Shun-Jen Cheng, Jae-Hoon Kwon, Ngau Lam
We construct a Bernstein-Gelfand-Gelfand type resolution in terms of direct sums of Kac modules for the finite-dimensional irreducible tensor representations of the general linear superalgebra. As a consequence it follows that the unique maximal submodule of a corresponding reducible Kac module is generated by its proper singular vector.
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We determine the Verma multiplicities of standard filtrations of projective modules for integral atypical blocks in the BGG category $\mathcal{O}$ for the orthosymplectic Lie superalgebras $\mathfrak{osp}(3|4)$ by way of translation functors. We then explicitly determine the composition factor multiplicities of Verma modules using BGG reciprocity.
We determine the Verma multiplicities and the characters of projective modules for atypical blocks in the BGG Category O for the general linear Lie superalgebras $\frak{gl}(2|2)$ and $\frak{gl}(3|1)$. We then explicitly determine the composition factor multiplcities of Verma modules in the atypicality 2 block of $\frak{gl}(2|2)$.
The main focus of this paper is Bott-Borel-Weil (BBW) theory for basic classical Lie superalgebras. We take a purely algebraic self-contained approach to the problem. A new element in this study is twisting functors, which we use in particular to prove that the top of the cohomology groups of BBW theory for generic weights is described by the recently introduced star action. We also study the algebra of regular functions, related to BBW theory. Then we introduce a weaker form of genericness, relative to the Borel subalgebra and show that the virtual BGG reciprocity of Gruson and Serganova beco…
Parabolic BGG categories and their block decomposition for Lie superalgebras of Cartan type
本文研究复数域上Cartan型阶化李超代数的抛物BGG范畴，证明仅有极小和极大两类抛物子代数，并将问题简化为极小抛物范畴，进而证明射影覆盖存在性以建立块理论，最终完成块的分类及不可分解倾斜模与射影模的特征标公式计算。
We develop a reduction procedure which provides an equivalence from an arbitrary block of the BGG category for the queer Lie superalgebra $\mathfrak{q}(n)$ to a "$\mathbb{Z}\pm s$-weights" ($s\in \mathbb{C}$) block of a BGG category for finite direct sum of queer Lie superalgebras. We give descriptions of blocks. We also establish equivalences between certain maximal parabolic subcategories for $\mathfrak{q}(n)$ and blocks of atypicality-one of the category of finite-dimensional modules for $\mathfrak{gl}(\ell|n-\ell)$.
We study tilting and projective-injective modules in a parabolic BGG category $\mathcal O$ for an arbitrary classical Lie superalgebra. We establish a version of Ringel duality for this type of Lie superalgebras which allows to express the characters of tilting modules in terms of those of simple modules in that category. We also obtain a classification of projective-injective modules in the full BGG category $\mathcal O$ for all simple classical Lie superalgebras. We then classify and give an explicit combinatorial description of parabolic subalgebras of the periplectic Lie superalgebras and …
We apply the super duality formalism recently developed by the authors to obtain new equivalences of various module categories of general linear Lie superalgebras. We establish the correspondence of standard, tilting, and simple modules, as well as the identification of the u-homology groups, under these category equivalences. As an application, we obtain a complete solution of the irreducible character problem for some new parabolic BGG categories of gl(m|n)-modules, including the full BGG category of gl(m|2)-modules, in terms of type A Kazhdan-Lusztig polynomials.
Simple character formulas for finite $W$-superalgebras of type $A$
本文提出了一种统一的特征公式，用于描述有限型A的有限W超代数中不可约模的结构，同时分类化了量子群的张量积模中的基。
We classify the blocks, compute the Verma flags of tilting and projective modules in the BGG category $\mathcal O$ for the exceptional Lie superalgebra $G(3)$. The projective injective modules in $\mathcal O$ are classified. We also compute the Jordan-Hölder multiplicities of the Verma modules in $\mathcal O$.
We develop a new method to solve the irreducible character problem for a wide class of modules over the general linear superalgebra, including all the finite-dimensional modules, by directly relating the problem to the classical Kazhdan-Lusztig theory. We further verify a parabolic version of a conjecture of Brundan on the irreducible characters in the BGG category $\mc{O}$ of the general linear superalgebra. We also prove the super duality conjecture.
Whittaker modules and representations of finite $W$-algebras of queer Lie superalgebras
本文研究了 queer Lie超代数的 Whittaker 模块及其与有限 $W$-超代数的表示关系，提出了 Whittaker 模块的分类方法，并建立了与 BGG 类别的等价关系。
We study a semisimple extension of a Takiff superalgebra which turns out to have a remarkably rich representation theory. We determine the blocks in both the finite-dimensional and BGG module categories and also classify the Borel subalgebras. We further compute all extension groups between two finite-dimensional simple objects and prove that all non-principal blocks in the finite-dimensional module category are Koszul.
We provide a new proof of the super duality equivalence between infinite-rank parabolic BGG categories of general linear Lie (super) algebras conjectured by Cheng and Wang and first proved by Cheng and Lam. We do this by establishing a new uniqueness theorem for tensor product categorifications motivated by work of Brundan, Losev, and Webster. Moreover we show that these BGG categories have Koszul graded lifts and super duality can be lifted to a graded equivalence.
For the BGG category of $\mathfrak{q}(n)$-modules of half-integer weights, a Kazhdan-Lusztig conjecture à la Brundan is formulated in terms of categorical canonical basis of the $n$th tensor power of the natural representation of the quantum group of type $C$. For the BGG category of $\mathfrak{q}(n)$-modules of congruent non-integral weights, a Kazhdan-Lusztig conjecture is formulated in terms of canonical basis of a mixed tensor of the natural representation and its dual of the quantum group of type $A$. We also establish a character formula for the finite-dimensional irreducible $\mathfrak{…
Whittaker categories of quasi-reductive Lie superalgebras and principal finite W-superalgebras
本文研究拟约化李超代数的Whittaker范畴，证明Backelin函子的不可约性，构造主有限W-超代数并建立Skryabin型等价，给出Soergel结构定理的超类比。
We develop a reduction procedure which provides an equivalence (as highest weight categories) from an arbitrary block (defined in terms of the central character and the integral Weyl group) of the BGG category O for a general linear Lie superalgebra to an integral block of O for (possibly a direct sum of) general linear Lie superalgebras. We also establish indecomposability of blocks of O.
We classify simple Whittaker modules for classical Lie superalgebras in terms of their parabolic decompositions. We establish a type of Miličić-Soergel equivalence of a category of Whittaker modules and a category of Harish-Chandra bimodules. For classical Lie superalgebras of type I, we reduce the problem of composition factors of standard Whittaker modules to that of Verma modules in their BGG categories $\mathcal O$. As a consequence, the composition series of standard Whittaker modules over the general linear Lie superalgebras $\mathfrak{gl}(m|n)$ and the ortho-symplectic Lie superalgebras…
We establish character formulae for representations of the one-parameter family of simple Lie superalgebras $D(2|1;ζ)$. We provide a complete description of the Verma flag multiplicities of the tilting modules and the projective modules in the BGG category $\mathcal O$ of $D(2|1;ζ)$-modules of integral weights, for any complex parameter $ζ$. The composition factors of all Verma modules in $\mathcal O$ are then obtained.
We study Kostant cohomology and Bernstein-Gelfand-Gelfand resolutions for finite dimensional representations of basic classical Lie superalgebras and reductive Lie superalgebras based on them. For each choice of parabolic subalgebra and irreducible representation of such a Lie superalgebra, there is a natural definition of the derivative and coderivative, which define the (co)homology groups. We prove that a necessary condition to have a resolution of an irreducible module in terms of Verma modules is complete reducibility of the cohomology groups. Essentially, if it exists, every such a resol…
Generalizing the super duality formalism for finite-dimensional Lie superalgebras of type $ABCD$, we establish an equivalence between parabolic BGG categories of a Kac-Moody Lie superalgebra and a Kac-Moody Lie algebra. The characters for a large family of irreducible highest weight modules over a symmetrizable Kac-Moody Lie superalgebra are then given in terms of Kazhdan-Lusztig polynomials for the first time. We formulate a notion of integrable modules over a symmetrizable Kac-Moody Lie superalgebra via super duality, and show that these integrable modules form a semisimple tensor subcategor…
Serre functors for Lie superalgebras and tensoring with $S^{\mathrm{top}}(\mathfrak{g}_{\overline{1}})$
本文证明了拟约化有限维李超代数抛物型BGG范畴中Serre函子在投射-内射模上的作用等价于与奇部对称幂顶端分量作张量积，并据此判定所有奇异李超代数中该子范畴何时对称。
We show that, for many Lie superalgebras admitting a compatible $\mathbb{Z}$-grading, Kac induction functor gives rise to a bijection between simple supermodules over a Lie superalgebra and simple supermodules over the even part of this Lie superalgebra. This reduces the classification problem for the former to the one for the latter. Our result applies to all classical Lie superalgebra of type $I$, in particular, to the general linear Lie superalgebra $\gl(m|n)$. In the latter case we also show that the rough structure of simple $\gl(m|n)$-supermodules and also that of Kac supermodules depend…
We establish a maximal parabolic version of the Kazhdan-Lusztig conjecture \cite[Conjecture 5.10]{CKW} for the BGG category $\mathcal{O}_{k,ζ}$ of $\mathfrak{q}(n)$-modules of "$\pm ζ$-weights", where $k\leq n$ and $ζ\in\mathbb{C} \setminus \frac{1}{2} \mathbb{Z}$. As a consequence, the irreducible characters of these $\mathfrak{q}(n)$-modules in this maximal parabolic category are given by the Kazhdan-Lusztig polynomials of type $A$ Lie algebras. As an application, closed character formulas for a class of $\mathfrak{q}(n)$-modules resembling polynomial and Kostant modules of the general linea…
We study the truncation functors and show the existence of projective cover of each irreducible module in parabolic BGG category $\mathcal O$ over infinite rank Lie algebra of types $\mathfrak{a,b,c,d}$. Moreover, $\mathcal O$ is a Koszul category. As a consequence, the corresponding parabolic BGG category $\overline{\mathcal O}$ over infinite rank Lie superalgebra of types $\mathfrak{a,b,c,d}$ through the super duality is also a Koszul category.
We classify blocks in the BGG category $\mathcal O$ of modules of non-integral weights for the exceptional Lie superalgebra $G(3)$. We compute the characters for tilting modules of non-integral weights in $\mathcal O$. Reduction methods are established to connect non-integral blocks of $G(3)$ with blocks of the special linear Lie algebra $\mathfrak{sl}(2)$, the exceptional Lie algebra $G_2$, the general linear Lie superalgebras $\mathfrak{gl}(1|1)$, $\mathfrak{gl}(2|1)$ and the ortho-symplectic Lie superalgebra $\mathfrak{osp}(3|2)$.
In the framework of canonical and dual canonical bases of Fock spaces, Brundan in 2003 formulated a Kazhdan-Lusztig type conjecture for the characters of the irreducible and tilting modules in the BGG category for the general linear Lie superalgebra for the first time. In this paper, we prove Brundan's conjecture and its variants associated to all Borel subalgebras in full generality.
Categorification of generic Su-Zhang character formula
本文提出了一种新的范畴化方法，用于泛泛苏-张特征公式，通过构造基于非共轭博尔子代物的Verma模同态图像的分解，扩展了经典的BGG分解。
We classify blocks in the BGG category $\mathcal O$ of modules of non-integral weights for the exceptional Lie superalgebra $D(2|1;ζ)$. We establish various reduction methods, which connect some types of non-integral blocks of $D(2|1;ζ)$ with integral blocks of general linear Lie superalgebras $\mathfrak{gl}{(1|1)}$ and $\mathfrak{gl}{(2|1)}$. We compute the characters for irreducible $D(2|1;ζ)$-modules of non-integral weights in $\mathcal O$.
Tensor product degenerate spin affine Hecke superalgebras
本文定义了高阶退化自旋仿射Hecke超代数，建立其与高阶退化仿射Hecke-Clifford超代数的同构关系，推广了退化情形下两类超代数的对应关系，丰富了超代数结构理论研究。
We study three related homological properties of modules in the BGG category O for basic classical Lie superalgebras, with specific focus on the general linear superalgebra. These are the projective dimension, associated variety and complexity. We demonstrate connections between projective dimension and singularity of modules and blocks. Similarly we investigate the connection between complexity and atypicality. This creates concrete tools to describe singularity and atypicality as homological, and hence categorical, properties of a block. However, we also demonstrate how two integral blocks i…
For semisimple Lie superalgebras over an algebraically closed field of characteristic zero, whose category of finite dimensional super representations is semisismple, we classify all irreducible super representations for which the alternating or symmetric square representation is irreducible or decomposes into an irreducible representation and a trivial representation.
We reformulate the Kazhdan-Lusztig theory for the BGG category $\mathcal{O}$ of Lie algebras of type D via the theory of canonical bases arising from quantum symmetric pairs initiated by Weiqiang Wang and the author. This is further applied to formulate and establish for the first time the Kazhdan-Lusztig theory for the BGG category $\mathcal{O}$ of the ortho-symplectic Lie superalgebra $\mathfrak{osp}(2m|2n)$.
In this paper, we study the tensor module $P\otimes M$ over the Witt superalgebra $W_{m,n}^+$ (resp. $W_{m,n}$), where $P$ is a simple module over the Weyl superalgebra $K_{m,n}^+$ (resp. $K_{m,n}$) and $M$ is simple weight module over the general linear Lie superalgebra $\mathfrak{gl}(m,n)$. We obtain the necessary and sufficient conditions for $P\otimes M$ to be simple, and determine all simple subquotient of $P\otimes M$ when it is not simple. All the work leads to completion of some classification problems on the weight representation theory of $W_{m,n}^+$ and $W_{m,n}$.
Tensor triangular geometry as introduced by Balmer is a powerful idea which can be used to extract the ambient geometry from a given tensor triangulated category. In this paper we provide a general setting for a compactly generated tensor triangulated category which enables one to classify thick tensor ideals and the Balmer spectrum. For a classical Lie superalgebra ${\mathfrak g}={\mathfrak g}_{\bar{0}}\oplus {\mathfrak g}_{\bar{1}}$, we construct a Zariski space from a detecting subalgebra of ${\mathfrak g}$ and demonstrate that this topological space governs the tensor triangular geometry f…
Revisiting Jacobi-Trudi identities via the BGG category $\mathcal{O}$
本文通过BGG范畴O中的张量积重数重新证明斜Schur多项式的Jacobi-Trudi恒等式，并确立截断展开的Schur正性，同时解释系数为张量积重数。
On tensor products of representations of Lie superalgebras
本文研究基本经典单李超代数典型表示张量积的唯一因子分解条件，并在额外假设下证明若干系列单非典型不可约模的张量积唯一因子分解，从而给出 Rajan 结果的李超代数类比。
We let S denote the ring of polynomial functions on the space of m x n matrices, and consider the action of the group GL = GL_m x GL_n via row and column operations on the matrix entries. For a GL-invariant ideal I in S we show that the linear strands of its minimal free resolution translate via the BGG correspondence to modules over the general linear Lie superalgebra gl(m|n). When I is the ideal generated by the GL-orbit of a highest weight vector, we give a conjectural description of the classes of these gl(m|n)-modules in the Grothendieck group, and prove that our prediction is correct for…
Using tensor categories, we present new constructions of several of the exceptional simple Lie superalgebras with integer Cartan matrix in characteristic $p = 3$ and $p = 5$ from the complete classification of modular Lie superalgebras with indecomposable Cartan matrix and their simple subquotients over algebraically closed fields by Bouarroudj, Grozman, and Leites in 2009. Specifically, let $\mathbfα_p$ denote the kernel of the Frobenius endomorphism on the additive group scheme $\mathbb{G}_a$ over an algebraically closed field of characteristic $p$. The Verlinde category $\mathrm{Ver}_p$ is …
We consider general linear superalgebra (type A) and tensor with Laurent polynomial ring in several variables. We then consider the universal central extension of this Lie superalgebra which we call toroidal superalgebra. We give a faithful representation of toroidal superalgebra using vertex operators and bosons.
We study stabilization of finite-dimensional representations of the periplectic Lie superalgebras $\mathfrak{p}(n)$ as $n \to \infty$. The paper gives a construction of the tensor category $Rep(\underline{P})$, possessing nice universal properties among tensor categories over the category $\mathtt{sVect}$ of finite-dimensional complex vector superspaces. First, it is the "abelian envelope" of the Deligne category corresponding to the periplectic Lie superalgebra, in the sense of arXiv:1511.07699. Secondly, given a tensor category $\mathcal{C}$ over $\mathtt{sVect}$, exact tensor functors $Rep(…
Let $A_{m,n}$ be the tensor product of the Laurient polynomial algebra in $m$ even variables and the exterior algebra in $n$ odd variables over the complex field $\bC$, and the Witt superalgebra $W_{m,n}$ be the Lie superalgebra of superderivations of $A_{m,n}$. In this paper, we classify the simple weight $W_{m,n}$ modules with finite-dimensional weight spaces with respect to the standard Cartan algebra of $W_{m,0}$. Every such module is either a simple quotient of a tensor module or a module of highest weight type.
Let $\mathfrak g$ be a simple complex Lie algebra. In this paper we study the BGG category $\mathcal O_q$ for the quantum group $U_q(\mathfrak g)$ with $q$ being a root of unity in a field $K$ of characteristic $p >0$. We first consider the simple modules in $\mathcal O_q$ and prove a Steinberg tensor product theorem for them. This result reduces the problem of determining the corresponding irreducible characters to the same problem for a finite subset of finite dimensional simple modules. Then we investigate more closely the Verma modules in $\mathcal O_q$. Except for the special Verma module…
Let $A_{m,n}$ be the tensor product of the polynomial algebra in $m$ even variables and the exterior algebra in $n$ odd variables over the complex field $\C$, and the Witt superalgebra $W_{m,n}$ be the Lie superalgebra of superderivations of $A_{m,n}$. In this paper, we classify the non-trivial simple bounded weight $W_{m,n}$ modules with respect to the standard Cartan algebra of $W_{m,n}$. Any such module is a simple quotient of a tensor module $F(P,L(V_1\otimes V_2))$ for a simple weight module $P$ over the Weyl superalgebra $\mathcal K_{m,n}$, a finite-dimensional simple $\gl_m$-module $V_1…
In this paper we construct a large class of modules for toroidal Lie superalgebras. Toroidal Lie superalgebras are universal central extension of G tensor A where G is a basic classical Lie superalgebra and A is a Laurent polynomial ring in sevaral variables. The case where G is simple finite dimensional Lie algebra is included.
Highest weight theory for minimal finite $W$-superalgebras and related Whittaker categories
本文研究最小有限 W-超代数的最高权理论，引入 Verma 模并给出有限维不可约模的完全分类，进而建立其 BGG 范畴 O，揭示了奇检测奇偶性带来的全新现象。
Hypercubic Decomposition of Verma Supermodules and Semibricks Realizing the Khovanov Algebra of Defect One
本文通过改变 Borel 子代数研究基本 Lie 超代数 Verma 模的变体，证明了 \\(\mathfrak{gl}(1|1)\\) 主块可实现为任意非典型 BGG 范畴 \\( \mathcal{O} \\) 块的满子范畴。
In this paper, we classify simple strong Harish-Chandra modules over the Lie superalgebra $W_{m,n}$ of vector fields on $\C^{m|n}$. Any such module is the unique simple submodule of some tensor module $F(P,V)$ for a simple weight module $P$ over the Weyl superalgebra $\mathcal K_{m,n}$ and a simple weight module $V$ over the general linear superalgebra $\gl_{m,n}$.
Representations of a class of infinite-dimensional primitive Lie superalgebras
本文针对无限维本原李超代数$\textbf{W}(m,n)$引入BGG范畴$\boldsymbol{\text{O}}$，通过同调分析与Skryabin理论的超化推广，完成了该范畴中不可约模的分类与特征标推导，并进一步得到了不可分解倾斜模的特征标公式。
In this paper, we begin the study of highest weight representations of the quantized enveloping superalgebra ${\mathfrak U}_q {\mathfrak p}_n$ of type $P$. We introduce a Drinfeld-Jimbo representation and establish a triangular-decomposition of ${\mathfrak U}_q {\mathfrak p}_n$. We explain how to relate modules over ${\mathfrak U}_q {\mathfrak p}_n$ to modules over ${\mathfrak p}_n$, the Lie superalgebra of type $P$, and we prove that the category of tensor modules over ${\mathfrak U}_q {\mathfrak p}_n$ is not semisimple.
In this paper, we develop the crystal basis theory for the quantum queer superalgebra $U_q(\mathfrak q(n))$. We define the notion of crystal bases and prove the tensor product rule for $U_q(\mathfrak q(n))$-modules in the category $O_int^{\geq 0}$. Our main theorem shows that every $U_q(\mathfrak q(n))$-module in the category $O_int^{\geq 0}$ has a unique crystal basis.
This article aims to contribute to the study of algebras with triangular decomposition over a Hopf algebra, as well as the BGG Category O. We study functorial properties of O across various setups. The first setup is over a skew group ring, involving a finite group $Γ$ acting on a regular triangular algebra $A$. We develop Clifford theory for $A \rtimes Γ$, and obtain results on block decomposition, complete reducibility, and enough projectives. O is shown to be a highest weight category when $A$ satisfies one of the "Conditions (S)"; the BGG Reciprocity formula is slightly different because t…
We prove that the tensor product of a simple and a finite dimensional $\mathfrak{sl}_n$-module has finite type socle. This is applied to reduce classification of simple $\mathfrak{q}(n)$-supermodules to that of simple $\mathfrak{sl}_n$-modules. Rough structure of simple $\mathfrak{q}(n)$-supermodules, considered as $\mathfrak{sl}_n$-modules, is described in terms of the combinatorics of category $\mathcal{O}$.
We introduce the orthosymplectic superalgebra osp(m|2n) as the algebra of Killing vector fields on Riemannian superspace R^{m|2n} which stabilize the origin. The Laplace operator and norm squared on R^{m|2n}, which generate sl(2), are orthosymplectically invariant, therefore we obtain the Howe dual pair (osp(m|2n),sl(2)). We study the osp(m|2n)-representation structure of the kernel of the Laplace operator. This also yields the decomposition of the supersymmetric tensor powers of the fundamental osp(m|2n)-representation under the action of sl(2) x osp(m|2n). As a side result we obtain informat…
This is Part I of a series of papers constructing intertwining operator superalgebras and vertex tensor categories associated to the superconformal minimal models and other related models. In this paper, we construct the intertwining operator superalgebras and vertex tensor categories in the N=1 case.
We construct the intertwining operator superalgebras and vertex tensor categories for the N=2 superconformal unitary minimal models and other related models.
We use the RTT realization of the quantum affine superalgebra associated with the Lie superalgebra $\mathfrak{gl}(M,N)$ to study its finite-dimensional representations and their tensor products. In the case $\mathfrak{gl}(1,1)$, the cyclicity condition of tensor products of finite-dimensional simple modules is determined completely in terms of zeros and poles of rational functions. This in turn induces cyclicity of some particular tensor products of Kirillov-Reshetikhin modules related to $\mathfrak{gl}(M,N)$.
In this paper, we used the free fields of Wakimoto to construct a class of irreducible representations for the general linear Lie superalgebra $\mathfrak{gl}_{m|n}(\mathbb{C})$. The structures of the representations over the general linear Lie superalgebra and the special linear Lie superalgebra are studied in this paper. Then we extend the construction to the affine Kac-Moody Lie superalgebra $\widehat{\mathfrak{gl}_{m|n}}(\mathbb{C})$ on the tensor product of a polynomial algebra and an exterior algebra with infinitely many variables involving one parameter $μ$, and we also obtain the necess…
We define global and local Weyl modules for $q \otimes A$, where $q$ is the queer Lie superalgebra and $A$ is an associative commutative unital $\mathbb{C}-$algebra. We prove that global Weyl modules are universal highest weight objects in certain category upto parity reversing functor $Π$. Then with the assumption that $A$ is finitely generated and with a special technical condition which simple root system of $q$ satisfy it is shown that the local Weyl modules are finite dimensional. Further they are universal highest map-weight objects in certain category upto $Π$. Finally we prove a tensor…
We generalize a construction in [BW18] (arXiv:1610.09271) by showing that the tensor product of a based $\textbf{U}^{\imath}$-module and a based $\textbf{U}$-module is a based $\textbf{U}^{\imath}$-module. This is then used to formulate a Kazhdan-Lusztig theory for an arbitrary parabolic BGG category $\mathcal{O}$ of the ortho-symplectic Lie superalgebras, extending a main result in [BW13] (arXiv:1310.0103).
Classifying thick subcategories over a Koszul complex via the curved BGG correspondence
本文利用弯曲 BGG 对应及完美弯曲 dg 模的厚张量子模分类，刻画了正则环中任意元素列表的 Koszul 复形上有界 dg 模导出范畴的厚子范畴，并恢复 Stevenson、Hopkins–Neeman 等结果。
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