What makes a complex a virtual resolution?

Virtual resolutions are homological representations of finitely generated Pic ( X ) \text {Pic}(X) -graded modules over the Cox ring of a smooth projective toric variety. In this paper, we identify two algebraic conditions that characterize when a chain complex of graded free modules over the Cox ring is a virtual resolution. We then turn our attention to the saturation of Fitting ideals by the irrelevant ideal of the Cox ring and prove some results that mirror the classical theory of Fitting ideals for Noetherian rings.

Designing Modular Cell-free Systems for Tunable Biotransformation of l-phenylalanine to Aromatic Compounds

Cell-free systems have been used to synthesize chemicals by reconstitution of in vitro expressed enzymes. However, coexpression of multiple enzymes to reconstitute long enzymatic pathways is often problematic due to resource limitation/competition (e.g., energy) in the one-pot cell-free reactions. To address this limitation, here we aim to design a modular, cell-free platform to construct long biosynthetic pathways for tunable synthesis of value-added aromatic compounds, using (S)-1-phenyl-1,2-ethanediol ((S)-PED) and 2-phenylethanol (2-PE) as models. Initially, all enzymes involved in the biosynthetic pathways were individually expressed by an E. coli-based cell-free protein synthesis (CFPS) system and their catalytic activities were confirmed. Then, three sets of enzymes were coexpressed in three cell-free modules and each with the ability to complete a partial pathway. Finally, the full biosynthetic pathways were reconstituted by mixing two related modules to synthesize (S)-PED and 2-PE, respectively. After optimization, the final conversion rates for (S)-PED and 2-PE reached 100 and 82.5%, respectively, based on the starting substrate of l-phenylalanine. We anticipate that the modular cell-free approach will make a possible efficient and high-yielding biosynthesis of value-added chemicals.

A class of non-weight modules of 𝑈𝑝(𝖘𝖑2) and Clebsch–Gordan type formulas

In this paper, we construct a class of new modules for the quantum group U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) which are free of rank 1 when restricted to C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] . The irreducibility of these modules and submodule structure for reducible ones are determined. It is proved that any C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] -free U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) -module of rank 1 is isomorphic to one of the modules we constructed, and their isomorphism classes are obtained. We also investigate the tensor products of the C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] -free modules with finite-dimensional simple modules over U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) , and for the generic cases, we obtain direct sum decomposition formulas for them, which are similar to the well-known Clebsch–Gordan formula for tensor products between finite-dimensional weight modules over U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) .