K-theory of n-coherent rings

Let [Formula: see text] be a strong [Formula: see text]-coherent ring such that each finitely [Formula: see text]-presented [Formula: see text]-module has finite projective dimension. We consider [Formula: see text] the full subcategory of [Formula: see text]-Mod of finitely [Formula: see text]-presented modules. We prove that [Formula: see text] is an exact category, [Formula: see text] for every [Formula: see text] and we obtain an expression of [Formula: see text].

On rings with envelopes and covers regarding to C3, D3 and flat modules

In this paper, by taking the class of all [Formula: see text] (or [Formula: see text]) right [Formula: see text]-modules for general envelopes and covers, we characterize a semisimple artinian ring (or a right perfect ring) via [Formula: see text]-covers (or [Formula: see text]-envelopes) and a right [Formula: see text]-ring (or a right noetherian [Formula: see text]-ring) via [Formula: see text]-covers (or [Formula: see text]-envelopes). By using isosimple-projective preenvelope, we obtained that if [Formula: see text] is a semiperfect right FGF ring (or left coherent ring), then every isosimple right [Formula: see text]-module has a projective cover. Moreover, we also characterize semihereditary serial rings (respectively, hereditary artinian serial rings) in terms of epic flat (respectively, projective) envelopes.

Closing the category of finitely presented functors under images made constructive

For an additive category P we provide an explicit construction of a category Q(P) whose objects can be thought of as formally representing im(γ)im(ρ)∩im(γ) for given morphisms γ:A→B and ρ:C→B in P, even though P does not need to admit quotients or images. We show how it is possible to calculate effectively within Q(P), provided that a basic problem related to syzygies can be handled algorithmically. We prove an equivalence of Q(P) with the smallest subcategory of the category of contravariant functors from P to the category of abelian groups Ab which contains all finitely presented functors and is closed under the operation of taking images. Moreover, we characterize the abelian case: Q(P) is abelian if and only if it is equivalent to fp(Pop,Ab), the category of all finitely presented functors, which in turn, by a theorem of Freyd, is abelian if and only if P has weak kernels.The category Q(P) is a categorical abstraction of the data structure for finitely presented R-modules employed by the computer algebra system Macaulay2, where R is a ring. By our generalization to arbitrary additive categories, we show how this data structure can also be used for modeling finitely presented graded modules, finitely presented functors, and some not necessarily finitely presented modules over a non-coherent ring.

Acyclic Complexes and Gorenstein Rings

For a given class of modules [Formula: see text], let [Formula: see text] be the class of exact complexes having all cycles in [Formula: see text], and dw([Formula: see text]) the class of complexes with all components in [Formula: see text]. Denote by [Formula: see text][Formula: see text] the class of Gorenstein injective R-modules. We prove that the following are equivalent over any ring R: every exact complex of injective modules is totally acyclic; every exact complex of Gorenstein injective modules is in [Formula: see text]; every complex in dw([Formula: see text][Formula: see text]) is dg-Gorenstein injective. The analogous result for complexes of flat and Gorenstein flat modules also holds over arbitrary rings. If the ring is n-perfect for some integer n ≥ 0, the three equivalent statements for flat and Gorenstein flat modules are equivalent with their counterparts for projective and projectively coresolved Gorenstein flat modules. We also prove the following characterization of Gorenstein rings. Let R be a commutative coherent ring; then the following are equivalent: (1) every exact complex of FP-injective modules has all its cycles Ding injective modules; (2) every exact complex of flat modules is F-totally acyclic, and every R-module M such that M+ is Gorenstein flat is Ding injective; (3) every exact complex of injectives has all its cycles Ding injective modules and every R-module M such that M+ is Gorenstein flat is Ding injective. If R has finite Krull dimension, statements (1)–(3) are equivalent to (4) R is a Gorenstein ring (in the sense of Iwanaga).

The trailing terms ideal

In this paper, we address the following question: for a nonzero finitely generated ideal [Formula: see text] of a multivariate polynomial ring [Formula: see text] over a coherent ring [Formula: see text], fixing a monomial order [Formula: see text] on [Formula: see text], is the trailing terms ideal [Formula: see text] of [Formula: see text] (that is, the ideal generated by the trailing terms of the nonzero polynomials in [Formula: see text]) finitely generated? We show that while [Formula: see text] can be nonfinitely generated, it is always countably generated when the monomial order is Noetherian (graded monomial orders as instances).

Completing perfect complexes

This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented modules over a right coherent ring is the completion of the category of perfect complexes. The result extends to non-affine noetherian schemes and gives rise to a direct construction of the singularity category. The parallel theory of completion for abelian categories is compatible with the completion of derived categories. There are three appendices. The first one by Tobias Barthel discusses the completion of perfect complexes for ring spectra. The second one by Tobias Barthel and Henning Krause refines for a separated noetherian scheme the description of the bounded derived category of coherent sheaves as a completion. The final appendix by Bernhard Keller introduces the concept of a morphic enhancement for triangulated categories and provides a foundation for completing a triangulated category.

Modules Whose Endomorphism Rings Are (m, n)-Coherent

Let M be a right R-module with endomorphism ring S. We study the left (m, n)-coherence of S. It is shown that S is a left (m, n)-coherent ring if and only if the left annihilator [Formula: see text] is a finitely generated left ideal of Mn(S) for any M-m-generated submodule X of Mn if and only if every M-(n, m)-presented right R-module has an add M-preenvelope. As a consequence, we investigate when the endomorphism ring S is left coherent, left pseudo-coherent, left semihereditary or von Neumann regular.

A non-commutative analogue of Costa’s first conjecture

Let [Formula: see text] and [Formula: see text] be arbitrary fixed integers. We prove that there exists a ring [Formula: see text] such that: (1) [Formula: see text] is a right [Formula: see text]-ring; (2) [Formula: see text] is not a right [Formula: see text]-ring for each non-negative integer [Formula: see text]; (3) [Formula: see text] is not a right [Formula: see text]-ring [Formula: see text]for [Formula: see text], for each non-negative integer [Formula: see text]; (4) [Formula: see text] is a right [Formula: see text]-coherent ring; (5) [Formula: see text] is not a right [Formula: see text]-coherent ring. This shows the richness of right [Formula: see text]-rings and right [Formula: see text]-coherent rings, and, in particular, answers affirmatively a problem posed by Costa in [D. L. Costa, Parameterizing families of non-Noetherian rings, Comm. Algebra 22 (1994) 3997–4011.] when the ring in question is non-commutative.

(m,n)-Injective and (m,n)-coherent endomorphism rings of modules

Let [Formula: see text] and [Formula: see text] be fixed positive integers. [Formula: see text] is called a right [Formula: see text]-injective ring if every right [Formula: see text]-homomorphism from an [Formula: see text]-generated submodule of the right [Formula: see text]-module [Formula: see text] to [Formula: see text] extends to one from [Formula: see text] to [Formula: see text]; [Formula: see text] is called a right [Formula: see text]-coherent ring if each [Formula: see text]-generated submodule of the right [Formula: see text]-module [Formula: see text] is a finitely presented right [Formula: see text]-module. Let [Formula: see text] be a right [Formula: see text]-module. We study the [Formula: see text]-injectivity and [Formula: see text]-coherence of the endomorphism ring [Formula: see text] of [Formula: see text]. Some applications are also given.

Vortex dynamics and sound emission in excited high-speed jets

This work aims to study the dynamics of and noise generated by large-scale structures in a Mach 0.9 turbulent jet of Reynolds number $6.2\times 10^{5}$ using plasma-based excitation of shear layer instabilities. The excitation frequency is varied to produce individual or periodic coherent ring vortices in the shear layer. First, two-point cross-correlations are used between the acoustic near field and far field in order to identify the dominant noise source region. The large-scale structure interactions are then investigated by stochastically estimating time-resolved velocity fields using time-resolved near-field pressure traces and non-time-resolved planar velocity snapshots (obtained by particle image velocimetry) by means of an artificial neural network. The estimated time-resolved velocity fields show multiple mergings of large-scale structures in the shear layer, and indicate that disintegration of coherent ring vortices is the dominant aeroacoustic source mechanism for the jet studied here. However, the merging of vortices in the initial shear layer is also identified as a non-trivial noise source mechanism.