Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.00913

Auslander-type grade conditions and subcategories related to syzygy modules

Yuya Otake
View original source ↗Read source document ↗Export metadata ↓

Abstract

The question of when a syzygy module or a module satisfying a Serre condition satisfies the cohomological vanishing conditions known as higher torsionfreeness has been extensively studied. In this paper, we address these questions using grade conditions introduced and studied by Auslander-Bridger and Auslander-Reiten. We also investigate homological conditions on Ext modules and, as an application, give a partial answer over commutative rings to a question of Auslander-Bridger concerning whether higher reflexive modules are higher torsionfree.

Explore related subjects

math.RTmath.ACmath.RA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yuya Otake. 2026-10-31. Auslander-type grade conditions and subcategories related to syzygy modules. https://arxiv.org/abs/2611.00913

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A lecture note on covering theory in representation theory of algebras

Covering theory is an important tool in representation theory of algebras, however, the results and the proofs are scattered in the literature. We give an introduction to covering theory at a level as elementary as possible and in a more general way on some basic notions than the existing literature does.

math.RT↗
arXiv2026

On the Orthogonal Projections

Let $A$ be a finite-dimensional algebra and $e$ a rigid presentation. We give an explicit formula for the left adjoint to the inclusion $\operatorname{rep}(e^\perp)\hookrightarrow \operatorname{rep}(A)$ and construct a bijection between presentations in $\operatorname{rep}(e^\perp)$ and presentations compatible with $e$. The induced bijection on weights preserves generic $\operatorname{E}$-invariants and canonical decompositions. If $A$ is Jacobian, the reduction algebra is Jacobian for every rigid $e$. We prove mutation formulas for the signed completions and for the dimension vectors of the simple objects of $\operatorname{rep}(e^\perp)$, enabling an algorithm to find the projected quiver with potential when $e$ is extended-reachable. Additionally, we introduce a modified projection for quivers with potentials that preserves general presentations. For applications to cluster algebras, we establish a connection to the stabilization functors.

math.RT↗
arXiv2026

Compatible gradings on dg models and cluster equivalence

Let $\La_1$ and $\La_2$ be finite-dimensional $\ta_2$-finite complex algebras of global dimension at most two. We prove that, if they are cluster equivalent, then they admit $\bbZ$-gradings for which the bounded derived categories of graded modules are triangle equivalent. The equivalence is not required to commute with the internal grading shifts. The proof uses common multigraded dg models, relative gradings on two-term silting objects, and covering quotients. We also consider tilting lifts of cluster-tilting objects with endomorphism algebras of global dimension at most two. In acyclic cluster type, the existence of such a lift for every cluster-tilting object implies piecewise heredity. We construct a counterexample without this assumption using a surface algebra associated with a one-holed torus.

math.RT↗