Lidia Angeleri Hügel, Michal Hrbek
Tilting modules over commutative rings were recently classified in [12]: they correspond bijectively to faithful Gabriel topologies of finite type. In this note we extend this classification by dropping faithfulness. The counterpart of an arbitrary Gabriel topology of finite type is obtained by replacing tilting with the more general notion of a silting module.
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On silting complexes associated to n-silting modules
本文研究了与n-silting模块相关的silting复形，并展示了其与tilting复形的关系，同时指出n-silting模块的有限类型性质可能失效。
Flat epimorphisms and silting epimorphisms coincide for commutative rings
本文研究 partial silting 模、Gabriel 拓扑与环满同态的关系，证明交换环上的环满同态为平坦当且仅当它是 silting 环满同态。
We will prove that over commutative rings the silting property of $2$-term complexes induced by morphisms between projective modules is preserved and reflected by faithfully flat extensions.
In the paper, we focus on the silting properties and the combinatorial properties of silting and Gorenstein, which is called Gorenstein silting, where the main tools used are recollements of module categories and tensor products. For a ring A and its idempotent ideal J, we show that an A/J-module T is a silting A-module if and only if T is a silting A/J-module. For the finite dimensional k-algebras, with k a field, we show that the tensor products of silting modules are still silting. We also show that the (partial) Gorenstein silting properties can be glued by the recollements of module categ…
Ring epimorphisms often induce silting modules and cosilting modules, termed minimal silting or minimal cosilting. The aim of this paper is twofold. Firstly, we determine the minimal tilting and minimal cotilting modules over a tame hereditary algebra. In particular, we show that a large cotilting module is minimal if and only if it has an adic module as a direct summand. Secondly, we discuss the behaviour of minimality under ring extensions. We show that minimal cosilting modules over a commutative noetherian ring extend to minimal cosilting modules along any flat ring epimorphism. Similar re…
The main goal of this paper is to compare the silting theory of an $R$-algebra $Λ$ over a Noetherian ring $R$ with that of its tensor product $Λ\otimes Γ$ with another $R$-algebra $Γ$. In the case that the $R$-algebra $Λ$ is Noetherian, $R$ a complete local ring and $\mathfrak{a}$ a certain ideal of the ring $R$, we obtain an isomorphism between the silting poset of $Λ$ and that of its quotient $Λ\otimes R/ \mathfrak{a}$. Furthermore, we study the restrictions of such a bijection to tilting complexes and deduce silting embedding and descent results for the algebra $Λ$ and a certain family of a…
There are well-known constructions relating ring epimorphisms and tilting modules. The new notion of silting module provides a wider framework for studying this interplay. To every partial silting module we associate a ring epimorphism which we describe explicitly as an idempotent quotient of the endomorphism ring of the Bongartz completion. For hereditary rings, this assignment is used to parametrise homological ring epimorphisms by silting modules. We further show that homological ring epimorphisms of a hereditary ring form a lattice which completes the poset of noncrossing partitions in the…
Silting modules are abundant. Indeed, they parametrise the definable torsion classes over a noetherian ring, and the hereditary torsion pairs of finite type over a commutative ring. Also the universal localisations of a hereditary ring, or of a finite dimensional algebra of finite representation type, can be parametrised by silting modules. In these notes, we give a brief introduction to the fairly recent concepts of silting and cosilting module, and we explain the classification results mentioned above.
Silting and tilting objects in cleft extensions of abelian categories
本文研究了在交换扩张abelian范畴中silting和tilting对象的构造与性质，并将其应用于模块范畴的扩展，扩展和加强了相关结果。
We give an overview of recent developments in silting theory. After an introduction on torsion pairs in triangulated categories, we discuss and compare different notions of silting and explain the interplay with t-structures and co-t-structures. We then focus on silting and cosilting objects in a triangulated category with coproducts and study the case of the unbounded derived category of a ring. We close the survey with some classification results over commutative noetherian rings and silting-discrete algebras.
Let $Λ,Γ$ be rings and $R=\left(\begin{array}{cc}Λ& 0 \\ M & Γ\end{array}\right)$ the triangular matrix ring with $M$ a $(Γ,Λ)$-bimodule. Let $X$ be a right $Λ$-module and $Y$ a right $Γ$-module. We prove that $(X, 0)$$\oplus$$(Y\otimes_ΓM, Y)$ is a silting right $R$-module if and only if both $X_Λ$ and $Y_Γ$ are silting modules and $Y\otimes_ΓM$ is generated by $X$. Furthermore, we prove that if $Λ$ and $Γ$ are finite dimensional algebras over an algebraically closed field and $X_Λ$ and $Y_Γ$ are finitely generated, then $(X, 0)$$\oplus$$(Y\otimes_ΓM, Y)$ is a support $τ$-tilting $R$-module i…
We develop silting theory of a noetherian algebra $Λ$ over a commutative noetherian ring $R$. We study mutation theory of $2$-term silting complexes of $Λ$, and as a consequence, we see that mutation exists. As in the case of finite dimensional algebras, functorially finite torsion classes of $Λ$ bijectively correspond to silting $Λ$-modules, if $R$ is complete local. We show a reduction theorem of $2$-term silting complexes of $Λ$, and by using this theorem, we study torsion classes of the module category of $Λ$. When $R$ has Krull dimension one, we describe the set of torsion classes of $Λ$ …
We study equivalences induced by a silting module $T$ or, equivalently, by a complex of projectives $\mathbb{P}$, concentrated in $-1$ and $0$ which is silting in the derived category $\mathbf{D}(R)$ of a ring $R$.
We study the classes of modules which are generated by a silting module. In the case of either hereditary or perfect rings it is proved that these are exactly the torsion $\mathcal{T}$ such that the regular module has a special $\mathcal{T}$-preenvelope. In particular every torsion enveloping class in $\textrm{Mod-} R$ are of the form $\mathrm{Gen}(T)$ for a minimal silting module $T$. For the dual case we obtain for general rings that the covering torsion-free classes of modules are exactly the classes of the form $\mathrm{Cogen}(T)$, where $T$ is a cosilting module.
An important result in tilting theory states that a class of modules over a ring is a tilting class if and only if it is the Ext-orthogonal class to a set of compact modules of bounded projective dimension. Moreover, cotilting classes are precisely the resolving and definable subcategories of the module category whose Ext-orthogonal class has bounded injective dimension. In this article, we prove a derived counterpart of the statements above in the context of silting theory. Silting and cosilting complexes in the derived category of a ring generalise tilting and cotilting modules. They give ri…
We introduce the new concept of silting modules. These modules generalise tilting modules over an arbitrary ring, as well as support $τ$-tilting modules over a finite dimensional algebra recently introduced by Adachi, Iyama and Reiten. We show that silting modules generate torsion classes that provide left approximations, and that every partial silting module admits an analogue of the Bongartz complement. Furthermore, we prove that silting modules are in bijection with 2-term silting complexes and with certain t-structures and co-t-structures in the derived module category. We also see how som…
Tilting complexes and codimension functions over commutative noetherian rings
本文在交换诺特环的导出范畴中，利用局部上同调构造与 sp-滤过关联的 silting 对象，并刻画其成为 tilting 的条件，特别与余维数函数及 Cohen-Macaulay 环同态像相关，同时给出 cosilting 对偶结果。
Let $R$ be a perfect ring and $T$ be an $R$-module. We study characterizations of sincere modules, sincere silting modules and tilting modules in terms of various vanishing conditions. It is proved that $T$ is sincere silting if and only if $T$ is presilting satisfing the vanishing condition $\mathrm{KerExt}^{0\le i\le 1}_R(T,-)=0$, and that $T$ is tilting if and only if $\mathrm{Ker}\mathrm{Ext}^{0\leqslant i\leqslant 1}_{R}(T,-)=0$ and $\mathrm{Gen}T\subseteq \mathrm{Ker}\mathrm{Ext}^{1\leqslant i\leqslant 2}_{R}(T,-)$. As an application, we prove that a sincere silting $R$-module $T$ of fin…
We show that silting modules are closely related with localisations of rings. More precisely, every partial silting module gives rise to a localisation at a set of maps between countably generated projective modules and, conversely, every universal localisation, in the sense of Cohn and Schofield, arises in this way. To establish these results, we further explore the finite-type classification of tilting classes and we use the morphism category to translate silting modules into tilting objects. In particular, we prove that silting modules are of finite type.
Large silting mutation in extriangulated categories
本文在具有集合指标(余)积的扩张三角范畴中发展了无限维倾斜突变理论，并应用于任意环上的n-余倾斜复形和有限整体维数环上的无限维n-(余)倾斜模的突变理论。
We classify compactly generated co-t-structures on the derived category of a commutative noetherian ring. In order to accomplish that, we develop a theory for compactly generated Hom-orthogonal pairs (also known as torsion pairs in the literature) in triangulated categories that resembles Bousfield localization theory. Finally, we show that the category of perfect complexes over a connected commutative noetherian ring admits only the trivial co-t-structures and (de)suspensions of the canonical co-t-structure and use this to describe all silting objects in the category.
A new characterization of silting subcategories in the stable category of a Frobenius extriangulated category
该研究给出Frobenius外三角范畴稳定范畴中silting子范畴的新刻画，建立其与原范畴共变有限子范畴的双射对应，推广了三角范畴的相关结果，并应用于同伦、导出等多类范畴。
We investigate parametrizations of compactly generated t-structures, or more generally, t-structures with a definable coaisle, in the unbounded derived category D(Mod-A) of a ring A. To this end, we provide a construction of t-structures from chains in the lattice of ring epimorphisms starting in A, which is a natural extension of the construction of compactly generated t-structures from chains of subsets of the Zariski spectrum known for the commutative noetherian case. We also provide constructions of silting and cosilting objects in D(Mod-A). This leads us to classification results over som…
We prove that a finite dimensional algebra is $τ$-tilting finite if and only if it does not admit large silting modules. Moreover, we show that for a $τ$-tilting finite algebra $A$ there is a bijection between isomorphism classes of basic support $τ$-tilting (that is, finite dimensional silting) modules and equivalence classes of ring epimorphisms $A\longrightarrow B$ with ${\rm Tor}_1^A(B,B)=0$. It follows that a finite dimensional algebra is $τ$-tilting finite if and only if there are only finitely many equivalence classes of such ring epimorphisms.
A lower bound for the Rouquier dimension of derived categories over commutative rings
本文证明交换诺特环上有限生成模的有界导出范畴的Rouquier维数不小于环的Krull维数。
We give a necessary condition of degeneration via matrix representations, and consider degenerations of indecomposable Cohen-Macaulay modules over hypersurface singularities of type ($A_\infty$). We also provide a method to construct degenerations of finitely generated modules over commutative rings.
We establish a link between trace modules and rigidity in modules over Noetherian rings. Using the theory of trace ideals we make partial progress on a question of Dao, and on the Auslander-Reiten conjecture over Artinian Gorenstein rings.
In this paper, as an analogue of the spectrum of a tensor triangulated category introduced by Balmer, we define a spectrum of a triangulated category which does not necessarily admit a tensor structure. We apply it for some triangulated categories associated to a commutative noetherian ring.
We show that any homotopically smashing t-structure in the derived category of a commutative noetherian ring is compactly generated. This generalizes the validity of the telescope conjecture for commutative noetherian rings due to Neeman. As another consequence, we obtain a cofinite type result for pure-injective cosilting objects. We also give a formulation of telescope conjecture for homotopically smashing t-structures in underlying triangulated categories of certain Grothendieck derivators.
We refine a well-known theorem of Auslander and Reiten about the extension closedness of n-th syzygies over noether algebras. Applying it, we obtain the converse of a celebrated theorem of Evans and Griffith on Serre's condition (S_n) and the local Gorensteiness of a commutative ring in height less than n. This especially extends a recent result of Araya and Iima concerning a Cohen-Macaulay local ring with canonical module to an arbitrary local ring.
Schur roots and tilting modules of acyclic quivers over commutative rings
本文将域上无环箭图支撑倾斜模与簇代数簇的双射推广至不可分解交换Noetherian环，建立2-term倾斜复形与簇的双射，并证明Dynkin箭图的挠类与Spec R到簇集的保序映射双射对应。
Stability over cDV singularities and other complete local rings
本文研究 cDV 奇点非交换极小模型的半稳定模范畴，利用 silting 理论与 Dynkin 图组合刻画稳定性，并将二项 silting–稳定性对应推广至完备局部 Noether 环上的代数。
Mutations and derived equivalences for commutative noetherian rings
本研究聚焦交换诺特环的表示理论，借助素谱描述其导出范畴内余倾斜对象的突变，深化了对该类环导出结构的认识。
Mutation and torsion pairs
本文在三角范畴的silting/cosilting t-结构框架下发展mutation理论，证明纯内射cosilting对象的mutation涵盖紧silting复形的经典mutation，并将阿廷环上有限生成模挠类极小包含对应不可约mutation的结果推广。
Two left noetherian rings $R$ and $S$ are said to be {\it singularly equivalent} if their singularity categories are equivalent as triangulated categories. The aim of this paper is to give a necessary condition for two commutative noetherian rings to be singularly equivalent. To do this, we develop the support theory for triangulated categories without tensor structure.
For a $Z$-graded Gorenstein ring $R$, we study the stable category $CM^ZR$ of $Z$-graded maximal Cohen-Macaulay $R$-modules, which is canonically triangle equivalent to the singularity category of Buchweitz and Orlov. Its thick subcategory given as the stable category of $CM_0^ZR$ is central in representation theory since it enjoys Auslander-Reiten-Serre duality and has almost split triangles. In the case $dim R=1$, we prove that the stable category of $CM_0^ZR$ always admits a silting object, and that it admits a tilting object if and only if either $R$ is regular or the $a$-invariant of $R$ …
AIR tilting subcategories of extended hearts
本文提出扩展心的AIR倾斜子类概念，推广了有限生成模的倾斜对，并建立了与硅ting子类的双射关系，同时探讨了相关类别的性质和关系。
We construct, for any finite commutative ring $R$, a family of representations of the general linear group $\mathrm{GL}_n(R)$ whose intertwining properties mirror those of the principal series for $\mathrm{GL}_n$ over a finite field.
Rigid integral representations of quivers over arbitrary commutative rings
本文将作者早期关于主理想环上由有限生成自由模构成的箭图刚性表示分类结果，推广至任意交换环上由有限生成投射模构成的箭图表示。
In this article we study the structure of highest weight modules for quantum groups defined over a commutative ring with particular emphasis on the structure theory for invariant bilinear forms on these modules.
Galois closures of commutative rank n ring extensions were introduced by Bhargava and the second author. In this paper, we generalize the construction to the case of non-commutative rings. We show that non-commutative Galois closures commute with base change and satisfy a product formula. As an application, we give a uniform construction of many of the representations arising in arithmetic invariant theory, including many Vinberg representations.
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