We approach the analysis of the extent of the projectivity of modules from a fresh perspective as we introduce the notion of relative subprojectivity. A module M and is said to be N -subprojective if for every epimorphism g : B → N and homomorphism f : M → N , there exists a homomorphism h : M → B such that gh = f . For a module M , the subprojectivity domain of M is defined to be the collection of all modules N such that M is N -subprojective. We consider, for every ring R, the subprojective profile of R, namely, the class of all subprojectivity domains for R modules. We show that the subprojective profile of R is a semilattice, and consider when this structure has coatoms or a smallest element. Modules whose subprojectivity domain is smallest as possible will be called subprojectively poor (sp-poor) or projectively indigent (pindigent) and those with co-atomic subprojectivy domain are said to be maximally subprojective. While we do not know if sp-poor modules and maximally subprojective modules exist over every ring, their existence is determined for various families. For example, we determine that artinian serial rings have sp-poor modules and attain the existence of maximally subprojective modules over the integers and for arbitrary V-rings. This work is a natural continuation to recent papers that have embraced the systematic study of the injective, projective and subinjective profiles of rings.
Given a ring R, we define its right i-profile (resp. right p-profile) to be the collection of injectivity domains (resp. projectivity domains) of its right R-modules. We study the lattice theoretic properties of these profiles and consider ways in which properties of the profiles may determine the structure of rings and viceversa. We show that the i-profile is isomorphic to an interval of the lattice of linear filters of right ideals of R, and is therefore modular and coatomic. In particular, we give a practical characterization of the profile of a right artinian ring. We show through an example that the p-profile of a ring is not necessarily a set, and also characterize the p-profile of a right perfect ring. The study of rings in terms of their (i or p-)profile was inspired by the study of rings with no right (i or p-)middle class, initiated in recent papers by Er, López-Permouth and Sökmez, and by Holston, López-Permouth and Orhan-Ertas. In this paper, we obtain further results about these rings and also we use our results to privede a characterization of a special class of QF rings in which the injectiviry and projectivity domains of all modules coincide.
In this manuscript we study braid varieties, a class of affine algebraic varieties associated to positive braids. Several geometric constructions are presented, including certain torus actions on braid varieties and holomorphic symplectic structures on their respective quotients. We also develop a diagrammatic calculus for correspondences between braid varieties and use these correspondences to obtain interesting stratifications of braid varieties and their quotients. It is shown that the maximal charts of these stratifications are exponential Darboux charts for the holomorphic symplectic structures, and we relate these strata to exact Lagrangian fillings of Legendrian links.
We study braid varieties and their relation to open positroid varieties. First, we construct a DG-algebra associated to certain braid words, possibly admitting negative crossings, show that its zeroth cohomology is an invariant under braid equivalence and positive Markov moves, and provide an explicit geometric model for its cohomology in terms of an affine variety and a set of locally nilpotent derivations. Second, we discuss four different types of braids associated to open positroid strata and show that their associated Legendrian links are all Legendrian isotopic. In particular, we prove that each open positroid stratum can be presented as the augmentation variety for different Legendrian fronts described in terms of either permutations, juggling patterns, cyclic rank matrices or Le diagrams. We also relate braid varieties to open Richardson varieties and brick manifolds, showing that the latter provide projective compactifications of braid varieties, with normal crossing divisors at infinity, and compatible stratifications. Finally, we state a conjecture on the existence and properties of cluster A-structures on braid varieties. Ona l bila Riqardsona Ne potomu, qtoby proqlaA. S. Puxkin, Evgeni i Onegin 1 varieties where only defined for positive braids words. The generalizations in this manuscript, which allow for negative crossings, will now relate the braid variety of a positroid braid for (u, w) with its positroid variety.3 ∆-equivalence of braids is imposed in this geometric context, rather than the standard notion of braid equivalence.Conjugations are forced to be βσ i ∼ σ n−i β, instead of βσ i ∼ σ i β, due to the necessary presence of a half-twist ∆.
We provide a homological construction of unitary simple modules of Cherednik and Hecke algebras of type A via BGG resolutions, solving a conjecture of Berkesch–Griffeth–Sam. We vastly generalize the conjecture and its solution to cyclotomic Cherednik and Hecke algebras over arbitrary ground fields, and calculate the Betti numbers and Castelnuovo–Mumford regularity of certain symmetric linear subspace arrangements.
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