L_d or of a subcategory L_d1 of L_d whose objects are all distributive -lattices and whose morphisms are -embeddings. This paper is dedicated to Walter Taylor. Received February 8, 2005; accepted in final form August 11, 2005. The work is a part of the research project MSM 0021620839 financed by MSMT and partly supported by INTAS project 03-51-4110, the grant GAUK 448/2004/B-MAT, and the post-doctoral grant GAČR 201/03/P140.     

The geometric realization of a simplicial Hausdorff space is Hausdorff

We show that the thin geometric realization of a simplicial Hausdorff space is Hausdorff. This proves a long-standing conjecture of Graeme Segal stating that the thin geometric realization of a simplicial k-space is a k-space.     

Tilting via torsion pairs and almost hereditary noetherian rings

We generalize the tilting process by Happel, Reiten and Smalø to the setting of finitely presented modules over right coherent rings. Moreover, we extend the characterization of quasi-tilted artin algebras as the almost hereditary ones to all right noetherian rings.     

A nondifferentiable extension of a theorem of Pucci and Serrin and applications

We study the multiplicity of critical points for functionals which are only differentiable along some directions. We extend to this class of functionals the three critical point theorem of Pucci and Serrin and we apply it to a one-parameter family of functionals Jλ, λ∈I⊂R. Under suitable assumptions, we locate an open subinterval of values λ in I for which Jλ possesses at least three critical points. Applications to quasilinear boundary value problems are also given.     

A Hardy field extension of Szemerédi's theorem

In 1975 Szemerédi proved that a set of integers of positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman showed in 1996 that the common difference of the arithmetic progression can be a square, a cube, or more generally of the form p(n) where p(n) is any integer polynomial with zero constant term. We produce a variety of new results of this type related to sequences that are not polynomial. We show that the common difference of the progression in Szemerédi's theorem can be of the form [nδ] where δ is any positive real number and [x] denotes the integer part of x. More generally, the common difference can be of the form [a(n)] where a(x) is any function that is a member of a Hardy field and satisfies a(x)/xk→∞ and a(x)/xk+1→0 for some non-negative integer k. The proof combines a new structural result for Hardy sequences, techniques from ergodic theory, and some recent equidistribution results of sequences on nilmanifolds.     

The formal theory of hopf algebras Part I: Hopf monoids in a monoidal category

Abstract

The category Hopf ? of Hopf monoids in a symmetric monoidal category ?, assumed to be locally finitely presentable as a category, is analyzed with respect to its categorical properties. Assuming that the functors “tensor squaring” and “tensor cubing” on ? preserve directed colimits one has the following results: (1) If, in ?, extremal epimorphisms are stable under tensor squaring, then Hopf C is locally presentable, coreflective in the category of bimonoids in ? and comonadic over the category of monoids in C. (2) If, in ?, extremal monomorphisms are stable under tensor squaring, then Hopf ? is locally presentable as well, reflective in the category of bimonoids in C and monadic over the category of comonoids in ?.     

On geometrically transitive Hopf algebroids

This paper contributes to the characterization of a certain class of commutative Hopf algebroids. It is shown that a commutative flat Hopf algebroid with a non zero base ring and a nonempty character groupoid is geometrically transitive if and only if any base change morphism is a weak equivalence (in particular, if any extension of the base ring is Landweber exact), if and only if any trivial bundle is a principal bi-bundle, and if and only if any two objects are fpqc locally isomorphic. As a consequence, any two isotropy Hopf algebras of a geometrically transitive Hopf algebroid (as above) are weakly equivalent. Furthermore, the character groupoid is transitive and any two isotropy Hopf algebras are conjugated. Several other characterizations of these Hopf algebroids in relation to transitive groupoids are also given.