Kevin Coulembier
We demonstrate equivalence between two definitions of lower finite highest weight categories. We also show that, in the presence of a duality, a lower finite highest weight structure on a category is unique. Finally, we give a new proof for the fact that any abelian category is extension full in its ind-completion.
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Bounding The Orlov Spectrum For A Completion Of Discrete Cluster Categories
本文通过引入同调连通对象及其分解，对 Dynkin 型 $A_{\infty}$ 离散簇范畴的 Paquette-Yıldırım 完备化中的厚子范畴与经典生成子进行分类，并利用同调长度界定了该范畴的 Orlov 谱上界与 Rouquier 维度。
Calabi-Yau algebras are particularly symmetric differential graded algebras. There is a construction called `Calabi-Yau completion' which produces a canonical Calabi-Yau algebra from any homologically smooth dg algebra. Homologically smooth dg algebras also form a 2-category to which the construction of `equivariant completion' can be applied. In this theory two objects are called `orbifold equivalent' if there is a 1-morphism $X$ with invertible quantum dimensions between them. Any such relation entails a whole family of equivalences between categories. We show that an orbifold equivalence be…
Let $\mathscr{A}$ be a small abelian category. For a closed subbifunctor $F$ of $\Ext_{\mathscr{A}}^{1}(-,-)$, Buan has generalized the construction of the Verdier's quotient category to get a relative derived category, where he localized with respect to $F$-acyclic complexes. In this paper, the homological properties of relative derived categories are discussed, and the relation with derived categories is given. For Artin algebras, using relatively derived categories, we give a relative version on derived equivalences induced by $F$-tilting complexes. We discuss the relationships between rela…
We describe the structure and homological properties of arbitrary generalized standard Auslander-Reiten components of artin algebras. In particular, we prove that for all but finitely many indecomposable modules in such components the Euler characteristic is defined and nonnegative. Further, we provide a handy criterion for an infinite Auslander-Reiten component of an artin algebra to be generalized standard. We solve also the long standing open problem concerning the structure of artin algebras admitting a~separating family of Auslander-Reiten components.
Let $\mathscr{C}$ be a 2-Calabi-Yau triangulated category, and let $\mathscr{T}$ be a cluster tilting subcategory of $\mathscr{C}$. An important result from Dehy and Keller tells us that a rigid object $c \in \mathscr{C}$ is uniquely defined by its index with respect to $\mathscr{T}$. The notion of triangulated categories extends to the notion of $(d+2)$-angulated categories. Thanks to a paper by Oppermann and Thomas, we now have a definition for cluster tilting subcategories in higher dimensions. This paper proves that under a technical assumption, an indecomposable object in a $(d+2)$-angula…
We prove that a certain homological epimorphism between two algebras induces a triangle equivalence between their singularity categories. Applying the result to a construction of matrix algebras, we describe the singularity categories of some non-Gorenstein algebras.
We study homological multiplicities of spherical varieties of reductive group $G$ over a $p$-adic field $F$. Based on Bernstein's decomposition of the category of smooth representations of a $p$-adic group, we introduce a sheaf that measures these multiplicities. We show that these multiplicities are finite whenever the usual mutliplicities are finite, in particular this holds for symmetric varieties, conjectured for all spherical varieties and known for a large class of spherical varieties. Furthermore, we show that the Euler-Poincaré characteristic is constant in families induced from admiss…
A graph theoretic model for the derived categories of gentle algebras and their homological bilinear forms
为温和代数的有界导出范畴定制图论模型，用于分析不可分解对象、Auslander-Reiten三角及其与同调双线性型和Coxeter变换的相互作用，并探索新的导出不变量。
We show, in full generality, that Lusztig's $\mathbf{a}$-function describes the projective dimension of both indecomposable tilting modules and indecomposable injective modules in the regular block of the BGG category $\mathcal{O}$, proving a conjecture from the first paper. On the way we show that the images of simple modules under projective functors can be represented in the derived category by linear complexes of tilting modules. These complexes, in turn, can be interpreted as the images of simple modules under projective functors in the Koszul dual of the category $\mathcal{O}$. Finally, …
We prove that a certain eventually homological isomorphism between module categories induces a triangle equivalence between their singularity categories, Gorenstein defect categories and the stable categories of Gorenstein projective modules. Further, we show that Auslander-Reiten conjecture and Gorenstein symmetry conjecture can be reduced by eventually homological isomorphisms. Applying the results to arrow removal and vertex removal, we describe the Gorenstein projective modules over some non-monomial algebras, and we verify the Auslander-Reiten conjecture for certain algebras.
An inductive Ext non-vanishing theorem for the $p$-adic general linear group
本文研究了 $p$-adic 一般线性群的抛物诱导 functor 的同调性质，并证明了非温顺 Gan-Gross-Prasad 猜想的变体。
We prove that among the finite dimensional algebras of finite representation type those that are string algebras are precisely the ones that have the property that the middle term of an arbitrary extension of indecomposable modules has at most two direct factors. On the other hand, we show that non-domestic string algebras are very far from having that property.
We consider filtrations of objects in an abelian category $\catA$ induced by a tilting object $T$ of homological dimension at most two. We define three disjoint subcategories with no maps between them in one direction, such that each object has a unique filtation with factors in these categories. This filtration coincides with the the classical two-step filtration induced by torsion pairs in dimension one. We also give a refined filtration, using the derived equivalence between the derived categories of $\catA$ and the module category of $End_\catA (T)^{op}$. The factors of this filtration con…
Cluster categories and cluster algebras encode two dimensional structures. For instance, the Auslander--Reiten quiver of a cluster category can be drawn on a surface, and there is a class of cluster algebras determined by surfaces with marked points. Cluster characters are maps from cluster categories (and more general triangulated categories) to cluster algebras. They have a tropical shadow in the form of so-called tropical friezes, which are maps from cluster categories (and more general triangulated categories) to the integers. This paper will define higher dimensional tropical friezes. One…
In this paper, we investigate properties of the bounded derived category of finite dimensional modules over a gentle or skew-gentle algebra. We show that the Rouquier dimension of the derived category of such an algebra is at most one. Using this result, we prove that the Rouquier dimension of an arbitrary tame projective curve is equal to one, too. Finally, we elaborate the classification of indecomposable objects of the (possibly unbounded) homotopy category of projective modules of a gentle algebra.
Let $k$ be a commutative Noetherian ring. In this paper we consider filtered modules of the category FI firstly introduced by Nagpal. We show that a finitely generated FI-module $V$ is filtered if and only if its higher homologies all vanish, and if and only if a certain homology vanishes. Using this homological characterization, we characterize finitely generated FI-modules $V$ whose projective dimension is finite, and describe an upper bound for it. Furthermore, we give a new proof for the fact that $V$ induces a finite complex of filtered modules, and use it as well as a result of Church an…
We continue the development of the homological theory of quantum general linear groups previously considered by the first author. The development is used to transfer information to the representation theory of quantised Schur algebras. The acyclicity of induction from some rank-one modules for quantised Borel-Schur subalgebras is deduced. This is used to prove the exactness of the complexes recently constructed by Boltje and Maisch, giving resolutions of the co-Specht modules for Hecke algebras.
In this paper we consider several homological dimensions of crossed products $A _α ^σ G$, where $A$ is a left Noetherian ring and $G$ is a finite group. We revisit the induction and restriction functors in derived categories, generalizing a few classical results for separable extensions. The global dimension and finitistic dimension of $A ^σ _α G$ are classified: global dimension of $A ^σ _α G$ is either infinity or equal to that of $A$, and finitistic dimension of $A ^σ _α G$ coincides with that of $A$. A criterion for skew group rings to have finite global dimensions is deduced. Under the hy…
Singular equivalences of $n$-adjoint type and standard eventually homological isomorphisms
本文修改最终同调同构定义，引入n-伴随型奇异等价，发展相关理论并应用于同调猜想简化与性质保持，同时建立其与三角函子的对应关系，纳入3-伴随型奇异等价框架。
Homological residue fields as comodules over coalgebras
本文研究张量三角范畴的同调剩余域，通过跨代数、几何与拓扑的多个例子将其明确刻画为余代数上的余模范畴，并揭示其编码谱点切向数据的自然意义。
We study the category of $\mathbb{Z}$-indexed sequences over an abelian category and certain generalized homology functors for this category of sequences which are indexed by positive integers $a$ and $b$. By looking at the corresponding derived category, we show that there is an "optimal" subcategory of sequences for every choice of our generalized homology functors, namely, the category of $N$-complexes (sequences for which the differential $d$ satisfies $d^N = 0$) where $N = a + b$. In this optimal case we show that our homology functors reduce to Kapranov's homology functors $\operatorname…
Actions of algebraic groups on DG categories provide a convenient, unifying framework in some parts of geometric representation theory, especially the representation theory of reductive Lie algebras. We extend this theory to loop groups and affine Lie algebras, extending previous work of Beraldo, Gaitsgory and the author. Along the way, we introduce some subjects of independent interest: ind-coherent sheaves of infinite type indschemes, topological DG algebras, and weak actions of group indschemes on categories. We also present a new construction of semi-infinite cohomology for affine Lie alge…
Ascent and descent of Gorenstein homological properties
本文研究有限代数环同态下 Gorenstein 同调性质的上升与下降，给出判据并用以检测环的 Gorenstein 性质及刻画稳定范畴的三角等价。
Let $n$ be a nonnegative integer. For each composition $α$ of $n$, Berg $\textit{et al.}$ introduced a cyclic indecomposable $H_n(0)$-module $\mathcal{V}_α$ with a dual immaculate quasisymmetric function as the image of the quasisymmetric characteristic. In this paper, we study $\mathcal{V}_α$'s from the homological viewpoint. To be precise, we construct a minimal projective presentation of $\mathcal{V}_α$ and a minimal injective presentation of $\mathcal{V}_α$ as well. Using them, we compute ${\rm Ext}^1_{H_n(0)}(\mathcal{V}_α, {\bf F}_β)$ and ${\rm Ext}^1_{H_n(0)}( {\bf F}_β, \mathcal{V}_α)$…
Stable equivalences and homological dimensions
本文研究了中心化矩阵代数之间的稳定等价，并证明了相关同调维数的保持以及Alperin--Auslander/Auslander--Reiten猜想的成立。
The completion of $d$-abelian categories
本文研究$d$-阿贝尔范畴的完备化Ind$(\u27e8M\u27e9)$的性质，证明其等价于左$d$-正合函子构成的满子范畴，探讨其成为$d$-丛倾斜子范畴的条件，并在$d=2$时建立与有限型的等价关系，关联Iyama的有限性问题。
Topology and monoid representations I: Foundations
本文用拓扑方法（在 E2 页坍缩的谱序列）计算有限幺半群膨胀表示间的 Ext 群，并应用于仿射变换幺半群代数的整体维数及同调满射刻画。
Homological properties of extensions of algebras
本文提出强投射有界扩张概念，证明左整体维数有限性、Hochschild 同调支集及大左有限维数有限性在该扩张下保持，并构造了有限相对整体维数的新扩张类，推广了 Cibils 等人的结果。
Categorical properties and homological conjectures for bounded extensions of algebras
本文定义了代数的有界扩张，证明其满足带有级别的 Morita 型奇异等价及相关稳定范畴等价，并探讨了多种同调猜想，给出了实用函子准则及在平凡扩张和三角矩阵代数上的应用。
The commutative and homological algebra of modules over posets is developed, as closely parallel as possible to the algebra of finitely generated modules over noetherian commutative rings, in the direction of finite presentations, primary decompositions, and resolutions. Interpreting this finiteness in the language of derived categories of subanalytically constructible sheaves proves two conjectures due to Kashiwara and Schapira concerning sheaves with microsupport in a given cone. The motivating case is persistent homology of arbitrary filtered topological spaces, especially the case of multi…
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