Relative FP-injective and FP-flat complexes and their model structures

In this paper, we introduce the notions of ${\rm FP}_n$-injective and ${\rm FP}_n$-flat complexes in terms of complexes of type ${\rm FP}_n$. We show that some characterizations analogous to that of injective, FP-injective and flat complexes exist for ${\rm FP}_n$-injective and ${\rm FP}_n$-flat complexes. We also introduce and study ${\rm FP}_n$-injective and ${\rm FP}_n$-flat dimensions of modules and complexes, and give a relation between them in terms of Pontrjagin duality. The existence of pre-envelopes and covers in this setting is discussed, and we prove that any complex has an ${\rm FP}_n$-flat cover and an ${\rm FP}_n$-flat pre-envelope, and in the case $n \geq 2$ that any complex has an ${\rm FP}_n$-injective cover and an ${\rm FP}_n$-injective pre-envelope. Finally, we construct model structures on the category of complexes from the classes of modules with bounded ${\rm FP}_n$-injective and ${\rm FP}_n$-flat dimensions, and analyze several conditions under which it is possible to connect these model structures via Quillen functors and Quillen equivalences.Comment: 41 page

Frobenius pairs in abelian categories: correspondences with cotorsion pairs, exact model categories, and Auslander-Buchweitz contexts

In this work, we revisit Auslander-Buchweitz Approximation Theory and find some relations with cotorsion pairs and model category structures. From the notions of relatives generators and cogenerators in Approximation Theory, we introduce the concept of left Frobenius pairs $(\mathcal{X},\omega)$ in an abelian category $\mathcal{C}$. We show how to construct from $(\mathcal{X},\omega)$ a projective exact model structure on $\mathcal{X}^\wedge$, as a result of Hovey-Gillespie Correspondence applied to two compatible and complete cotorsion pairs in $\mathcal{X}^\wedge$. These pairs can be regarded as examples of what we call cotorsion pairs relative to a thick subcategory of $\mathcal{C}$. We establish some correspondences between Frobenius pairs, relative cotorsion pairs, exact model structures and Auslander-Buchweitz contexts. Finally, some applications of these results are given in the context of Gorenstein homological algebra by generalizing some existing model structures on the categories of modules over Gorenstein and Ding-Chen rings, and by encoding the stable module category of a ring as a certain homotopy category. We also present some connections with perfect cotorsion pairs, covering classes, and cotilting modules.Comment: 54 pages, 10 figures. The statement and proof of 2.6.21 was corrected. Typos corrected. Section 4 was improved, and new results in Section 5 were adde

STB-VMM: Swin Transformer Based Video Motion Magnification

The goal of video motion magnification techniques is to magnify small motions in a video to reveal previously invisible or unseen movement. Its uses extend from bio-medical applications and deepfake detection to structural modal analysis and predictive maintenance. However, discerning small motion from noise is a complex task, especially when attempting to magnify very subtle, often sub-pixel movement. As a result, motion magnification techniques generally suffer from noisy and blurry outputs. This work presents a new state-of-the-art model based on the Swin Transformer, which offers better tolerance to noisy inputs as well as higher-quality outputs that exhibit less noise, blurriness, and artifacts than prior-art. Improvements in output image quality will enable more precise measurements for any application reliant on magnified video sequences, and may enable further development of video motion magnification techniques in new technical fields.Comment: Code available at: https://github.com/RLado/STB-VM

Cut cotorsion pairs

We present the concept of cotorsion pairs cut along subcategories of an abelian category. This provides a generalization of complete cotorsion pairs, and represents a general framework to find approximations restricted to certain subcategories. We also exhibit some connections between cut cotorsion pairs and Auslander-Buchweitz approximation theory, by considering relative analogs for Frobenius pairs and Auslander-Buchweitz contexts. Several applications are given in the settings of relative Gorenstein homological algebra, chain complexes and quasi-coherent sheaves, but also to characterize some important results on the Finitistic Dimension Conjecture, the existence of right adjoints of quotient functors by Serre subcategories, and the description of cotorsion pairs in triangulated categories as co-$t$-structures.Comment: 48 page

Balanced pairs, cotorsion triplets and quiver representations

©2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ This document is the Accepted version of a Published Work that appeared in final form in Proceedings of the Edinburgh Mathematical Society. To access the final edited and published work seehttps://doi.org/10.1017/S0013091519000270Balanced pairs appear naturally in the realm of relative homological algebra associated with the balance of right-derived functors of the Hom functor. Cotorsion triplets are a natural source of such pairs. In this paper, we study the connection between balanced pairs and cotorsion triplets by using recent quiver representation techniques. In doing so, we find a new characterization of abelian categories that have enough projectives and injectives in terms of the existence of complete hereditary cotorsion triplets. We also provide a short proof of the lack of balance for derived functors of Hom computed using flat resolutions, which extends the one given by Enochs in the commutative case