On the hierarchy of cohomological dimensions of groups

Using Auslander’s G-dimension, we assign a numerical invariant to any group Γ. It provides a refinement of the cohomological dimension and fits well into the well-known hierarchy of dimensions assigned already to Γ. We study this dimension and show its power in reflecting the properties of the underlying group. We also discuss its connections to relative and Tate cohomology of groups.     

On the Auslander–Reiten conjecture for Cohen–Macaulay rings and path algebras

In this note, it is shown that the validity of the Auslander–Reiten conjecture for a given d-dimensional Cohen–Macaulay local ring R depends on its validity for all direct summands of d-th syzygy of R-modules of finite length, provided R is an isolated singularity. Based on this result, it is shown that under a mild assumption on the base ring R, satisfying the Auslander–Reiten conjecture behaves well under completion and reduction modulo regular elements. In addition, it will turn out that, if R is a commutative Noetherian ring and 𝒬 a finite acyclic quiver, then the Auslander–Reiten conjecture holds true for the path algebra R𝒬, whenever so does R. Using this result, examples of algebras satisfying the Auslander–Reiten conjecture are presented.     

Spanier–Whitehead Categories of Resolving Subcategories and Comparison with Singularity Categories

Algebras and Representation Theory - Let $\mathcal {A}$ be an abelian category with enough projective objects, and let $\mathcal {X}$ be a quasi-resolving subcategory of $\mathcal {A}$ . In this...     

New results related to a conjecture of Moore

Let Γ be a group, Γ′ be a subgroup of Γ of finite index, and R be a ring with identity. Assume that M is an RΓ-module whose restriction to RΓ′ is projective. Moore’s conjecture: Assume that, for all ${x \in (\Gamma-\Gamma^{\prime})}$ , either there is an integer n such that ${1 \neq x^{n} \in \Gamma^{\prime}}$ or x has finite order and is invertible in R. Then M is also projective over RΓ. In this paper, we consider an analogue of this conjecture for injective modules. It turns out that the validity of the conjecture for injective modules implies the validity of it on projective and flat modules. It is also shown that the conjecture for injective modules is true whenever Γ belongs to Kropholler’s hierarchy ${{\bf LH}\mathfrak{F}}$ . In addition, assume that M is an RΓ-module whose restriction to RΓ′ is Gorenstein projective (resp. injective), it is proved that M is Gorenstein projective (resp. injective) over RΓ whenever Γ′ is a subgroup of Γ of finite index.     

Chitosan-Placental ECM Composite Thermos-Responsive Hydrogel as a Biomimetic Wound Dressing with Angiogenic Property

Attempts are being made to develop an ideal wound dressing with excellent biomechanical and biological properties. Here, a thermos-responsive hydrogel is fabricated using chitosan (CTS) with various concentrations (1%, 2.5%, and 5% w/v) of solubilized placental extracellular matrix (ECM) and 20% β-glycerophosphate to optimize a smart wound dressing hydrogel with improved biological behavior. The thermo-responsive CTS (TCTS) alone or loaded with ECMs (ECM-TCTS) demonstrate uniform morphology using SEM. TCTS and ECM1%-TCTS and ECM2.5%-TCTS show a gelation time of 5 min at 37 °C, while no gel formation is observed at 4 and 25 °C. ECM5%-TCTS forms gel at both 25 and 37 °C. The degradation and swelling ratios increase as the ECM content of the hydrogel increase. All the constructs show excellent biocompatibility in vitro and in vivo, however, the hydrogels with a higher concentration of ECM demonstrate better cell adhesion for fibroblast cells and induce expression of angiogenic factors (VEGF and VEGFR) from HUVEC. Only the ECM5%-TCTS has antibacterial activity against Acinetobacter baumannii ATCC 19606. The data obtained from the current study suggest the ECM2.5%-TCTS as an optimized smart biomimetic wound dressing with improved angiogenic properties now promises to proceed with pre-clinical and clinical investigations.