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We investigate the notions of weak annihilator and nilpotent associated prime defined by Ouyang and Birkenmeier (Weak annihilator over extension rings, 2012) in the setting of skew PBW extensions. We extend several results formulated in the literature concerning annihilators and associated primes of commutative rings and skew polynomial rings to a more general setting of algebras not considered before. We exemplify our results with families of algebras appearing in the theory of enveloping algebras, noncommutative algebraic geometry, and theoretical physics. Finally, we present some ideas for future research. [ABSTRACT FROM AUTHOR]

The wild McKay correspondence, a variant of the McKay correspondence in positive characteristics, shows that stringy motives of quotient varieties equal some motivic integrals on the moduli space of the Galois covers of a formal disk. In this paper, we determine when the integral converges for the case of cyclic groups of prime power order. As an application, we give a criterion for the quotient variety being canonical or log canonical. [ABSTRACT FROM AUTHOR]

Polstra showed that the cardinality of the torsion subgroup of the divisor class group of a local strongly F-regular ring is finite. We expand upon this result and prove that the reciprocal of the F-signature of a local strongly F-regular ring R bounds the cardinality of the torsion subgroup of the divisor class group of R. [ABSTRACT FROM AUTHOR]

Let C be a nodal curve, and let E be a union of semistable subcurves of C. We consider the problem of contracting the connected components of E to singularities in a way that preserves the genus of C and makes sense in families. In order to do this, we introduce the notion of mesa curve, a nodal curve C with a logarithmic structure and a piecewise linear function λ ¯ on the tropicalization of C. We then construct a contraction of the support of λ ¯ inside of C for families of mesa curves. [ABSTRACT FROM AUTHOR]

In this article, we check that Fano schemes of lines on certain rational cubic fourfolds are birational to Hilbert schemes of two points on K3 surfaces. [ABSTRACT FROM AUTHOR]

The study of invariants of group actions on commutative polynomial rings has motivated many developments in commutative algebra and algebraic geometry. It has been of particular interest to understand what conditions on the group result in an invariant ring satisfying useful properties. In particular, Watanabe's Theorem states that the invariant subring of k [ x 1 , ... , x n ] under the natural action of a finite subgroup of S L n (k) is always Gorenstein. In this paper, we study this question in the more general setting of group actions on noncommutative non-connected algebras A. We develop the notion of a homological determinant of an automorphism of A, then use the homological determinant to study actions of finite groups G on A. We give a sufficient condition so that the invariant ring AG has finite injective dimension and satisfies the generalized Gorenstein condition. More precisely, let A be a noetherian N -graded generalized Gorenstein algebra with finite global dimension. Suppose all elements of G fix the idempotents of A and act with trivial homological determinant. Then the invariant ring AG is generalized Gorenstein. [ABSTRACT FROM AUTHOR]

The Isbell duality tells us a dual equivalence between spatial frames (aka. locales) and sober spaces; it is induced by the prime spectrum functor on frames. In the present paper, we give another dual equivalence induced by the maximal spectrum functor. The Isbell duality subsumes all sober spaces, but not all T 1 spaces; the duality shown in this paper subsumes all T 1 spaces, but not all sober spaces. Non-sober T 1 spaces are particularly important in classical algebraic geometry; they include, inter alia, algebraic varieties in the traditional sense, the points of which can be recovered from their open set frames via the maximal spectrum functor (and cannot via the prime spectrum functor). The duality in this paper is particularly useful for those spaces in algebraic geometry. In addition to the duality induced by maximal spectra, we give a dual adjunction lurking behind it, and an algebraic characterization of having enough points in terms of maximal spectra. Communicated by Jason P. Bell [ABSTRACT FROM AUTHOR]

Let Gr (k , n) be the Grassmannian. The quantum multiplication by the first Chern class c 1 (Gr (k , n)) induces an endomorphism c ̂ 1 of the finite-dimensional vector space QH * (Gr (k , n)) | q = 1 specialized at q = 1. Our main result is a case that a conjecture by Galkin holds. It states that the largest real eigenvalue of c ̂ 1 is greater than or equal to dim Gr (k , n) +1 with equality if and only if Gr (k , n) = P n − 1. [ABSTRACT FROM AUTHOR]

In this paper, we consider the wide class of subvarieties of the variety of all representation of Lie algebras over a field k of characteristic 0. We study the relation between the geometric equivalence and automorphic equivalence of the representations from these subvarieties. [ABSTRACT FROM AUTHOR]

In this article, we define the notion of Brauer-Clifford group for H-locally finite (S, H)-Azumaya algebras, when H is a cocommutative Hopf algebra and S is an H-locally finite commutative H-module algebra over a commutative noetherian ring k. This is the situation that arises in applications with connections to algebraic geometry. This Brauer-Clifford group turns out to be an example of a Brauer group of a symmetric monoidal category. [ABSTRACT FROM AUTHOR]

In this article, we investigate, for varieties over with trivial group of 0-cycles, the gap between essential -dimension 2 and essential -dimension 0. In particular, we present sufficient (and necessary) conditions for a variety with trivial group of 0-cycles and essential -dimension to have, in fact, essential -dimension 0. [ABSTRACT FROM AUTHOR]

For a formal scheme over a complete discrete valuation ring with a good action of a finite group, we define equivariant motivic integration, and we prove a change of variable formula for that. To do so, we construct and examine an induced group action on the Greenberg scheme of such a formal scheme. Using this equivariant motivic integration, we define an equivariant volume Poincaré series, from which we deduce Denef and Loeser's motivic zeta function including the action of the profinite group of roots of unity. [ABSTRACT FROM AUTHOR]

A submodule N of a module M is δ-small in M if N+X≠M for any proper submodule X of M with M∕X singular. A projective δ-cover of a module M is a projective module P with an epimorphism to M whose kernel is δ-small in P. A module M is called δ-semiperfect if every factor module of M has a projective δ-cover. In this paper, we prove various properties, including a structure theorem and several characterizations, for δ-semiperfect modules. Our proofs can be adapted to generalize several results of Mares [8] and Nicholson [11] from projective semiperfect modules to arbitrary semiperfect modules. [ABSTRACT FROM AUTHOR]

Let M be a hypermodule over a hyperring R such that the intersection of any two subhypermodules of M, is a subhypermodule of M. We introduce the concept of an essential subhypermodule in M relative to an arbitrary subhypermodule T of M, which is called a T-essential subhypermodule of M. Our main goal in this work is to investigate properties of (relative) essential subhypermodules. We apply this concept to introduce extending hypermodules. Examples are provided to illustrate different concepts. [ABSTRACT FROM AUTHOR]

Nikolai Durov introduced the theory of generalized rings and schemes to study Arakelov geometry in an alternative algebraic framework, and introduced the residue field at the infinite place, 픽∞. We show an elementary algebraic approach to modules and algebras over this object, define prime congruences, show that the polynomial ring of n variables is of Krull dimension n, and derive a prime decomposition theorem for these primes. [ABSTRACT FROM AUTHOR]

In this paper, we will classify normal Gorenstein stable log surfaces with . [ABSTRACT FROM AUTHOR]

We study the problem of lifting of polynomial symplectomorphisms in characteristic zero to automorphisms of the Weyl algebra by means of approximation by tame automorphisms. In 1983, Anick proved the fundamental result on approximation of polynomial automorphisms. We obtain similar approximation theorems for symplectomorphisms and Weyl algebra authomorphisms. We then formulate the lifting problem. More precisely, we prove the possibility of lifting of a symplectomorphism to an automorphism of the power series completion of the Weyl algebra of the corresponding rank. The lifting problem has its origins in the context of deformation quantization of the affine space and is closely related to several major open problems in algebraic geometry and ring theory. This paper is a continuation of the study [19]. [ABSTRACT FROM AUTHOR]

In [4], Mezzetti and Miró-Roig proved that the minimal number of generators μ(I) of a minimal (smooth) monomial Togliatti system satisfies and they classify all smooth minimal monomial Togliatti systems with 2n+1≤μ(I)≤2n+2. In this paper, we address the first open case. We classify all smooth monomial Togliatti systems of forms of degree d≥4 with μ(I) = 2n+3 and n≥2 and all monomial Togliatti systems of forms of degree d≥6 with μ(I) = 7. [ABSTRACT FROM AUTHOR]

A depiction of a nonnoetherian integral domain R is a special coordinate ring that provides a framework for describing the geometry of R. We show that if R is noetherian in codimension 1, then R has a unique maximal depiction T. In this case, the geometric dimensions of the points of Spec R may be computed directly from T. If in addition R has a normal depiction S, then S is the unique maximal depiction of R. [ABSTRACT FROM AUTHOR]

In this paper, we use Block’s results to classify irreducible modules over the diamond Lie algebra𝔇. As a corollary, we also give a classification of irreducible modules over the Euclidean algebra𝔢(2). [ABSTRACT FROM PUBLISHER]

Letbe a connected reductive algebraic group defined over an algebraically closed field. Letbe a maximal torus ofand letbe a Borel subgroup ofcontaining. LetWdenote the Weyl group ofand letẇdenote an element in the preimage of the natural map fromtoW. We investigate what conditions are necessary and sufficient for the double cosets(called Bruhat cells) to contain only regular elements. An application is provided on how this information about Bruhat cells can be used to determine character values for certain Hecke algebras. [ABSTRACT FROM PUBLISHER]

Let 𝒳 be an irreducible algebraic curve defined over a finite field 𝔽q of characteristic p>2. Assume that the 𝔽q-automorphism group of 𝒳 admits a subgroup isomorphic to the direct product of two cyclic groups Cm and Cn of orders m and n prime to p, such that both quotient curves 𝒳∕Cn and 𝒳∕Cm are rational. In this paper, we provide a complete classification of such curves as well as a characterization of their full automorphism groups. [ABSTRACT FROM AUTHOR]

We provide an explicit algorithm for computing intersection numbers between basis elements of complementary codimension in the Hilbert scheme ofNpoints in the projective plane. [ABSTRACT FROM AUTHOR]

LetGbe a finite group,Ea normal subgroup ofGandpa fixed prime. We say thatEisp-hypercyclically embeddedinGif everyp-G-chief factor ofEis cyclic. A subgroupHofGis said to satisfy Π-propertyinGif |G∕K:NG∕K((H∩L)K∕K)| is aπ((H∩L)K∕K)-number for any chief factorL∕KinG; we say thatHhas Π -property inGifH∩Oπ(H)(G) has Π-property inG. In this paper, we prove thatEisp-hypercyclically embedded inGif and only if some classes ofp-subgroups ofEhave Π -property inG. Some recent results are extended. [ABSTRACT FROM AUTHOR]

Differential algebraic geometry seeks to extend the results of its algebraic counterpart to objects defined by differential equations. Many notions, such as that of a projective algebraic variety, have close differential analogues but their behavior can vary in interesting ways. Workers in both differential algebra and model theory have investigated the property of completeness of differential varieties. After reviewing their results, we extend that work by proving several versions of a “differential valuative criterion" and using them to give new examples of complete differential varieties. We conclude by analyzing the first examples of incomplete, finite-rank projective differential varieties, demonstrating a clear difference from projective algebraic varieties. [ABSTRACT FROM PUBLISHER]

Let Λ be a non semisimple symmetric artin algebra over a commutative artinian ring R and let repdim(Λ) be its representation dimension. Assume also that Λ is a finite quiver whose connected components are amalgamations of single cycles possibly with multiple arrows, and that ℐ is an admissible ideal in the generalized path algebra ΛΩ. We prove that repdim... ≤ (Λ)+1. [ABSTRACT FROM AUTHOR]

A famous theorem of Hopf [14] states that: Every finite-dimensional real commutative division algebraAis at most two-dimensional. To this day, no elementary proof of Hopf’s theorem is known. In the present paper, we extend this result to more general situation. Indeed, we show that ifAis a real commutative algebraic algebra without divisors of zero, thenAis finite dimensional. Moreover, we prove by a simple algebraic method, there is no four-dimensional real commutative division algebra. Finally, we prove that every third-power associative real algebra, with unit element, without divisors of zero, and algebraic of degree ≠8, is quadratic. This last generalizes previously known results of Diankha et al. [6]. [ABSTRACT FROM PUBLISHER]

For any skewed-gentle algebra, we characterize its indecomposable Gorenstein projective modules explicitly and describe its Cohen–Macaulay Auslander algebra. We prove that skewed-gentle algebras are always Gorenstein, which is independent of the characteristic of the ground field, and the Cohen–Macaulay Auslander algebras of skewed-gentle algebras are also skewed-gentle algebras. [ABSTRACT FROM PUBLISHER]

The class ofextended Lie-type algebrascontains the ones of associative algebras, Lie algebras, Leibniz algebras, dual Leibniz algebras, pre-Lie algebras, and Lie-type algebras, etc. We focus on the class of extended Lie-type algebras graded by an Abelian groupGand study its structure, by stating, under certain conditions, a second Wedderburn-type theorem for this class of algebras. [ABSTRACT FROM PUBLISHER]

The rate of a graded moduleMover a standard graded algebraR,rateR(M), is a measure of the growth of the shifts in the minimal graded free resolution ofM. In this paper, we study the rate of Veronese modules ofM. More precisely, it is shown thatfor allc≥1. This extends a result of Aramova et al. [2] As a consequence of this, ifMis generated in degree zero, then, for allc≥max{rateR(M),Rate(R)}. Also, it is shown that, for all integerss,c≥1. In particular case, we give a simple proof to a result of Backelin. [ABSTRACT FROM PUBLISHER]

We show how the recently again discussedN-point Witt, Virasoro, and affine Lie algebras are genus zero examples of the multipoint versions of Krichever–Novikov-type algebras as introduced and studied by Schlichenmaier. Using this more general point of view, useful structural insights and an easier access to calculations can be obtained. The concept of almost-grading will yield information about triangular decompositions which are of importance in the theory of representations. As examples, the algebra of functions, vector fields, differential operators, current algebras, affine Lie algebras, Lie superalgebras, and their central extensions are studied. Very detailed calculations for the three-point case are given. [ABSTRACT FROM PUBLISHER]

In this paper, we study a new Lie superalgebra constructed by a 2|2-dimensional Balinsky–Novikov superalgebra, which is called the superalgebra ofW(2,2). It can be realized from semi product of theW-algebraW(2,2) and its module of the intermediate series. Finally, we determine all modules of the intermediate series over this superalgebra. Since it is difficult to do so directly, we make it by using modules of the intermediate series over the trivial super extension of the Witt algebra. [ABSTRACT FROM PUBLISHER]

In this paper, we investigate commutativity of ringRwith involution ∗ which admits a derivation satisfying certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Finally, we provide examples to show that various restrictions imposed in the hypotheses of our theorems are not superfluous. [ABSTRACT FROM PUBLISHER]

In this paper, using Larsson’s functor with irreducible𝔰𝔩2-modulesV, we construct a class ofℤ2-graded modules for the Virasoro-like algebra and itsq-analogs. We determine the irreducibility of these modules for finite-dimensional or infinite-dimensionalVusing a unified method. In particular, these modules provide new irreducible weight modules with infinite-dimensional weight spaces for the corresponding algebras. [ABSTRACT FROM PUBLISHER]

Motivated by a well-known theorem of Mathieu’s on Harish–Chandra modules over the Virasoro algebra and its super version, we show that an irreducible quasifinite module over two classes of Lie superalgebras𝒮(q) of Block type is either a highest or lowest weight module or else a module of the intermediate series ifq≠−1. For such a module over𝒮(−1), we give a rough classification. [ABSTRACT FROM PUBLISHER]

LetTbe a totally ordered set and letD(T) denotes the set of all cuts ofT. We prove the existence of a discrete valuation domainOvsuch thatTis order isomorphic to two special subsets of Spec(Ov). We prove that ifAis a ring (not necessarily commutative), whose prime spectrum is totally ordered and satisfies (K2), then there exists a totally ordered setU⊆Spec(A) such that the prime spectrum ofAis order isomorphic toD(U). We also present equivalent conditions for a totally ordered set to be a Dedekind totally ordered set. At the end, we present an algebraic geometry point of view. [ABSTRACT FROM AUTHOR]

The notion of𝒟-mutation pairs of subcategories in an abelian category is defined in this article. When (𝒵,𝒵) is a𝒟-mutation pair in an abelian category𝒜, the quotient category𝒵∕𝒟carries naturally a triangulated structure. Moreover, our result generalize the construction of the quotient triangulated category by Happel [10, Theorem 2.6]. Finally, we find a one-to-one correspondence between cotorsion pairs in𝒜and cotorsion pairs in the quotient category𝒵∕𝒟, and study homological finiteness of subcategories in a mutation pair. [ABSTRACT FROM AUTHOR]

In classical projective algebraic geometry, ℙnwas seen mainly as a linear subspace. The modern setting has produced in the last 40 years several remarkable abstract characterizations of projective space. We survey some interaction between these two points of view. [ABSTRACT FROM AUTHOR]

We investigate properties and describe examples of tilt-stable objects on a smooth complex projective threefold. We give a structure theorem on slope semistable sheaves of vanishing discriminant, and describe certain Chern classes for which every slope semistable sheaf yields a Bridgeland semistable object of maximal phase. Then, we study tilt stability as the polarization ω gets large, and give sufficient conditions for tilt-stability of sheaves of the following two forms: 1) twists of ideal sheaves or 2) torsion-free sheaves whose first Chern class is twice a minimum possible value. [ABSTRACT FROM PUBLISHER]

Toric geometry provides a bridge between the theory of polytopes and algebraic geometry: one can associate to each lattice polytope a polarized toric variety. In this article, we explore this correspondence to classify smooth lattice polytopes having small degree, extending a classification provided by Dickenstein, Di Rocco, and Piene. We follow their approach of interpreting the degree of a polytope as a geometric invariant of the corresponding polarized variety, and then apply techniques from Adjunction Theory and Mori Theory. [ABSTRACT FROM AUTHOR]

Let ⟨ K, ν ⟩ be a real closed valued field, and letS ⊆ Knbe an open semialgebraic set. Using tools from model theory, we find an algebraic characterization of rational functions which admit, onS, only values in the valuation ring. We use this result to deduce a criterion for a rational function to be bounded on an open semialgebraic subset of some irreducible variety over a real closed field or over an ordered field which is dense in its real closure. [ABSTRACT FROM AUTHOR]

Kempf proved that the theta divisor of a smooth projective curveChas rational singularities. In this paper we estimate the dimensions of the jet schemes of the theta divisor and show that all these schemes are irreducible. In particular, we recover Kempf's theorem in this way. For general projective smooth curves, our method also gives a formula for the log canonical threshold of the pair. [ABSTRACT FROM PUBLISHER]

We compute the dimensions and cohomology classes of the loci on a general hypersurface where the second fundamental form has rank at mostr. We also determine the number of hypersurfaces in a general pencil in ℙn, with, that contain a point where the second fundamental form has rankn − 1 − q. These results generalize many classical formulae. [ABSTRACT FROM PUBLISHER]

We prove boundedness of rationally-connected threefolds in ℙ6under some extra-assumptions. [ABSTRACT FROM PUBLISHER]

In this paper, we present a classification of quadratic homogeneous automorphisms in dimension 5 up to linear conjugation. As a consequence, we give an affirmative answer to the Rusek Conjecture in dimension 5. [ABSTRACT FROM AUTHOR]

In the mid 1970s, Hartshorne conjectured that, for alln > 7, any rank 2 vector bundles on ℙnis a direct sum of line bundles. This conjecture remains still open. In this paper, we construct indecomposable rank two vector bundles on a large class of Fano toric varieties. Unfortunately, this class does not contain ℙn. [ABSTRACT FROM AUTHOR]

Friedlander and Mazur proposed a conjecture of hard Lefschetz type for Lawson homology. We shall relate this conjecture to Suslin's conjecture for Lawson homology and hence to Grothendieck's standard conjectures. Applying theSL2-action on abelian varieties observed by Beauville, we show that for abelian varieties, this conjecture is equivalent to a weak form of a vanishing conjecture of Beauville type for Lawson homology. [ABSTRACT FROM AUTHOR]

B-construction is a way of obtaining a graded algebra from the triple consisting of an additive category, an object, and an autoequivalence, while C-construction is a way of obtaining an algebra (without unity) from the pair consisting of an additive category and a set of objects. In this article, we study and compare three important classes of algebras in noncommutative algebraic geometry and representation theory of finite dimensional algebras, namely, quantum polynomial algebras, preprojetive algebras and trivial extensions, via these constructions. [ABSTRACT FROM AUTHOR]

A clean ring is one in which every element is a sum of an idempotent and a unit. It is shown that every ring can be embedded in a clean ring as an essential ring extension. It is seen that the centre of a clean ring need not be a clean ring. There is no “clean hull” of a ring. A family of examples is given where there is a ringR, not a clean ring, embedded in a commutative clean ringSso that there is no clean ringT,R ⊆ T ⊆ S, minimal with that property. It is also shown that a commutative pm ring (each prime ideal is contained in a unique maximal ideal) cannot be extended to a clean ring by the adjunction of finitely many central idempotents. [ABSTRACT FROM PUBLISHER]