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Let \pi \colon Y\to X denote the canonical resolution of the two dimensional Kleinian singularity X of type ADE. In the present paper, we establish isomorphisms between the cohomological and K-theoretical Hall algebras of \omega-semistable properly supported sheaves on Y with fixed slope \mu and \zeta-semistable finite-dimensional representations of the preprojective algebra of affine type ADE of slope zero respectively, under some conditions on \zeta depending on the polarization \omega and \mu. These isomorphisms are induced by the derived McKay correspondence. In addition, they are interpreted as decategorified versions of a monoidal equivalence between the corresponding categorified Hall algebras. In the type A case, we provide a finer description of the cohomological, K-theoretical and categorified Hall algebra of \omega-semistable properly supported sheaves on Y with fixed slope \mu: for example, in the cohomological case, the algebra can be given in terms of Yangians of finite type ADE Dynkin diagrams. [ABSTRACT FROM AUTHOR]

Symmetric homology is a natural generalization of cyclic homology, in which symmetric groups play the role of cyclic groups. In the case of associative algebras, the symmetric homology theory was introduced by Z. Fiedorowicz (1991) and was further developed in the work of S. Ault (2010). In this paper, we show that, for algebras defined over a field of characteristic 0, the symmetric homology is naturally equivalent to the (one-dimensional) representation homology introduced by the authors in joint work with G. Khachatryan (2013). Using known results on representation homology, we compute symmetric homology explicitly for basic algebras, such as polynomial algebras and universal enveloping algebras of (DG) Lie algebras. As an application, we prove two conjectures of Ault and Fiedorowicz, including their main conjecture (2007) on topological interpretation of symmetric homology of polynomial algebras. [ABSTRACT FROM AUTHOR]

In the classical Batalin–Vilkovisky formalism, the BV operator Δ is a differential operator of order two with respect to the commutative product. In the differential graded setting, it is known that if the BV operator is homotopically trivial, then there is a tree level cohomological field theory induced on the homology; this is a manifestation of the fact that the homotopy quotient of the operad of BV algebras by Δ is represented by the operad of hypercommutative algebras. In this paper, we study generalized Batalin–Vilkovisky algebras where the operator Δ is of the given finite order. In that case, we unravel a new interesting algebraic structure on the homology whenever Δ is homotopically trivial. We also suggest that the sequence of algebraic structures arising in the higher order formalism is a part of a "trinity" of remarkable mathematical objects, fitting the philosophy proposed by Arnold in the 1990s. [ABSTRACT FROM AUTHOR]

Hyperoctahedral homology is the homology theory associated to the hyperoctahedral crossed simplicial group. It is defined for involutive algebras over a commutative ring using functor homology and the hyperoctahedral bar construction of Fiedorowicz. The main result of the paper proves that hyper- octahedral homology is related to equivariant stable homotopy theory: for a discrete group of odd order, the hyperoctahedral homology of the group algebra is isomorphic to the homology of the fixed points under the involution of an equivariant infinite loop space built from the classifying space of the group. [ABSTRACT FROM AUTHOR]

In this paper we introduce a class of noncommutative (finitely generated) monomial algebras whose Hilbert series are algebraic functions. We use the concept of graded homology and the theory of unambiguous context-free grammars for this purpose. We also provide examples of finitely presented graded algebras whose corresponding leading monomial algebras belong to the proposed class and hence possess algebraic Hilbert series. [ABSTRACT FROM AUTHOR]

Let ℓ be a commutative ring with unit. Garkusha constructed a functor from the category of ℓ -algebras into a triangulated category D , that is a universal excisive and homotopy invariant homology theory. Later on, he provided different descriptions of D , as an application of his motivic homotopy theory of algebras. Using these, it can be shown that D is triangulated equivalent to a category, denote it by K , whose objects are pairs (A , m) with A an ℓ -algebra and m an integer, and whose Hom-sets can be described in terms of homotopy classes of morphisms. All these computations, however, require a heavy machinery of homotopy theory. In this paper, we give a more explicit construction of the triangulated category K and prove its universal property, avoiding the homotopy-theoretic methods and using instead the ones developed by Cortiñas-Thom for defining kk -theory. Moreover, we give a new description of the composition law in K , mimicking the one in the suspension-stable homotopy category of bornological algebras defined by Cuntz-Meyer-Rosenberg. We also prove that the triangulated structure in K can be defined using either extension or mapping path triangles. [ABSTRACT FROM AUTHOR]

In this paper, we study an excision theorem of the dihedral and reflexive (co)homology theory of associative algebras. That is, for such an extension, we obtain a six-term exact sequence in the dihedral cohomology. Also, we present and prove the relation between cyclic and dihedral cohomology of algebras and some examples. [ABSTRACT FROM AUTHOR]

Let A be a finite-dimensional k -algebra. In this paper, we mainly study the representation type of subcategories of the bounded derived category D b (A). First, we define the representation type and some homological invariants including cohomological length, width, range for subcategories. In this framework, we provide a characterization for derived discrete algebras. Moreover, for a finite-dimensional algebra A , we establish the first Brauer–Thrall type theorem of certain contravariantly finite subcategories 𝒞 of D b (A) , that is, 𝒞 is of finite type if and only if its cohomological range is finite. [ABSTRACT FROM AUTHOR]

Abstract In this paper we compute Gröbner bases for determinantal ideals of the form I 1 (X Y) , where X and Y are both matrices whose entries are indeterminates over a field K. We use the Gröbner basis structure to determine Betti numbers for such ideals. [ABSTRACT FROM AUTHOR]

Based on Kadeishvili's original theorem inducing A ∞ {A_{\infty}} -algebra structures on the homology of dg-algebras, several directions of algorithmic research in A ∞ {A_{\infty}} -algebras have been pursued. In this paper, we survey the work done on calculating explicit A ∞ {A_{\infty}} -algebra structures from homotopy retractions, in group cohomology and in persistent homology. [ABSTRACT FROM AUTHOR]

Abstract Let L be a finite dimensional nilpotent Leibniz algebra such that dim ⁡ (L) = n and dim ⁡ (L 2) = m ≠ 0. In this paper, we prove dim ⁡ (H L 2 (L)) ≤ (n + m − 2) (n − m) − m + 2 , where H L 2 (L) is the second Leibniz homology of L. As a consequence, for a non-abelian nilpotent Leibniz algebra L , we find that s (L) = (n − 1) 2 + 1 − dim ⁡ (H L 2 (L)) ≥ 0. Furthermore, we determine all finite dimensional nilpotent Leibniz algebras with s (L) less than or equal to three. [ABSTRACT FROM AUTHOR]

The derivatives of the identity functor on spaces in Goodwillie calculus form an operad in spectra. Antolin-Camarena computed the mod 2 homology of free algebras over this operad for 1-connected spectra. In this present paper we carry out similar computations for mod p homology for odd primes p, also for non-connected spectra. [ABSTRACT FROM AUTHOR]

We define additional gradings on two generalizations of Khovanov homology (one due to the first author, the other due to the second), and use them to define invariants of various kinds of embeddings. These include invariants of links in thickened surfaces and of surfaces embedded in thickened 3 -manifolds. In particular, the invariants of embedded surfaces are expressed in terms of certain diagrams related to the thickened 3 -manifold, so that we refer to them as picture-valued invariants. This paper contains the first instance of such invariants for 2 -dimensional objects. The additional gradings are defined using cohomological and homotopic information of surfaces: using this information we decorate the smoothings of the standard Khovanov cube, before transferring the decorations into algebra. [ABSTRACT FROM AUTHOR]

We study the preprojective cohomological Hall algebra (CoHA) introduced by the authors in Yang and Zhao (The cohomological Hall algebra of a preprojective algebra. arXiv: 1407.7994v5, 2015) for any quiver Q and any one-parameter formal group G. In this paper, we construct a comultiplication on the CoHA, making it a bialgebra. We also construct the Drinfeld double of the CoHA. The Drinfeld double is a quantum affine algebra of the Lie algebra gQ associated to Q, whose quantization comes from the formal group G. We prove, when the group G is the additive group, the Drinfeld double of the CoHA is isomorphic to the Yangian. [ABSTRACT FROM AUTHOR]

In this paper we continue our study of logarithmic topological Hochschild homology. We show that the inclusion of the connective Adams summand I into the p-local complex connective K-theory spectrum ku(p), equipped with suitable log structures, is a formally log THH-étale map, and compute the V(1)-homotopy of their logarithmic topological Hochschild homology spectra. As an application, we recover Ausoni's computation of the V(1)-homotopy of the ordinary THH of ku. [ABSTRACT FROM AUTHOR]

R. W. Carey and J. Pincus in [] proposed an index theory for non-Fredholm bounded operators T on a separable Hilbert space such that is in the trace class. We showed in [] using Dirac-type operators acting on sections of bundles over that we could construct bounded operators T satisfying the more general condition that the operator is in the trace class. We proposed there a 'homological index' for these Dirac-type operators given by . In this paper we show that the index introduced in [] represents the result of a paring between a cyclic homology theory for the algebra generated by T and and its dual cohomology theory. This leads us to establish the homotopy invariance of our homological index (in the sense of cyclic theory). We are then able to define in a very general fashion a homological index for certain unbounded operators and prove invariance of this index under a class of unbounded perturbations. [ABSTRACT FROM AUTHOR]

Let G be the simple algebraic group Sp(2; 2), to be defined over Q. It is a non-quasi-split, Q-rank-two inner form of the split symplectic group Sp8 of rank four. The cohomology of the space of automorphic forms on G has a natural subspace, which is spanned by classes represented by residues and derivatives of cuspidal Eisenstein series. It is called Eisenstein cohomology. In this paper we give a detailed description of the Eisenstein cohomology HEisq (G, E) of G in the case of regular coefficients E. It is spanned only by holomorphic Eisenstein series. For non-regular coefficients E we really have to detect the poles of our Eisenstein series. Since G is not quasi-split, we are out of the scope of the so-called 'Langlands-Shahidi method' (cf. F. Shahidi, On certain L-functions, Amer. J. Math. 103 (1981), 297-355; F. Shahidi, On the Ramanujan conjecture and finiteness of poles for certain L-functions, Ann. of Math. (2) 127 (1988), 547-584). We apply recent results of Grbac in order to find the double poles of Eisenstein series attached to the minimal parabolic P0 of G. Having collected this information, we determine the square-integrable Eisenstein cohomology supported by P0 with respect to arbitrary coefficients and prove a vanishing result. This will exemplify a general theorem we prove in this paper on the distribution of maximally residual Eisenstein cohomology classes. [ABSTRACT FROM AUTHOR]

The present paper continues a cycle of papers, in which the Yoneda algebras were calculated for several families of algebras of dihedral and semidihedral type in the classification by K. Erdmann. Using the technique of a previous paper, a description of the Yoneda algebras for both families of local algebras occurring in this classification is given. Namely, a conjecture about the structure of the minimal free resolution of a (unique) simple module is stated, which is based on some empirical observations, and after establishing this conjecture, “cohomology information" is derived from the resolution discovered, and, as a result, this allows us to describe the Yoneda algebras of the algebras under consideration, It is noted that a similar technique was applied in computation of the Hochschild cohomology algebra for some finite-dimensional algebras. Bibliography: 23 titles. [ABSTRACT FROM AUTHOR]

This paper is a continuation of the study begun in the previous paper with the same title. We analyze ER(2)16*+8(RP2n) and compute ER(2)*(RP16K+1), and use these to prove more nonimmersion theorems for RPn, including many in fairly low dimensions. In particular, we get 12 new nonimmersion results for RPn where n < 192, the range included in the tables Don Davis keeps. These complement the 10 already found in the first paper. [ABSTRACT FROM AUTHOR]

Abstract: In a previous paper [Avner Ash, Paul E. Gunnells, Mark McConnell, Cohomology of congruence subgroups of , J. Number Theory 94 (2002) 181–212] we computed cohomology groups , where is a certain congruence subgroup of , for a range of levels N. In this note we update this earlier work by extending the range of levels and describe cuspidal cohomology classes and additional boundary phenomena found since the publication of [Avner Ash, Paul E. Gunnells, Mark McConnell, Cohomology of congruence subgroups of , J. Number Theory 94 (2002) 181–212]. The cuspidal cohomology classes in this paper are the first cuspforms for concretely constructed in terms of Betti cohomology. [Copyright &y& Elsevier]

Abstract: We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov–Schechtman–Vaintrob of a manifold with a group action. The main idea in this paper is to synthesize the algebraic approach to classical equivariant cohomology due to H. Cartan, 2 with [2] Cartan''s theory was further developed by Duflo–Kumar–Vergne [M. Duflo, S. Kumar, M. Vergne, Sur la cohomologie équivariante des variétés différentiables, Astérisque 215 (1993)] and Guillemin–Sternberg [V. Guillemin, S. Sternberg, Supersymmetry and Equivariant de Rham Theory, Springer, 1999]. This paper follows closely the latter approach. the theory of differential vertex algebras, by using an appropriate notion of invariant theory. We also construct the vertex algebra analogues of the Mathai–Quillen isomorphism, the Weil and the Cartan models for equivariant cohomology, and the Chern–Weil map. We give interesting cohomology classes in the new theory that have no classical analogues. [Copyright &y& Elsevier]

The present paper continues a series of papers by the author (some of them are written in collaboration) in which the Yoneda algebra is calculated for several families of algebras of dihedral and semidihedral type (in K. Erdmann’s classification). In the present paper, the Yoneda algebra is described (in terms of quivers with relations) for algebras of semidihedral type, namely, of the families SD(2 A)1, SD(2 A)2, and SD(3 A)2. Bibliography: 13 titles. [ABSTRACT FROM AUTHOR]

Strong flatness and condition (P) property of right S-acts over a monoid S have been the topic of many papers on homological classification of monoids. In this paper, we continue to investigate the monoids over which every right S-act satisfying condition (P) is strongly flat and we obtain some new classes of monoids. [ABSTRACT FROM AUTHOR]

The connections between braid groups, algebraic functions and spaces of generic polynomials have been described by Arnol''d in his famous paper on topological invariants of algebraic functions. In this seminal paper Arnol''d studied the cohomology rings of the spaces of polynomials without multiple roots and showed that these rings satisfy three theorems: the stabilization theorem (the rings stabilize has the degree of the polynomials tends to infinity), the repetition theorem (the rings associated to spaces of polynomials of successive even and odd degrees are isomorphic) and the finiteness theorem (all cohomology groups are finite except the first two). We generalize the results of Arnol''d by considering cohomology rings dual to arbitrary Young strata of the space of polynomials. We show that the stabilization, repetition and finiteness theorems still hold mutatis mutandis in this more general context. [Copyright &y& Elsevier]

Persistent homology with coefficients in a field $$\mathbb {F}$$ coincides with the same for cohomology because of duality. We propose an implementation of a recently introduced algorithm for persistent cohomology that attaches annotation vectors with the simplices. We separate the representation of the simplicial complex from the representation of the cohomology groups, and introduce a new data structure for maintaining the annotation matrix, which is more compact and reduces substantially the amount of matrix operations. In addition, we propose a heuristic to simplify further the representation of the cohomology groups and improve both time and space complexities. The paper provides a theoretical analysis, as well as a detailed experimental study of our implementation and comparison with state-of-the-art software for persistent homology and cohomology. [ABSTRACT FROM AUTHOR]

This paper is concerned with demonstrating that the twistor realization of the ladder representation of U(p,q) can be generalized to super representations of U(p,q|N). This is achieved by considering the generalization of twistor elementary states to a super algebraic category. Unitarity of these super representations follows from the positive definiteness of a super twistor scalar product constructed in this paper. Although generalizations of the ladder representations have been well studied by other means, it will be shown that the super twistor generalization is especially natural and merits special investigation. [ABSTRACT FROM AUTHOR]

Classical homological algebra studies chain complexes, resolutions, and derived functors in additive categories. In this paper we define higher order chain complexes, resolutions, and derived functors in the context of a new type of algebraic structure, called an algebra of left cubical balls . We show that higher order resolutions exist in these algebras, and that they determine higher order Ext-groups. In particular, the E m -term of the Adams spectral sequence ( m > 2 ) is such a higher Ext-group, providing a new way of constructing its differentials. [ABSTRACT FROM AUTHOR]

We give the dimension formula of the cyclic homology of truncated quiver algebras over a field of positive characteristic. This is done by using a mixed complex due to Cibils. We consider a spectral sequence associated with the Cibils' mixed complex, which converges to the cyclic homology. The -term of this spectral sequence, that is the Hochschild homology, is calculated by Sköldberg. In this paper, by means of chain maps between the projective resolution constructed by Sköldberg and one by Cibils, we calculate the -term, and we have the dimension formula of the cyclic homology. [ABSTRACT FROM AUTHOR]

In this paper, the authors develop an approach to construct a q-deformed Heisenberg-Virasoro algebra which is a Hom-Lie algebra, and investigate its central extensions and second cohomology group. Finally, quantum deformations of the Heisenberg-Virasoro algebra which provide a non-trivial Hopf structure are presented. [ABSTRACT FROM AUTHOR]

Bi-Koszul algebras, including two classes of non-Koszul Artin-Schelter regular algebras of global dimension 4, were a class of graded algebras with non-pure resolutions, introduced in [8]. There is a natural question: can we construct bi-Koszul algebras from algebras with pure resolutions? In this paper, we study this question in terms of normal extensions and Ore extensions. More precisely, we attempt to obtain bi-Koszul algebras from algebras with pure resolutions by these two kinds of extensions. Furthermore, some homological properties of bi-Koszul algebras obtained in such ways are discussed. [ABSTRACT FROM AUTHOR]

The zeroth coefficient polynomial is a one variable polynomial contained in the HOM-FLYPT polynomial. In this paper, we give a basic computation of the zeroth coefficient polynomial of a 2-cable knot. In particular, we compute the zeroth coefficient polyno-mials of the 2-cable knots of the Kanenobu knots. It is known that the Kanenobu knots have the same HOMFLYPT polynomial and the same Khovanov-Rozansky homology. As a result, we distinguish the Kanenobu knots completely. Moreover, we estimate the braid indices of the Kanenobu knots. [ABSTRACT FROM AUTHOR]

Effective Homology is an algebraic-topological method based on the computational concept of chain homotopy equivalence on a cell complex. Using this algebraic data structure, Effective Homology gives answers to some important computability problems in Algebraic Topology. In a discrete context, Effective Homology can be seen as a combinatorial layer given by a forest graph structure spanning every cell of the complex. In this paper, by taking as input a pixel-based 2D binary object, we present a logarithmic-time uniform solution for describing a chain homotopy operator $$\phi $$ for its adjacency graph. This solution is based on Membrane Computing techniques applied to the spanning forest problem and it can be easily extended to higher dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we study a natural extension of Kontsevich's characteristic class construction for A∞- and L∞-algebras to the case of curved algebras. These define homology classes on a variant of his graph homology that allows vertices of valence at least 1. We compute this graph homology, which is governed by star-shaped graphs with odd-valence vertices. We also classify non-trivially curved cyclic A∞- and L∞- algebras over a field up to gauge equivalence, and show that these are essentially reduced to algebras of dimension at most 2 with only even-ary operations. We apply the reasoning to compute stability maps for the homology of Lie algebras of formal vector fields. Finally, we explain a generalization of these results to other types of algebras, using the language of operads. A key observation is that operads governing curved algebras are closely related to the Koszul dual of those governing unital algebras. [ABSTRACT FROM AUTHOR]

Let G be a finite group or a compact connected Lie group and let BG be its classifying space. Let ℒ BG ≔ map( S1, BG) be the free loop space of BG, i.e. the space of continuous maps from the circle S1 to BG. The purpose of this paper is to study the singular homology H*(ℒ BG) of this loop space. We prove that when taken with coefficients in a field the homology of ℒ BG is a homological conformal field theory. As a byproduct of our Main Theorem, we get a Batalin-Vilkovisky algebra structure on the cohomology H*(ℒ BG). We also prove an algebraic version of this result by showing that the Hochschild cohomology HH*( S*( G), S*( G)) of the singular chains of G is a Batalin-Vilkovisky algebra. [ABSTRACT FROM AUTHOR]

In this paper we provide a construction which produces sequences of pseudo-Anosov mapping classes on surfaces with decreasing Euler characteristic. The construction is based on Penner's examples used in the proof that the minimal dilatation, δg,0, for a closed surface of genus g behaves asymptotically like 1/g. We give a bound for the dilatation of the pseudo-Anosov elements of each sequence produced by the construction and use this bound to show that if gi = rni for some rational number r > 0 then Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. behaves like Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. where Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. is the Euler characteristic of the genus gi surface with ni punctures. [ABSTRACT FROM AUTHOR]

In this paper, we consider the module-relative-Hochschild homology and cohomology of tensor products of algebras and relate them to those of the factor algebras. Moreover, we show that the tensor product is formally smooth if and only if one of its factor algebras is formally smooth and the other is separable. [ABSTRACT FROM AUTHOR]

This paper presents the first complete calculation of the cohomology of any nontrivial quandle, establishing that this cohomology exhibits a very rich and interesting algebraic structure. Rack and quandle cohomology have been applied in recent years to attack a number of problems in the theory of knots and their generalizations like virtual knots and higher-dimensional knots. An example of this is estimating the minimal number of triple points of surface knots [E. Hatakenaka, An estimate of the triple point numbers of surface knots by quandle cocycle invariants, Topology Appl139(1-3) (2004) 129-144.]. The theoretical importance of rack cohomology is exemplified by a theorem [R. Fenn, C. Rourke and B. Sanderson, James bundles and applications, Proc. London Math. Soc. (3)89(1) (2004) 217-240] identifying the homotopy groups of a rack space with a group of bordism classes of high-dimensional knots. There are also relations with other fields, like the study of solutions of the Yang-Baxter equations. [ABSTRACT FROM AUTHOR]

The chiral equivariant cohomology contains and generalizes the classical equivariant cohomology of a manifold M with an action of a compact Lie group G. For any simple G, there exist compact manifolds with the same classical equivariant cohomology, which can be distinguished by this invariant. When M is a point, this cohomology is an interesting conformal vertex algebra whose structure is still mysterious. In this paper, we scratch the surface of this object in the case G = SU(2). [ABSTRACT FROM AUTHOR]

Abstract: A toric arrangement is a finite set of hypersurfaces in a complex torus, every hypersurface being the kernel of a character. In the present paper, we build a CW-complex homotopy equivalent to the arrangement complement, with a combinatorial description similar to that of the well-known Salvetti complex. If the toric arrangement is defined by a Weyl group, we also provide an algebraic description, very handy for cohomology computations. In the last part, we give a description in terms of tableaux for a toric arrangement appearing in robotics. [Copyright &y& Elsevier]

Abstract: In this paper we study monomial ideals attached to posets, introduce generalized Hibi rings and investigate their algebraic and homological properties. The main tools to study these objects are Gröbner basis theory, the concept of sortability due to Sturmfels and the theory of weakly polymatroidal ideals. [Copyright &y& Elsevier]

Abstract: Let R be a commutative Noetherian ring and M a finitely generated R-module. We show in this paper that, for an integer t, if the local cohomology module with respect to an ideal is finitely generated for all , then for all -filter regular elements x contained in a enough large power of and all . As consequences we obtain generalizations, by very short proofs, of the main results of M. Brodmann and A.L. Faghani [M. Brodmann, A.L. Faghani, A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc. 128 (2000) 2851–2853] and of H.L. Truong and the first author [N.T. Cuong, H.L. Truong, Asymptotic behavior of parameter ideals in generalized Cohen–Macaulay module, J. Algebra 320 (2008) 158–168]. [Copyright &y& Elsevier]

In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian category, which for the sake of computability need to be turned into constructive ones. We do this explicitly for the often-studied example Abelian category of finitely presented modules over a so-called computable ring R, i.e. a ring with an explicit algorithm to solve one-sided (in)homogeneous linear systems over R. For a finitely generated maximal ideal 픪 in a commutative ring R, we show how solving (in)homogeneous linear systems over R픪 can be reduced to solving associated systems over R. Hence, the computability of R implies that of R픪. As a corollary, we obtain the computability of the category of finitely presented R픪-modules as an Abelian category, without the need of a Mora-like algorithm. The reduction also yields, as a byproduct, a complexity estimation for the ideal membership problem over local polynomial rings. Finally, in the case of localized polynomial rings, we demonstrate the computational advantage of our homologically motivated alternative approach in comparison to an existing implementation of Mora's algorithm. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring, 픞 be an ideal of R and M be an arbitrary R-module. In this paper, among other things, we show that if, for a non-negative integer t, the R-module $\textrm{Ext}^t_R(R/{\mathfrak a}, M)$ is weakly Laskerian and $H^i_{\mathfrak a}(M)$ is 픞-weakly cofinite for all i < t, then, for any weakly Laskerian submodule U of $H^t_{\mathfrak a}(M)$, the R-module $\textrm{Hom}_R(R/{\mathfrak a}, H^t_\mathfrak{a}(M)/U)$ is weakly Laskerian. As a consequence the set of associated primes of $H^t_\mathfrak{a}(M)/U$ is finite. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove that Λ is an -Auslander Artinian algebra with admitting a trivial maximal -orthogonal subcategory of mod Λ if and only if it is Morita equivalent to a finite product of F and , where F is a division algebra. In addition, we obtain a necessary condition for an Auslander Artinian algebra admitting a non-trivial maximal 1-orthogonal subcategory. [Copyright &y& Elsevier]

We give a rigorous mathematical proof for the validity of the toric sheaf cohomology algorithm conjectured in the recent paper by Blumenhagen, Jurke, Rahn, and Roschy (arXiv:1003.5217). We actually prove not only the original algorithm but also a speed-up version of it. Our proof is independent from (in fact appeared earlier on the arXiv than) the proof by Roschy and Rahn (arXiv:1006.2392), and has several advantages such as being shorter and cleaner and can also settle the additional conjecture on 'Serre duality for Betti numbers' which was raised but unresolved in arXiv:1006.2392. [ABSTRACT FROM AUTHOR]

Abstract: Penrose hyperbolic tilings are tilings of the hyperbolic plane which admit, up to affine transformations a finite number of prototiles. In this paper, we give a complete description of the -algebras and of the -theory for such tilings. Since the continuous hull of these tilings have no transversally invariant measure, these -algebras are traceless. Nevertheless, harmonic currents give rise to -cyclic cocycles and we discuss in this setting a higher-order version of the gap-labeling. [Copyright &y& Elsevier]

In this paper we show that the systems introduced in [[Eelbode, Adv. Appl. Clifford Algebr. 17: 635--649, 2007]] and [[Peñña-Peñña, Sabadini, Sommen, Complex Anal. Oper. Theory 1: 97--113, 2007]] are equivalent, both giving the notion of quaternionic Hermitian monogenic functions. This makes it possible to prove that the free resolution associated to the system is linear in any dimension, and that the first cohomology module is nontrivial, thus generalizing the results in [[Peñña-Peñña, Sabadini, Sommen, Complex Anal. Oper. Theory 1: 97--113, 2007]]. Furthermore, exploiting the decomposition of the spinor space into ( m)-irreducibles, we find a certain number of ''algebraic'' compatibility conditions for the system, suggesting that the usual spinor reduction is not applicable. [ABSTRACT FROM AUTHOR]

Let f0, f1, f2, f3 be linearly independent nonzero homogeneous polynomials in the standard ℤ-graded ring R ≔ 핂[s, t, u] of the same degree d, and gcd(f0, f1, f2, f3) = 1. This defines a rational map ℙ2 → ℙ3. The Rees algebra Rees(I) = R ⊕ I ⊕ I2 ⊕ ⋯ of the ideal I = 〈f0, f1, f2, f3〉 is the graded R-algebra which can be described as the image of the R-algebra homomorphism h: R[x, y, z, w ] → Rees(I). This paper discusses one result concerning the structure of the kernel of the map h when I is a saturated local complete intersection ideal with V(I) ≠ ∅ and μ-basis of degrees (1,1,d - 2). [ABSTRACT FROM AUTHOR]