Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA new characterization of silting subcategories in the stable category of a Frobenius extriangulated category
76
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We give a new characterization of silting subcategories in the stable category of a Frobenius extriangulated category, generalizing the result of Di et al. (J. Algebra 525 (2019) 42-63) about the Auslander-Reiten type correspondence for silting subcategories over triangulated categories. More specifically, for any Frobenius extriangulated category $mathcal{C}$, we establish a bijective correspondence between silting subcategories of the stable category $underline{mathcal{C}}$ and certain covariantly finite subcategories of $mathcal{C}$. As a consequence, a characterization of silting subcategories in the stable category of a Frobenius exact category is given. This result is applied to homotopy categories over abelian categories with enough projectives, derived categories over Grothendieck categories with enough projectives as well as to the stable category of Gorenstein projective modules over a ring $R$.
rate research
Read More
For a simple Lie algebra, over $mathbb{C}$, we consider the weight which is the sum of all simple roots and denote it $tilde{alpha}$. We formally use Kostants weight multiplicity formula to compute the dimension of the zero-weight space. In type $A_r$, $tilde{alpha}$ is the highest root, and therefore this dimension is the rank of the Lie algebra. In type $B_r$, this is the defining representation, with dimension equal to 1. In the remaining cases, the weight $tilde{alpha}$ is not dominant and is not the highest weight of an irreducible finite-dimensional representation. Kostants weight multiplicity formula, in these cases, is assigning a value to a virtual representation. The point, however, is that this number is nonzero if and only if the Lie algebra is classical. This gives rise to a new characterization of the exceptional Lie algebras as the only Lie algebras for which this value is zero.
We provide an intrinsic definition of the fundamental group of a linear category over a ring as the automorphism group of the fibre functor on Galois coverings. If the universal covering exists, we prove that this group is isomorphic to the Galois group of the universal covering. The grading deduced from a Galois covering enables us to describe the canonical monomorphism from its automorphism group to the first Hochschild-Mitchell cohomology vector space.
Starting with a k-linear or DG category admitting a (homotopy) Serre functor, we construct a k-linear or DG 2-category categorifying the Heisenberg algebra of the numerical K-group of the original category. We also define a 2-categorical analogue of the Fock space representation of the Heisenberg algebra. Our construction generalises and unifies various categorical Heisenberg algebra actions appearing in the literature. In particular, we give a full categorical enhancement of the action on derived categories of symmetric quotient stacks introduced by Krug, which itself categorifies a Heisenberg algebra action proposed by Grojnowski.
We introduce pre-silting and silting subcategories in extriangulated categories and generalize the silting theory in triangulated categories. We prove that the silting reduction $mathcal B/({rm thick}mathcal W)$ of an extriangulated category $mathcal B$ with respect to a pre-silting subcategory $mathcal W$ can be realized as a certain subfactor category of $mathcal B$. This generalizes the result by Iyama-Yang. In particular, for a Gorenstein algebra, we get the relative version of the description of the singularity category due to Happel and Chen-Zhang by this reduction.
A well-known theorem of Buchweitz provides equivalences between three categories: the stable category of Gorenstein projective modules over a Gorenstein algebra, the homotopy category of acyclic complexes of projectives, and the singularity category. To adapt this result to $N$-complexes, one must find an appropriate candidate for the $N$-analogue of the stable category. We identify this $N$-stable category via the monomorphism category and prove Buchweitzs theorem for $N$-complexes over a Frobenius exact abelian category. We also compute the Serre functor on the $N$-stable category over a self-injective algebra and study the resultant fractional Calabi-Yau properties.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
