Two homological proofs of the Noetherianity of $FI_G$
Liping Li
我们给出了两个同调证明的Noetherianity的范畴$FI$，一个基本结果发现教会，Ellenberg，Farb，和Nagpal。
We give two homological proofs of the Noetherianity of the category $FI$, a fundamental result discovered by Church, Ellenberg, Farb, and Nagpal.
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Noetherianity and rooted trees
设T是对象是有根树且态射是保根序嵌入的范畴。我们证明了由Sam和Snowden开发的技术，推广经典的Grobner理论，T的生成表示是诺特的。证明使用了Kruskals树定理的相对版本。
We prove the (equivariant) noetherian property for a wide class of varieties generalizing the class of Plucker varieties (Theorem 1). It improves previous results of Draisma-Eggermont who treated the case of bounded Plucker varieties. Key ingredient of our proof is the constructive proof of the equivariant noetherianity for the hyper-Pfaffians (Theorem 36) which implies the equivariant noetherianity of the dual infinite wedge.
Detailed proof of a theorem on coincidence of homological dimensions of Fréchet algebras of smooth functions on a manifold with the dimension of the manifold
给定的工作包含全文的证明以下断言：对于拓扑代数$C^{\infty}(\mathcal{M})$的光滑函数上的光滑$m$ -维实流形$\mathcal{M}$的小全球维度$(\mathop{\mathrm{ds}} C^\infty (\mathcal{M}))$，全球同调维度$(\mathop{\mathrm{dg}} C^\infty (\mathcal{M}))$和二维$(\mathop{\mathrm{db}} C^\infty(\mathcal{M}))$等于$m$（所有维度的理解意义上的同源性拓扑（局部凸）代数）。
I give an elementary proof of the known fact that the category $\mathfrak{F} \left( Δ\right)$ of $Δ-$filtered modules, associated to a given finite homological system $\left( Δ; Ω, \leq \right) ,$ is closed under direct summands.
Homological properties of representations of p-adic groups related to geometry of the group at infinity
建筑物的几何被用来证明约化p进群的光滑表示范畴的一些同调性质（Kazhdan的“配对猜想”，Bernstein的Deligne-Lusztig对偶的同调对偶描述）。施耐德和施图勒稍早一点得到了一个不同的证明。
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.
A second note on homological systems
本文针对b-同调系统，证明扩展$Δ$-过滤模范畴$\widetilde{\cal F}(Δ)$对直和项封闭。
We show that the base spaces of the semiuniversal unfoldings of some weighted homogeneous singularities can be identified with moduli spaces of $A_\infty$-structures on the trivial extension algebras of the endomorphism algebras of the tilting objects. The same algebras also appear in the Fukaya categories of their mirrors. Based on these identifications, we discuss applications to homological mirror symmetry for Milnor fibers, and give a proof of homological mirror symmetry for an $n$-dimensional affine hypersurface of degree $n + 2$ and the double cover of the $n$-dimensional affine space br…
We show that, up to Morita equivalence, any finite-dimensional algebra with a suitable homological system, admits an exact Borel subalgebra. This generalizes a theorem by Koenig, Külshammer and Ovsienko, which holds for quasi-hereditary algebras. Our proof follows the same general scheme proposed by these authors, in a more general context: we associate a differential graded tensor algebra with relations, using the structure of $A_{\infty}-$algebra of a suitable Yoneda algebra, and use its category of modules to describe the category of filtered modules associated to the given homological syst…
The homological spectrum via definable subcategories
本文提出用可定义子范畴研究张量三角范畴的同调谱，证明其同胚于 Ziegler 谱的商，并刻画同调剩余域内射对象，从而以纯性视角联系同调谱与 Balmer 谱。
A simple proof of Bernstein-Lunts equivalence
给出了Harish-Chandra模的普通和等变导出范畴的Bernstein-Lunts等价性的一个简单证明.这个证明不需要有界性假设。在附录中，我们收集了一些必要的，但不完全标准的事实，从同调代数。
On a proof of the Gorenstein Symmetry Conjecture
本研究通过同调方法给出Gorenstein对称猜想的证明，明确Artin代数自内射维数单侧有限性是其具备Gorenstein性的充分条件，完善了相关代数理论的证明体系。
On $q$-deformed Farey sum and a homological interpretation of $q$-deformed real quadratic irrational numbers
本文研究负连分数下的左 $q$-形变有理数，建立其 $q$-形变 Farey 和计算公式，并据此给出其与有理纽结 Jones 多项式关系的组合证明及同调解释。
The purpose of this paper is to give information on the FI-homology of the standard injective cogenerators of the category of FI-modules, where FI is the category of finite sets and injections. Working over a field k of characteristic zero, a full calculation is given in homological degree zero and a conjectural description in higher homological degree. The proof of the main theorem reduces to a calculation in representation theory of the symmetric groups, exploiting the Young orthonormal basis.
Coinduction functor in representation stability theory
研究了FI-模范畴及其变体上的余诱导函子。利用余归纳函子，我们给出了FI-模同调性质的各种结果的新的和更简单的证明。我们还证明了特征为0的域上的任意n-生成投射VI-模都是内射的。
Torsion pairs and filtrations in abelian categories with tilting objects
给定一个同调维数为2的Noether阿贝尔范畴$\mathcal Z$和一个倾斜对象$T$，阿贝尔范畴$\mathcal Z$和$\text{End} (T)^{\textit{op}}$上的交换模范畴通过一个两个倾斜的序列联系起来，我们给出了所涉及的挠对的一个明确的描述.然后，我们使用我们的技术，以获得一个简化的证明定理的詹森-马德森-苏，即$\mathcal Z$有一个三步过滤的扩展封闭的子范畴。最后，我们将Jensen-Madsen-Su的过滤推广到任意有限同调维数的Noether阿贝尔范畴。
Gröbner methods for representations of combinatorial categories
给定一个组合性质的范畴C，我们研究以下基本问题：C的组合行为如何影响C的代数表示行为？我们证明了两个一般结果。第一个给出了一个组合的标准表示的C承认理论的Gröbner基地。由此，我们得到一个标准noetherianity的表示。第二类给出了C的Hilbert表示级数的一般“合理性”结果的组合判据。这个准则与形式语言的理论相联系，并充分利用了语言生成函数的结果，如转移矩阵方法和Chomsky-Schützenberger定理。我们的工作是出于最近的工作在文献中的各种具体类别的表示。我们的一般准则恢复了许多已经被特别方法证明的关于这些范畴的结果，并且经常产生更清晰的证明和更有力的陈述。举例来说：我们给出了一个新的，更强大的，证明FI-模（最初由Church-Ellenberg-Farb引入）和一族自然推广是Noether的;我们从有限域的一般表示论的研究中给出了Lannes-Schwartz artinian猜想的推广的简单证明;我们显著改进了Snowden引入的与Segre嵌入合冲有关的$Δ$ -模理论;建立了正特征扭交换代数的基本性质。
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