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Let G be a finite group, p a prime, and k a field of characteristic p. We introduce the notion of an endotrivial chain complex of p -permutation kG -modules, and study the corresponding group E k (G) of endotrivial complexes. Such complexes are shown to induce splendid Rickard autoequivalences of kG. The elements of E k (G) are determined uniquely by integral invariants arising from the Brauer construction and a degree one character G → k ×. Using ideas from Bouc's theory of biset functors, we provide a canonical decomposition of E k (G) , and as an application, give complete descriptions of E k (G) for abelian groups and p -groups of normal p -rank 1. Taking Lefschetz invariants of endotrivial complexes induces a group homomorphism Λ : E k (G) → O (T (k G)) , where O (T (k G)) is the orthogonal unit group of the trivial source ring. Using recent results of Boltje and Carman, we give a Frobenius stability condition elements in the image of Λ must satisfy. [ABSTRACT FROM AUTHOR]

Let G be a finite group and X be a subgroup of G. We investigate the topological properties of the poset C X (G) of cosets Hx in G with X ≤ H < G. We show that C X (G) is non-contractible if G is solvable or N G (X) contains a Sylow 2-subgroup and a Sylow 3-subgroup of G. This result follows J. Shareshian and R. Woodroofe's work in [8] (2016). We also give some divisibility properties of the Euler characteristic of C X (G) when X is a p -group, which follows K. S. Brown's classical result in [3] (2000). [ABSTRACT FROM AUTHOR]

This paper is a commutative algebra introduction to the homological theory of quasi-coherent sheaves and contraherent cosheaves over quasi-compact semi-separated schemes. Antilocality is an alternative way in which global properties are locally controlled in a finite affine open covering. For example, injectivity of modules over non-Noetherian commutative rings is not preserved by localizations, while homotopy injectivity of complexes of modules is not preserved by localizations even for Noetherian rings. The latter also applies to the contraadjustedness and cotorsion properties. All the mentioned properties of modules or complexes over commutative rings are actually antilocal. They are also colocal, if one presumes contraadjustedness. Generally, if the left class in a (hereditary complete) cotorsion theory for modules or complexes of modules over commutative rings is local and preserved by direct images with respect to affine open immersions, then the right class is antilocal. If the right class in a cotorsion theory for contraadjusted modules or complexes of contraadjusted modules is colocal and preserved by such direct images, then the left class is antilocal. As further examples, the class of flat contraadjusted modules is antilocal, and so are the classes of acyclic, Becker-coacyclic, or Becker-contraacyclic complexes of contraadjusted modules. The same applies to the classes of homotopy flat complexes of flat contraadjusted modules and acyclic complexes of flat contraadjusted modules with flat modules of cocycles. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the dimension of dual modules of local cohomology of Stanley–Reisner rings to obtain a new vector. This vector contains important information on the Serre's condition (S r) and the CM t property as well as the depth of Stanley–Reisner rings. We prove some results in this regard including lower bounds for the depth of Stanley–Reisner rings. Further, we give a characterization of (d − 1) -dimensional simplicial complexes with codimension two which are (S d − 3) but they are not Cohen–Macaulay. By using this characterization, we obtain a condition to equality of projective dimension of the Stanley–Reisner rings and the arithmetical rank of their Stanley–Reisner ideals. Moreover, our characterization allows us to compute the h -vectors and give a negative answer to a known question regarding these vectors. [ABSTRACT FROM AUTHOR]

We examine virtual resolutions of Stanley–Reisner ideals for a product of projective spaces by providing sufficient conditions for a simplicial complex to be virtually Cohen–Macaulay (to have a virtual resolution with length equal to its codimension). In particular, we show that all balanced simplicial complexes are virtually Cohen–Macaulay. [ABSTRACT FROM AUTHOR]

Let R be a ring such that (GP , GP ⊥) forms a cotorsion pair cogenerated by a set, where GP denotes the category of all Gorenstein projective R -modules. Recently, the first author defined for any complex X the relative cohomology functors Ext GP ⁎ (X , −) as H − ⁎ (Hom (G , −)) in which G is a special Gorenstein projective precover of X. In the present paper, we introduce the dimension of X related to Gorenstein projective precovers, and show that such a dimension of X is equal to the least integer n for which Ext GP i (X , Q) = 0 for all i > n and all R -modules Q ∈ GP ⊥. This result gives a "Gorenstein" version of the relationship between the projective dimension of complexes introduced by Avramov and Foxby and the absolutely cohomology functors Ext R ⁎ (− , −). [ABSTRACT FROM AUTHOR]

We reformulate the problem of bounding the total rank of the homology of perfect chain complexes over the group ring F p [ G ] of an elementary abelian p -group G in terms of commutative algebra. This extends results of Carlsson for p = 2 to all primes. As an intermediate step, we construct an embedding of the derived category of perfect chain complexes over F p [ G ] into the derived category of p -DG modules over a polynomial ring. [ABSTRACT FROM AUTHOR]

We study 2-term tilting complexes of Brauer tree algebras in terms of simplicial complexes. We show the symmetry and convexity of the lattice polytope corresponding to the simplicial complex of 2-term tilting complexes. Via a geometric interpretation of derived equivalences, we show that the f -vector of the simplicial complexes of Brauer tree algebras only depends on the number of the edges of the Brauer trees and hence it is a derived invariant. In particular, this result implies that the number of 2-term tilting complexes, which is in bijection with support τ -tilting modules, is a derived invariant. Moreover, we apply our result to the enumeration problem of Coxeter-biCatalan combinatorics. [ABSTRACT FROM AUTHOR]

Let R be a Noetherian commutative ring and M a R -module with pd R M ≤ 1 that has rank. Necessary and sufficient conditions were provided in [1] for an exterior power ∧ k M to be torsion free. When M is an ideal of R similar necessary and sufficient conditions were provided in [2] for a symmetric power S k M to be torsion free. We extend these results to a broad class of Schur modules L λ / μ M. En route, for any map of finite free R modules ϕ : F → G we also study the general structure of the Schur complexes L λ / μ ϕ , and provide necessary and sufficient conditions for the acyclicity of any given L λ / μ ϕ by computing explicitly the radicals of the ideals of maximal minors of all its differentials. [ABSTRACT FROM AUTHOR]

Abstract Let A be a Grothendieck category. We prove that there is an infinite regular cardinal λ such that A has λ -pure injective envelopes. Furthermore, we show that any λ -pure acyclic complex in A is a directed union of contractible subcomplexes. This provides an answer to a question which is conjectured by Simson in [14, Conjecture 3.4]. [ABSTRACT FROM AUTHOR]

Abstract We prove that any derived equivalence between derived-discrete algebras of finite global dimension is standard, that is, isomorphic to the derived tensor functor by a two-sided tilting complex. [ABSTRACT FROM AUTHOR]

Abstract We apply the Auslander–Buchweitz approximation triangles to study (pre)silting subcategories in a triangulated catego- ry T. An Auslander–Reiten type correspondence between the class of silting subcategories of T and that of certain covariantly finite subcategories of T is established. We introduce a relation on presilting subcategories of T , which can be used to obtain another silting subcategory from a given one. We also give a Bazzoni's characterization for two presilting subcategories satisfying such a relation. The results can be used to improve the Auslander–Reiten type correspondence and Bazzoni's characterization for small silting complexes established by Wei. [ABSTRACT FROM AUTHOR]

Let k be a field and let Λ be a finite dimensional k -algebra. We prove that every bounded complex V • of finitely generated Λ-modules has a well-defined versal deformation ring R ( Λ , V • ) which is a complete local commutative Noetherian k -algebra with residue field k . We also prove that nice two-sided tilting complexes between Λ and another finite dimensional k -algebra Γ preserve these versal deformation rings. Additionally, we investigate stable equivalences of Morita type between self-injective algebras in this context. We apply these results to the derived equivalence classes of the members of a particular family of algebras of dihedral type that were introduced by Erdmann and shown by Holm to be not derived equivalent to any block of a group algebra. [ABSTRACT FROM AUTHOR]

For an associative ring R , the projective level of a complex F is the smallest number of mapping cones needed to build F from projective R -modules. We establish lower bounds for the projective level of F in terms of the vanishing of homology of F . We then use these bounds to derive a new version of The New Intersection Theorem for level when R is a commutative Noetherian local ring. [ABSTRACT FROM AUTHOR]

We introduce and study the new concepts of cosilting complexes, cosilting modules and AIR-cotilting modules. We prove that the three concepts AIR-cotilting modules, cosilting modules and quasi-cotilting modules coincide with each other, in contrast with the dual fact that AIR-tilting modules, silting modules and quasi-tilting modules are different. Further, we show that there are bijections between the following four classes (1) equivalence classes of AIR-cotilting (resp., cosilting, quasi-cotilting) modules, (2) equivalence classes of 2-term cosilting complexes, (3) torsion-free cover classes and (4) torsion-free special precover classes. We also extend a classical result of Auslander and Reiten on the correspondence between certain contravariantly finite subcategories and cotilting modules to the case of cosilting complexes. [ABSTRACT FROM AUTHOR]

We prove a special case of a conjecture of Dao–Kurano concerning the vanishing of Hochster's theta pairing. The proof uses Adams operations on both topological K -theory and perfect complexes with support. [ABSTRACT FROM AUTHOR]

We consider twisted standard filtrations of Soergel bimodules associated to arbitrary Coxeter groups and show that the graded multiplicities in these filtrations can be interpreted as structure constants in the Hecke algebra. This corresponds to the positivity of the polynomials occurring when expressing an element of the canonical basis in a generalized standard basis twisted by a biclosed set of roots in the sense of Dyer, and comes as a corollary of Soergel's conjecture. We then show the positivity of the corresponding inverse polynomials in the case where the biclosed set is an inversion set of an element or its complement by generalizing a result of Elias and Williamson on the linearity of the Rouquier complexes associated to lifts of these basis elements in the Artin–Tits group. These lifts turn out to be generalizations of Mikado braids as introduced in a joint work with Digne. This second positivity property generalizes a result of Dyer and Lehrer from finite to arbitrary Coxeter groups. [ABSTRACT FROM AUTHOR]

We define two different simplicial complexes, the common divisor simplicial complex and the prime divisor simplicial complex, from a set of integers, and explore their similarities. We will define a map between the two simplicial complexes, and use this map to show that for any set of integers, the fundamental groups of the resulting simplicial complexes are isomorphic. [ABSTRACT FROM AUTHOR]

We prove a duality result for the parabolic Kazhdan–Lusztig R -polynomials of a finite Coxeter system. This duality is similar to, but different from, the one obtained in [9] . As a consequence of our duality we obtain an identity between the parabolic Kazhdan–Lusztig and inverse Kazhdan–Lusztig polynomials of a finite Coxeter system. We also obtain applications to certain modules defined by Deodhar and derive a result that gives evidence in favor of Marietti's combinatorial invariance conjecture for parabolic Kazhdan–Lusztig polynomials. [ABSTRACT FROM AUTHOR]

It is known that over an Iwanaga–Gorenstein ring the Gorenstein injective (Gorenstein projective, Gorenstein flat) modules are simply the cycles of acyclic complexes of injective (projective, flat) modules. We consider the question: are these characterizations only working over Iwanaga–Gorenstein rings? We prove that if R is a commutative noetherian ring of finite Krull dimension then the following are equivalent: 1. R is an Iwanaga–Gorenstein ring. 2. Every acyclic complex of injective modules is totally acyclic. 3. The cycles of every acyclic complex of Gorenstein injective modules are Gorenstein injective. 4. Every acyclic complex of projective modules is totally acyclic. 5. The cycles of every acyclic complex of Gorenstein projective modules are Gorenstein projective. 6. Every acyclic complex of flat modules is F-totally acyclic. 7. The cycles of every acyclic complex of Gorenstein flat modules are Gorenstein flat. Thus we improve slightly on a result of Iyengar and Krause; in [22] they proved that for a commutative noetherian ring R with a dualizing complex, the class of acyclic complexes of injectives coincides with that of totally acyclic complexes of injectives if and only if R is Gorenstein. We replace the dualizing complex hypothesis by the finiteness of the Krull dimension, and add more equivalent conditions. In the second part of the paper we focus on the noncommutative case. We prove that for a two sided noetherian ring R of finite finitistic flat dimension that satisfies the Auslander condition the following are equivalent: 1. Every complex of injective (left and respectively right) R -modules is totally acyclic. 2. R is Iwanaga–Gorenstein. [ABSTRACT FROM AUTHOR]

We study tangent cones to Schubert subvarieties of the flag variety of a complex reductive group G . Let T be a maximal torus of G , B be a Borel subgroup of G containing T , Φ be the root system of G with respect to T , W be the Weyl group of Φ, and F = G / B be the flag variety. We prove that if every irreducible component of Φ is of type B n or C n , and w 1 , w 2 are two distinct involutions in W , then the tangent cones at the point p = e B to the corresponding Schubert subvarieties X w 1 , X w 2 of F do not coincide as subschemes of the tangent space T p F . We also show that if every irreducible component of Φ is of type A n or C n , then the reduced tangent cones to X w 1 and X w 2 do not coincide as subvarieties of T p F . [ABSTRACT FROM AUTHOR]

In this paper, we study the pure acyclic complexes of modules. We obtain several characterizations of these complexes, extending results that are known for the pure acyclic complexes of flat modules. In particular, we extend Neeman's characterization [18] of the pure acyclic complexes of flat modules in terms of complexes of projective modules, by considering more generally complexes of pure projective modules. As a consequence, we obtain Simson's result [22] on the pure projectivity of pure periodic modules. [ABSTRACT FROM AUTHOR]

Let R be a local commutative noetherian ring and HKR the homology ring of the corresponding Koszul complex. We study the homological properties of HKR in particular the Avramov spectral sequence. When the embedding dimension of R is four and when R can be presented with quadratic relations we have found 101 cases where this spectral sequence degenerates and only three cases where it does not degenerate. We also determine completely the Hilbert series of the bigraded Tor of these HKR in Tables A–D at the end of the paper. We also study some higher embedding dimensions. Among the methods used are the programme BERGMAN by Jörgen Backelin et al., the Macaulay2 -package DGAlgebras by Frank Moore, combined with results by Govorov, Clas Löfwall, Victor Ufnarovski and others. [ABSTRACT FROM AUTHOR]

Let Φ be an f × g matrix with entries from a commutative Noetherian ring R , with g ≤ f . Recall the family of generalized Eagon–Northcott complexes { C Φ i } associated to Φ. (See, for example, Appendix A2 in “Commutative Algebra with a View Toward Algebraic Geometry” by D. Eisenbud.) For each integer i , C Φ i is a complex of free R -modules. For example, C Φ 0 is the original “Eagon–Northcott” complex with zero-th homology equal to the ring R / I g ( Φ ) defined by ideal generated by the maximal order minors of Φ; and C Φ 1 is the “Buchsbaum–Rim” complex with zero-th homology equal to the cokernel of the transpose of Φ. If Φ is sufficiently general, then each C Φ i , with − 1 ≤ i , is acyclic; and, if Φ is generic, then these complexes resolve half of the divisor class group of R / I g ( Φ ) . The family { C Φ i } exhibits duality; and, if − 1 ≤ i ≤ f − g + 1 , then the complex C Φ i exhibits depth-sensitivity with respect to the ideal I g ( Φ ) in the sense that the tail of C Φ i of length equal to grade ( I g ( Φ ) ) is acyclic. The entries in the differentials of C Φ i are linear in the entries of Φ at every position except at one, where the entries of the differential are g × g minors of Φ. This paper expands the family { C Φ i } to a family of complexes { C Φ i , a } for integers i and a with 1 ≤ a ≤ g . The entries in the differentials of { C Φ i , a } are linear in the entries of Φ at every position except at two consecutive positions. At one of the exceptional positions the entries are a × a minors of Φ, at the other exceptional position the entries are ( g − a + 1 ) × ( g − a + 1 ) minors of Φ. The complexes { C Φ i } are equal to { C Φ i , 1 } and { C Φ i , g } . The complexes { C Φ i , a } exhibit all of the properties of { C Φ i } . In particular, if − 1 ≤ i ≤ f − g and 1 ≤ a ≤ g , then C Φ i , a exhibits depth-sensitivity with respect to the ideal I g ( Φ ) . [ABSTRACT FROM AUTHOR]

In this paper, we study simplicial complexes whose Stanley–Reisner rings are almost Gorenstein and have a -invariant zero. We call such a simplicial complex an almost Gorenstein* simplicial complex. To study the almost Gorenstein* property, we introduce a new class of simplicial complexes which we call uniformly Cohen–Macaulay simplicial complexes. A d -dimensional simplicial complex Δ is said to be uniformly Cohen–Macaulay if it is Cohen–Macaulay and, for any facet F of Δ, the simplicial complex Δ ∖ { F } is Cohen–Macaulay of dimension d . We investigate fundamental algebraic, combinatorial and topological properties of these simplicial complexes, and show that almost Gorenstein* simplicial complexes must be uniformly Cohen–Macaulay. By using this fact, we show that every almost Gorenstein* simplicial complex can be decomposed into those of having one dimensional top homology. Also, we give a combinatorial criterion of the almost Gorenstein* property for simplicial complexes of dimension ≤2. [ABSTRACT FROM AUTHOR]

A monomial ideal I admits a Betti splitting I = J + K if the Betti numbers of I can be determined in terms of the Betti numbers of the ideals J , K and J ∩ K . Given a monomial ideal I , we prove that I = J + K is a Betti splitting of I , provided J and K are componentwise linear, generalizing a result of Francisco, Hà and Van Tuyl. If I has a linear resolution, the converse also holds. We apply this result recursively to the Alexander dual of vertex-decomposable, shellable and constructible simplicial complexes. Moreover we determine the graded Betti numbers of the defining ideal of three general fat points in the projective space. [ABSTRACT FROM AUTHOR]

It is shown that the category of enriched functors [ C , V ] is Grothendieck whenever V is a closed symmetric monoidal Grothendieck category and C is a category enriched over V . Localizations in [ C , V ] associated to collections of objects of C are studied. Also, the category of chain complexes of generalized modules Ch ( C R ) is shown to be identified with the Grothendieck category of enriched functors [ mod R , Ch ( Mod R ) ] over a commutative ring R , where the category of finitely presented R -modules mod R is enriched over the closed symmetric monoidal Grothendieck category Ch ( Mod R ) as complexes concentrated in zeroth degree. As an application, it is proved that Ch ( C R ) is a closed symmetric monoidal Grothendieck model category with explicit formulas for tensor product and internal Hom-objects. Furthermore, the class of unital algebraic almost stable homotopy categories generalizing unital algebraic stable homotopy categories of Hovey–Palmieri–Strickland [14] is introduced. It is shown that the derived category of generalized modules D ( C R ) over commutative rings is a unital algebraic almost stable homotopy category which is not an algebraic stable homotopy category. [ABSTRACT FROM AUTHOR]

Let a ⊂ O X be a coherent ideal sheaf on a normal complex variety X , and let c ≥ 0 be a real number. De Fernex and Hacon associated a multiplier ideal sheaf to the pair ( X , a c ) which coincides with the usual notion whenever the canonical divisor K X is Q -Cartier. We investigate the properties of the jumping numbers associated to these multiplier ideals. We show that the set of jumping numbers of a pair is unbounded, countable and satisfies a certain periodicity property. We then prove that the jumping numbers form a discrete set of real numbers if the locus where K X fails to be Q -Cartier is zero-dimensional. It follows that discreteness holds whenever X is a threefold with rational singularities. Furthermore, we show that the jumping numbers are rational and discrete if one removes from X a closed subset W ⊂ X of codimension at least three, which does not depend on a . We also obtain that outside of W , the multiplier ideal reduces to the test ideal modulo sufficiently large primes p ≫ 0 . [ABSTRACT FROM AUTHOR]

We describe a “cellular” approach to the computation of the cohomology of a poset with coefficients in a presheaf. A cellular cochain complex is constructed, described explicitly, and shown to compute the cohomology under certain circumstances. The descriptions are refined further for certain classes of posets including the cell posets of regular CW-complexes and geometric lattices. [ABSTRACT FROM AUTHOR]

We conjecture what the cone of hypercohomology tables of bounded complexes of coherent sheaves on projective spaces is, when we have specified regularity conditions on the cohomology sheaves of this complex and its dual. There is an injection from this cone into the cone of homological data sets of squarefree modules over a polynomial ring k [ x 1 , … , x n ] , and we conjecture that this is an isomorphism: The Tate resolutions of a complex of coherent sheaves on projective space P ( W ) , and the exterior coalgebra on 〈 x 1 , … , x n 〉 may be amalgamated together to form a complex of free Sym ( ⊕ i x i ⊗ W ⁎ ) -modules, a procedure introduced by Cox and Materov. Via a reduction ⊕ i x i ⊗ W ⁎ → ⊕ i x i ⊗ k we get a complex of free modules over k [ x 1 , … , x n ] The extremal rays in the cone of squarefree complexes are conjecturally given by triplets of pure free squarefree complexes introduced in [15] . We describe the corresponding classes of hypercohomology tables, a class which generalizes vector bundles with supernatural cohomology. We also show how various pure resolutions in the literature, like resolutions of modules supported on determinantal varieties, and tensor complexes, may be obtained by the first part of the procedure. [ABSTRACT FROM AUTHOR]

We show that every exact fully faithful functor from the category of perfect complexes on the spectrum of dual numbers to the bounded derived category of a noetherian separated scheme is of Fourier–Mukai type. The kernel turns out to be an object of the bounded derived category of coherent complexes on the product of the two schemes. We also study the space of stability conditions on the derived category of the spectrum of dual numbers. [ABSTRACT FROM AUTHOR]

For a complex semisimple Lie algebra g with Hermitian real form g R = k R + p R , there exists a positive system of roots such that the adjoint k -representation on p stabilizes the positive and negative root spaces. In this article, we extend this result to contragredient Lie superalgebras g , and study the number of irreducible components of the k -representation. We also discuss the complex structure on g R / k R . [ABSTRACT FROM AUTHOR]

We define several homology theories for central hyperplane arrangements, categorifying well-known polynomial invariants including the characteristic polynomial, Poincaré polynomial, and Tutte polynomial. We consider basic algebraic properties of such chain complexes, including long-exact sequences associated to deletion–restriction triples and dg-algebra structures. We also consider signed hyperplane arrangements, and generalize the odd Khovanov homology of Ozsváth–Rasmussen–Szabó from link projections to signed arrangements. We define hyperplane Reidemeister moves which generalize the usual Reidemeister moves from framed link projections to signed arrangements, and prove that the chain homotopy type associated to a signed arrangement is invariant under hyperplane Reidemeister moves. [ABSTRACT FROM AUTHOR]

We make a detailed study of idempotent ideals that are traces of countably generated projective right modules. We associate to such ideals an ascending chain of finitely generated left ideals and, dually, a descending chain of cofinitely generated right ideals. The study of the first sequence allows us to characterize trace ideals of projective modules and to show that projective modules can always be lifted modulo the trace ideal of a projective module. As a consequence we give some new classification results of (countably generated) projective modules over particular classes of semilocal rings. The study of the second sequence leads us to consider projective modules over noetherian FCR-algebras; we make some constructions of non-trivial projective modules showing that over such rings the behavior of countably generated projective modules that are not direct sum of finitely generated ones is, in general, quite complex. [ABSTRACT FROM AUTHOR]

Abstract: This article is about the foundation of the Mayer–Vietoris exact sequence in Čech cohomology, with respect to the augmented Čech complex, often called “the stable Koszul complex”. Our treatment is elementary and uses neither the local cohomology of Grothendieck nor the widespread noetherian local cohomology. We use explicit objects giving us an elementary algebraic treatment in a general nonnoetherian context, i.e. two finite sequences , of a commutative ring and some Čech complexes built from these sequences. More precisely, our strategy consists in producing a short exact sequence of “type Čech“ complexes having the expected cohomologies. Then arises a simplicial complex on the set with a relatively surprising combinatory. [Copyright &y& Elsevier]

Abstract: We generalize Bulitko's Lemma to equations over (or homomorphisms into) groups that have κ-acylindrical splittings. This is a key technical lemma used in the author's algorithm to find tracks in 2-complexes. [Copyright &y& Elsevier]

Abstract: We show that for any two proper monomial ideals I and J in the polynomial ring the ring is Golod. We also show that if I is squarefree then for large enough k the quotient of S by the kth symbolic power of I is Golod. As an application we prove that the multiplication on the cohomology algebra of some classes of moment-angle complexes is trivial. [Copyright &y& Elsevier]

Abstract: We give an explicit proof of a Bogomolov-type inequality for of reflexive sheaves on . Then, using resolutions of rank-two reflexive sheaves on , we prove that the closed points of some strata of the moduli of rank-two complexes that are both PT-stable and dual-PT-stable can be given by the structure of quotient stacks. Using monads, we apply the same techniques to and obtain similar results for some strata of the moduli of Bridgeland-stable complexes. [Copyright &y& Elsevier]

Abstract: Let R be a ring with identity and denote the category of complexes of R-modules. In this paper we study the homotopy categories arising from projective (resp. injective) complexes as well as Gorenstein projective (resp. Gorenstein injective) modules. We show that the homotopy category of projective complexes over R, denoted , is compactly generated provided is so. Based on this result, it will be proved that the class of Gorenstein projective complexes is precovering, whenever R is a commutative noetherian ring of finite Krull dimension. Furthermore, it turns out that over such rings the inclusion functor has a right adjoint , where is the homotopy category of Gorenstein projective R modules. Similar, or rather dual, results for the injective (resp. Gorenstein injective) complexes will be provided. If R has a dualising complex, a triangle-equivalence between homotopy categories of projective and of injective complexes will be provided. As an application, we obtain an equivalence between the triangulated categories and , that restricts to an equivalence between and , whenever R is commutative, noetherian and admits a dualising complex. [Copyright &y& Elsevier]

Abstract: We call product generator of an additive category a fixed object satisfying the property that every other object is a direct factor of a product of copies of it. In this paper we start with an additive category with products and images, e.g. a module category, and we are concerned with the homotopy category of complexes with entries in that additive category. We prove that the Brown representability theorem is valid for the dual of the homotopy category if and only if the initial additive category has a product generator. [Copyright &y& Elsevier]

Abstract: Many key invariants in the representation theory of classical groups (symmetric groups , matrix groups , , ) are polynomials in n (e.g., dimensions of irreducible representations). This allowed Deligne to extend the representation theory of these groups to complex values of the rank n. Namely, Deligne defined generically semisimple families of tensor categories parametrized by , which at positive integer n specialize to the classical representation categories. Using Deligneʼs work, Etingof proposed a similar extrapolation for many non-semisimple representation categories built on representation categories of classical groups, e.g., degenerate affine Hecke algebras (dAHA). It is expected that for generic such extrapolations behave as they do for large integer n (“stabilization”). The goal of our work is to provide a technique to prove such statements. Namely, we develop an algebro-geometric framework to study categories indexed by a parameter n, in which the set of values of n for which the category has a given property is constructible. This implies that if a property holds for integer n, it then holds for generic complex n. We use this to give a new proof that Deligneʼs categories are generically semisimple. We also apply this method to Etingofʼs extrapolations of dAHA, and prove that when n is transcendental, “finite-dimensional” simple objects are quotients of certain standard induced objects, extrapolating Zelevinskyʼs classification of simple dAHA-modules for . Finally, we obtain similar results for the extrapolations of categories associated to wreath products of the symmetric group with associative algebras. [Copyright &y& Elsevier]

Abstract: We present a constructive analysis of Laplaceʼs proof that the field of complex numbers is algebraically closed. This proof was criticized by Gauss, who presented later a rigorous and constructive version of this proof. Our argument is different, and closer to the one of Laplace. [Copyright &y& Elsevier]

Abstract: Herrlich showed that a Mumford curve of genus over the p-adic complex field has at most , , or automorphisms as or . The Mumford curves attaining these bounds are uniformised by normal subgroups of finite index in certain p-adic triangle groups for , or in a p-adic quadrangle group for . The finite groups attaining these bounds are p-adic analogues of the Hurwitz groups arising from curves over . We construct explicit infinite families of such groups as quotients of and . These include groups of type and , all arising as congruence quotients when , and also various alternating and symmetric groups arising as noncongruence quotients of the groups . [Copyright &y& Elsevier]

Abstract: Let G be an ℓ-group (which is short for “lattice-ordered abelian group”). Baker and Beynon proved that G is finitely presented iff it is finitely generated and projective. In the category of unital ℓ-groups, those ℓ-groups having a distinguished order-unit u, only the -direction holds in general. We show that a unital ℓ-group is finitely presented iff it has a basis. A large class of projectives is constructed from bases having special properties. [Copyright &y& Elsevier]

Abstract: Let G be a classical group and suppose that G does not contains non-trivial graph automorphisms. In this paper we prove that the order complex of the coset poset of G is non-contractible. In order to prove it, we show that does not vanish, where is the Dirichlet polynomial associated to the group G. [Copyright &y& Elsevier]

Abstract: This paper studies properties of simplicial complexes Δ with the equality for a given . The main results are combinatorial characterizations of such complexes in the two-dimensional case. It turns out that there exists only a finite number of complexes with this property and that these complexes can be described completely. As a consequence we are able to determine all complexes for which is Cohen–Macaulay for some . In particular, there are complexes with or but for all and that if for some , then for all . Similarly, there are complexes for which is Cohen–Macaulay but is not Cohen–Macaulay for all and if is Cohen–Macaulay for some , then is a complete intersection. [ABSTRACT FROM AUTHOR]

Abstract: Let be a polynomial ring. Let I be a Stanley–Reisner ideal in S of a pure simplicial complex of dimension one. In this paper, we study the Buchsbaum property of for any integer . Our first purpose is giving a characterization of Ext-modules for any monomial ideal J, where , in terms of certain simplicial complexes. Then we consider the Buchsbaum property of . The main tool to check the Buchsbaumness is the surjectivity criterion. We see the behavior of the canonical map from to from the view point of reduced cohomology groups of simplicial complexes. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we investigate the description of the complex Leibniz superalgebras with nilindex , where n and m () are dimensions of even and odd parts, respectively. In fact, such superalgebras with characteristic sequence equal to (where , ) for and were classified in works by Ayupov et al. (2009) , Camacho et al. (2010) , Camacho et al. (in press) , Camacho et al. (in press) . Here we prove that in the case of , where and the Leibniz superalgebras have nilindex less than . Thus, we complete the classification of Leibniz superalgebras with nilindex . [ABSTRACT FROM AUTHOR]

Abstract: We prove that there exists just one pair of complex four-dimensional Lie algebras such that a well-defined contraction among them is not equivalent to a generalized IW-contraction (or to a one-parametric subgroup degeneration in conventional algebraic terms). Over the field of real numbers, this pair of algebras is split into two pairs with the same contracted algebra. The example we constructed demonstrates that even in the dimension four generalized IW-contractions are not sufficient for realizing all possible contractions, and this is the lowest dimension in which generalized IW-contractions are not universal. Moreover, this is also the first example of nonexistence of generalized IW-contraction for the case when the contracted algebra is not characteristically nilpotent and, therefore, admits nontrivial diagonal derivations. The lower bound (equal to three) of nonnegative integer parameter exponents which are sufficient to realize all generalized IW-contractions of four-dimensional Lie algebras is also found. [ABSTRACT FROM AUTHOR]

Abstract: We introduce the notion of balanced pair of additive subcategories in an abelian category. We give sufficient conditions under which a balanced pair of subcategories gives rise to a triangle-equivalence between two homotopy categories of complexes. As an application, we prove that for a left-Gorenstein ring, there exists a triangle-equivalence between the homotopy category of its Gorenstein projective modules and the homotopy category of its Gorenstein injective modules, which restricts to a triangle-equivalence between the homotopy category of projective modules and the homotopy category of injective modules. In the case of commutative Gorenstein rings we prove that up to a natural isomorphism our equivalence extends Iyengar–Krause''s equivalence. [Copyright &y& Elsevier]