Search

Let Λ be a Noetherian algebra and T ∈ Λ - mod be an n -tilting module with Γ = End Λ (T). Then Λ satisfies Auslander–Reiten conjecture if and only if so does Γ. As a consequence, we obtain that Auslander–Reiten conjecture is left and right symmetric for a certain Noetherian algebra. [ABSTRACT FROM AUTHOR]

We study zero-cycles in families of rationally connected varieties. We show that for a smooth projective scheme over a henselian discrete valuation ring the restriction of relative zero cycles to the special fiber induces an isomorphism on Chow groups if the special fiber is separably rationally connected. We further extend this result to certain higher Chow groups and develop conjectures in the non-smooth case. Our main results generalise a result of Kollár (Publ. Res. Inst. Math. Sci. 40(3):689–708, 2004). [ABSTRACT FROM AUTHOR]

We study the structure of Jacobians of geometrically reduced curves over arbitrary (i.e., not necessarily perfect) fields. We show that, while such a group scheme cannot in general be decomposed into an affine and an Abelian part as over perfect fields, several important structural results for these group schemes nevertheless have close analoga over imperfect fields. We apply our results to prove two conjectures due to Bosch-Lütkebohmert-Raynaud about the existence of Néron models and Néron lft-models over excellent Dedekind schemes in the special case of Jacobians of geometrically reduced curves. Finally, we prove some existence results for semi-factorial models and related objects for general geometrically integral curves in the local case. [ABSTRACT FROM AUTHOR]

Let Γ be a congruence subgroup of level N in GL n (Z). Let P be a maximal Q -parabolic subgroup of GL n / Q , with unipotent radical U , and let Q = (P ∩ Γ) / (U ∩ Γ). Let p > dim Q ⁡ (U (Q)) + 1 be a prime number that does not divide N. Let M be a (U , p) -admissible Γ-module. Consider the Lyndon-Hochschild-Serre spectral sequence arising from the exact sequence 1 → U ∩ Γ → P ∩ Γ → Q → 1 , which abuts to H ⁎ (P ∩ Γ , M). We show that if M is a trivial U ∩ Γ -module, then certain classes in the E 2 page survive to E ∞. We use this to obtain information about classes in H ⁎ (P ∩ Γ , M) even if M is not a trivial U ∩ Γ -module. This information will be used in future work to prove a Serre-type conjecture for sums of two irreducible Galois representations. [ABSTRACT FROM AUTHOR]

We consider stratifying ideals of finite dimensional algebras in relation with Morita contexts. A Morita context is an algebra built on a data consisting of two algebras, two bimodules and two morphisms. For a strongly stratifying Morita context - or equivalently for a strongly stratifying ideal - we show that Han's conjecture holds if and only if it holds for the diagonal subalgebra. The main tool is the Jacobi-Zariski long exact sequence. One of the main consequences is that Han's conjecture holds for an algebra admitting a strongly (co-)stratifying chain whose steps verify Han's conjecture. If Han's conjecture is true for local algebras and an algebra Λ admits a primitive strongly (co-)stratifying chain, then Han's conjecture holds for Λ. [ABSTRACT FROM AUTHOR]

We prove that if B\subseteq A is an extension of finite dimensional algebras such that the projective dimension of A/B as a B-bimodule is finite, then the finitistic dimension of B is finite whenever the finitistic dimension of A is finite. We exhibit examples demonstrating that the algebra B appearing in such an extension can be more complicated than A. [ABSTRACT FROM AUTHOR]

In this paper, we show that the conjectural mod p local Langlands correspondence can be realized in the mod p cohomology of the Lubin–Tate towers. The proof utilizes a well-known conjecture of Buzzard–Diamond–Jarvis [8, Conjecture 4.9], a study of completed cohomology of the ordinary and supersingular locus of the Shimura curves for a totally real field F and of mod l (≠ p) local Langlands correspondence as given by Emerton–Helm [20]. In the case of modular curves, a similar theorem was obtained by Chojecki [13]. [ABSTRACT FROM AUTHOR]

Motivated by the L-space conjecture, we investigate various notions of order-detection of slopes on knot manifolds. These notions are designed to characterise when rational homology 3-spheres, obtained by gluing compact manifolds along torus boundary components, have leftorderable fundamental groups and when a Dehn filling of a knot manifold has a left-orderable fundamental group. Our developments parallel the results by Hanselman et al. (2020) in the case of Heegaard Floer slope detection and by Boyer et al. (2021) in the case of foliation slope detection, leading to several conjectured structure theorems that connect relative Heegaard Floer homology and the boundary behaviour of co-oriented taut foliations with the set of left-orders supported by the fundamental group of a 3-manifold. The dynamics of the actions of 3-manifold groups on the real line play a key role in our constructions and proofs. Our analysis leads to conjectured dynamical constraints on such actions in the case where the underlying manifold is Floer simple. [ABSTRACT FROM AUTHOR]

We obtain a homological characterisation of virtually free-by-cyclic groups among groups that are hyperbolic and virtually compact special. As a consequence, we show that many groups known to be coherent actually possess the stronger property of being virtually free-by-cyclic. In particular, we show that all one-relator groups with torsion are virtually free-by-cyclic, solving a conjecture of Baumslag. [ABSTRACT FROM AUTHOR]

Co-Gorenstein algebras were introduced by Beligiannis in [A. Beligiannis, The homological theory of contravariantly finite subcategories: Auslander–Buchweitz contexts, Gorenstein categories and co-stabilization, Comm. Algebra28(10) (2000) 4547–4596]. In [S. Kvamme and R. Marczinzik, Co-Gorenstein algebras, Appl. Categorical Struct.27(3) (2019) 277–287], the authors propose the following conjecture (co-GC): if Ω n (m o d A) is extension closed for all n ≤ 1 , then A is right co-Gorenstein, and they prove that the generalized Nakayama conjecture implies the co-GC, also that the co-GC implies the Nakayama conjecture. In this paper, we characterize the subcategory Ω ∞ (m o d A) for algebras of Ω n -finite representation type. As a consequence, we characterize when a truncated path algebra is a co-Gorenstein algebra in terms of its associated quiver. We also study the behavior of Artin algebras of Ω ∞ -infinite representation type. Finally, an example of a non-Gorenstein algebra of Ω ∞ -infinite representation type and an example of a finite dimensional algebra with infinite ϕ -dimension are presented. [ABSTRACT FROM AUTHOR]

Let G be a classical group defined over a local field F of characteristic zero. For any irreducible admissible representation \pi of G(F), which is of Casselman-Wallach type if F is archimedean, we extend the study of spectral decomposition of local descents by Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535] for special orthogonal groups over non-archimedean local fields to more general classical groups over any local field F. In particular, if \pi has a generic local L-parameter, we introduce the spectral first occurrence index {\mathfrak {f}}_{\mathfrak {s}}(\pi) and the arithmetic first occurrence index {\mathfrak {f}}_{{\mathfrak {a}}}(\pi) of \pi and prove in this paper that {\mathfrak {f}}_{\mathfrak {s}}(\pi)={\mathfrak {f}}_{{\mathfrak {a}}}(\pi). Based on the theory of consecutive descents of enhanced L-parameters developed by Jiang, Liu, and Zhang [Arithmetic wavefront sets and generic L-packets, arXiv:2207.04700], we are able to show in this paper that the first descent spectrum consists of all discrete series representations, which determines explicitly the branching decomposition problem by means of the relevant arithmetic data and extends the main result (Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535], Theorem 1.7) to broader generality. [ABSTRACT FROM AUTHOR]

A long standing conjecture in Hamiltonian Dynamics states that every contact form on the standard contact sphere S^{2n+1} has at least n+1 simple periodic Reeb orbits. In this work, we consider a refinement of this problem when the contact form has a suitable symmetry and we ask if there are at least n+1 simple symmetric periodic orbits. We show that there is at least one symmetric periodic orbit for any contact form and at least two symmetric closed orbits whenever the contact form is dynamically convex. [ABSTRACT FROM AUTHOR]

We prove that a virtually periodic object in an abelian category gives rise to a non-vanishing result on certain Hom groups in the singularity category. Consequently, for any artin algebra with infinite global dimension, its singularity category has no silting subcategory, and the associated differential graded Leavitt algebra has a non-vanishing cohomology in each degree. We verify the Singular Presilting Conjecture for singularly-minimal algebras and ultimately-closed algebras. We obtain a trichotomy on the Hom-finiteness of the cohomologies of differential graded Leavitt algebras. [ABSTRACT FROM AUTHOR]

Let \Gamma \subseteq \mathbb {N} be a numerical semigroup. In this paper, we prove an upper bound for the Betti numbers of the semigroup ring of \Gamma which depends only on the width of \Gamma, that is, the difference between the largest and the smallest generator of \Gamma. In this way, we make progress towards a conjecture of Herzog and Stamate [J. Algebra 418 (2014), pp. 8–28]. Moreover, for 4-generated numerical semigroups, the first significant open case, we prove the Herzog-Stamate bound for all but finitely many values of the width. [ABSTRACT FROM AUTHOR]

It is well known that the Eisenbud-Goto regularity conjecture is true for arithmetically Cohen-Macaulay varieties, projective curves, smooth surfaces, smooth threefolds in \mathbb {P}^5, and toric varieties of codimension two. After J. McCullough and I. Peeva constructed counterexamples in 2018, it has been an interesting question to find the categories such that the Eisenbud-Goto conjecture holds. So far, surface counterexamples have not been found while counterexamples of any dimension greater or equal to 3 are known. In this paper, we construct counterexamples to the Eisenbud-Goto conjecture for projective surfaces in \mathbb {P}^4 and investigate projective invariants, cohomological properties, and geometric properties. The counterexamples are constructed via binomial rational maps between projective spaces. [ABSTRACT FROM AUTHOR]

We develop techniques to construct explicit symplectic Lefschetz fibrations over the 2-sphere with any prescribed signature σ and any spin type when σ is divisible by 16. This solves a long-standing conjecture on the existence of such fibrations with positive signature. As applications, we produce symplectic 4-manifolds that are homeomorphic but not diffeomorphic to connected sums of S² ×S², with the smallest topology known to date, as well as larger examples as symplectic Lefschetz fibrations. [ABSTRACT FROM AUTHOR]

We show that counterexamples of Iyengar and Walker to the algebraic version of Gunnar Carlsson's conjecture on the rank of the homology of a free complex can be extended to examples over any finite group with many choices of the complex. [ABSTRACT FROM AUTHOR]

Given a space with a circle action, we study certain cocyclic chain complexes and prove a theorem relating cyclic homology to S¹ -equivariant homology, in the spirit of celebrated work of Jones. As an application, we describe a chain level refinement of the gravity algebra structure on the (negative) S¹ -equivariant homology of the free loop space of a closed oriented smooth manifold, based on work of Irie on chain level string topology and work of Ward on an S¹ -equivariant version of operadic Deligne’s conjecture. [ABSTRACT FROM AUTHOR]

We address the problem of when the tensor product of two finitely generated modules over a Cohen-Macaulay local ring is Ulrich in the generalized sense of Goto et al., and in particular in the original sense from the 80's. As applications, besides freeness criteria for modules, characterizations of complete intersections, and an Ulrich-based approach to the long-standing Berger's conjecture, we give simple proofs that two celebrated homological conjectures, namely the Huneke-Wiegand and the Auslander-Reiten problems, are true for the class of Ulrich modules. [ABSTRACT FROM AUTHOR]

Our main goal in this note is to use a version of a lemma by Hanes and Huneke to provide characterizations of when certain one-dimensional reduced local rings are regular. This is of interest in view of the long-standing Berger's Conjecture (the ring is predicted to be regular if its universally finite differential module is torsion-free), which in fact we show to hold under suitable additional conditions, mostly toward the G-regular case of the conjecture. Furthermore, applying the same lemma to a Cohen-Macaulay local ring which is locally Gorenstein on the punctured spectrum but of arbitrary dimension, we notice a numerical characterization of when an ideal is strongly non-obstructed and of when a given semidualizing module is free. [ABSTRACT FROM AUTHOR]

We study \mathbb {R}^k \times \mathbb {Z}^\ell actions on arbitrary compact manifolds with a projectively dense set of Anosov elements and 1-dimensional coarse Lyapunov foliations. Such actions are called totally Cartan actions. We completely classify such actions as built from low-dimensional Anosov flows and diffeomorphisms and affine actions, verifying the Katok-Spatzier conjecture for this class. This is achieved by introducing a new tool, the action of a dynamically defined topological group describing paths in coarse Lyapunov foliations, and understanding its generators and relations. We obtain applications to the Zimmer program. [ABSTRACT FROM AUTHOR]

Given an odd prime number p and a p-stabilized Artin representation \rho over \mathbb {Q}, we introduce a family of p-adic Stark regulators and we formulate an Iwasawa-Greenberg main conjecture and a p-adic Stark conjecture which can be seen as an explicit strengthening of conjectures by Perrin-Riou and Benois in the context of Artin motives. We show that these conjectures imply the p-part of the Tamagawa number conjecture for Artin motives at s=0 and we obtain unconditional results on the torsionness of Selmer groups. We also relate our new conjectures with various main conjectures and variants of p-adic Stark conjectures that appear in the literature. In the case of monomial representations, we prove that our conjectures are essentially equivalent to some newly introduced Iwasawa-theoretic conjectures for Rubin-Stark elements. We derive from this a p-adic Beilinson-Stark formula for finite-order characters of an imaginary quadratic field in which p is inert. Along the way, we prove that the Gross-Kuz'min conjecture unconditionally holds for abelian extensions of imaginary quadratic fields. [ABSTRACT FROM AUTHOR]

Let (R,\mathfrak {m}_R,\mathbb {k}) be a one-dimensional complete local reduced \mathbb {k}-algebra over a field of characteristic zero. The ring R is said to be quasihomogeneous if there exists a surjection \Omega _R\twoheadrightarrow \mathfrak {m} where \Omega _R denotes the module of differentials. We present two characterizations of quasihomogeniety of R in the case when R is a domain. The first one on the valuation semigroup of R and the other on the trace ideal of the module \Omega _R. [ABSTRACT FROM AUTHOR]

Let R be an associative Noetherian unital noncommutative ring R. We introduce the functor PΓP over the category of R-modules and use it to characterize P-semiprime. P-semisecond R-modules also characterized by the functor PΛP. We also show that the Greenless-May type Duality (GM) and Matlis Greenless-May Equality(MGM) hold over the full subcategory of R-Mod consisting of P-semiprime and P-semisecond modules. Finally, we generate a one-sided right ideal PΓP(R), which gives an equivalent formulation to solve Köthe conjecture positively or negatively. [ABSTRACT FROM AUTHOR]

Let G be the fundamental group of a three-manifold. By piecing together many known facts about three manifold groups, we establish two properties of the group ring \mathbb {C}G. We show that if G has rational cohomological dimension two, then \mathbb {C}G is coherent. We also show that if G is torsion-free, then G satisfies the Strong Atiyah Conjecture over \mathbb {C} and hence that \mathbb {C}G satisfies Kaplansky's Zero-divisor Conjecture. [ABSTRACT FROM AUTHOR]

Kim gave a new proof of Siegel's Theorem that there are only finitely many S-integral points on \mathbb {P}^1_\mathbb {Z}\setminus \{0,1,\infty \}. One advantage of Kim's method is that it in principle allows one to actually find these points, but the calculations grow vastly more complicated as the size of S increases. In this paper, we implement a refinement of Kim's method to explicitly compute various examples where S has size 2 which has been introduced by Betts and Dogra. In so doing, we exhibit new examples of a natural generalization of a conjecture of Kim. [ABSTRACT FROM AUTHOR]

Over a field of characteristic 0, we construct a minimal set of generators of the defining ideals of closures of nilpotent conjugacy class in the set of n \times n matrices. This modifies a conjecture of Weyman and provides a complete answer to it. [ABSTRACT FROM AUTHOR]

We construct a co-t-structure on the derived category of coherent sheaves on the nilpotent cone \mathcal {N} of a reductive group, as well as on the derived category of coherent sheaves on any parabolic Springer resolution. These structures are employed to show that the push-forwards of the "exotic parity objects" (considered by Achar, Hardesty, and Riche [Transform. Groups 24 (2019), pp. 597–657]), along the (classical) Springer resolution, give indecomposable objects inside the coheart of the co-t-structure on \mathcal {N}. We also demonstrate how the various parabolic co-t-structures can be related by introducing an analogue to the usual translation functors. As an application, we give a proof of a scheme-theoretic formulation of the relative Humphreys conjecture on support varieties of tilting modules in type A for p>h. [ABSTRACT FROM AUTHOR]

Let A be a finite dimensional hereditary algebra over an algebraically closed field k. In this paper, we study the tilting quiver of A from the viewpoint of τ -tilting theory. First, we prove that there exists an isomorphism between the support τ -tilting quiver Q(s τ -tilt A) of A and the tilting quiver Q(tilt A ¯ ) of the duplicated algebra A ¯ . Then, we give a new method to calculate the number of arrows in the tilting quiver Q(tilt A) when A is representation-finite. Finally, we study the conjecture given by Happel and Unger, which claims that each connected component of Q(tilt A) contains only finitely many non-saturated vertices. We provide an example to show that this conjecture does not hold for some algebras whose quivers are wild with at least four vertices. [ABSTRACT FROM AUTHOR]

We develop a strategy to classify the components of the space of sections of a del Pezzo fibration over P¹ . In particular, we prove the Movable Bend-and-Break lemma for del Pezzo fibrations. Our approach is motivated by Geometric Manin’s Conjecture and proves upper bounds on the associated counting function. We also give applications to enumerativity of Gromov–Witten invariants and to the study of the Abel–Jacobi map. [ABSTRACT FROM AUTHOR]

We prove the p-part of the strong Stark conjecture for every totally odd character and every odd prime p. Let L/K be a finite Galois CM-extension with Galois group G, which has an abelian Sylow p-subgroup for an odd prime p. We give an unconditional proof of the minus p-part of the equivariant Tamagawa number conjecture for the pair (h^0(\operatorname {Spec}(L)), \mathbb {Z}[G]) under certain restrictions on the ramification behavior in L/K. [ABSTRACT FROM AUTHOR]

In this note, we show that any epimorphism originating at a von Neumann regular ring (not necessary commutative) is a universal localization. As an application, we prove that the Telescope Conjecture holds for the unbounded derived categories of von Neumann regular rings (not necessary commutative). [ABSTRACT FROM AUTHOR]

Following the recent advances in the study of groups of circle diffeomorphisms, we describe an efficient way of classifying the topological dynamics of locally discrete, finitely generated, virtually free subgroups of the group Diff+ω(S¹) of orientation-preserving real-analytic circle diffeomorphisms, which include all subgroups of Diff+ω(S¹) acting with an invariant Cantor set. An important tool that we develop, of independent interest, is the extension of classical ping-pong lemma to actions of fundamental groups of graphs of groups. Our main motivation is an old conjecture by Dippolito [Ann. of Math. (2) 107 (1978), 403-453] from foliation theory, which we solve in this restricted but significant setting: this and other consequences of the classification will be treated in more detail in a companion work (by a slightly different list of authors). [ABSTRACT FROM AUTHOR]

Let f_t(z)=z^2+t. For any z\in \mathbb {Q}, let S_z be the collection of t\in \mathbb {Q} such that z is preperiodic for f_t. In this article, assuming a well-known conjecture of Flynn, Poonen, and Schaefer [Duke Math. J. 90 (1997), pp. 435–463], we prove a uniform result regarding the size of S_z over z\in \mathbb {Q}. In order to prove it, we need to determine the set of rational points on a specific non-hyperelliptic curve C of genus 4 defined over \mathbb {Q}. We use Chabauty's method, which requires us to determine the Mordell-Weil rank of the Jacobian J of C. We give two proofs that the rank is 1: an analytic proof, which is conditional on the BSD rank conjecture for J and some standard conjectures on L-series, and an algebraic proof, which is unconditional, but relies on the computation of the class groups of two number fields of degree 12 and degree 24, respectively. We finally combine the information obtained from both proofs to provide a numerical verification of the strong BSD conjecture for J. [ABSTRACT FROM AUTHOR]

We prove that a hyper-Kähler fourfold satisfying a mild topological assumption is of K3^{[2]} deformation type. This proves in particular a conjecture of O'Grady stating that hyper-Kähler fourfolds of K3^{[2]} numerical type are of K3^{[2]} deformation type. Our topological assumption concerns the existence of two integral degree-2 cohomology classes satisfying certain numerical intersection conditions. There are two main ingredients in the proof. We first prove a topological version of the statement, by showing that our topological assumption forces the Betti numbers, the Fujiki constant, and the Huybrechts–Riemann–Roch polynomial of the hyper-Kähler fourfold to be the same as those of K3^{[2]} hyper-Kähler fourfolds. The key part of the article is then to prove the hyper-Kähler SYZ conjecture for hyper-Kähler fourfolds for divisor classes satisfying the numerical condition mentioned above. [ABSTRACT FROM AUTHOR]