A non-commutative differential module approach to Alexander modules

The theory of R. Crowell on derived modules is approached within the theory of non-commutative differential modules. We also seek analogies to the theory of cotangent complex from differentials in the commutative ring setting. Finally, we give examples motivated from the theory of Galois coverings of curves.

Improved neuroendocrine intelligent control for a permanent magnet linear synchronous motor

Given the complex characteristics of permanent magnet linear synchronous motors and the external interference encountered during their operation, controlled precision and efficiency are necessary. In this paper, the mechanism of biological endocrine hormone regulation is analyzed, and an intelligent controller based on the obtained neuroendocrine algorithm is implemented in the control system of a permanent magnet linear synchronous motor. The controller mainly includes a hypothalamic regulation module, a single-neuron proportion integration differential module, and an ultrashort feedback module. It is designed and referenced to the long feedback, short feedback, and ultrashort feedback loop mechanisms of neuroendocrine hormone regulation and abides by the principle of human neuroendocrine hormone regulation. The antagonistic hormone regulation module achieves rapid and stable elimination of errors through the fusion of enhanced regulation with regulation inhibition, and the single-neuron proportion integration differential module enhances the adaptive and self-learning capabilities of the control system. The proposed control is successfully used in a permanent magnet linear synchronous motor, and the experimental results show that the controller presents many advantages, such as fast dynamic responses, strong online adjustment ability, and good running stability in the control system, all of which improve the robustness of the control system.

On Fitting Ideals of Kahler Differential Module

Let k be an algebraically closed field of characteristic zero, and R / I and S / J be algebras over k . Ω 1 ( R / I ) and Ω 1 ( S / J ) denote universal module of first order derivation over k . The main result of this paper asserts that the first nonzero Fitting ideal Ω 1 ( R / I ⊗ k S / J ) is an invertible ideal, if the first nonzero Fitting ideals Ω 1 ( R / I ) and Ω 1 ( S / J ) are invertible ideals. Then using this result, we conclude that the projective dimension of Ω 1 ( R / I ⊗ k S / J ) is less than or equal to one.

A Presentation of the Kähler Differential Module for a Fat Point Scheme in ℙn1 × … × ℙnk

Let 핐 be a fat point scheme in ℙn1 × … × ℙnk over a field K of characteristic zero. In this paper we introduce the multi-graded Kähler differential module for 핐 and we establish a short exact sequence of this module in terms of the thickening of 핐.

Gorenstein homological theory for differential modules

We show that a differential module is Gorenstein projective (injective, respectively) if and only if its underlying module is Gorenstein projective (injective, respectively). We then relate the Ringel–Zhang theorem on differential modules to the Avramov–Buchweitz–Iyengar notion of projective class of differential modules and prove that for a ring R there is a bijective correspondence between projectively stable objects of split differential modules of projective class not more than 1 and R-modules of projective dimension not more than 1, and this is given by the homology functor H and stable syzygy functor ΩD. The correspondence sends indecomposable objects to indecomposable objects. In particular, we obtain that for a hereditary ring R there is a bijective correspondence between objects of the projectively stable category of Gorenstein projective differential modules and the category of all R-modules given by the homology functor and the stable syzygy functor. This gives an extended version of the Ringel–Zhang theorem.