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Xuefeng Mao

Publications and source records attributed to Xuefeng Mao.

9 recordsLinked to original sources
arXiv2025

Quillen-Suslin Theorem for connected cochain DG algebras

Let $\mathscr{A}$ be a connected cochain DG algebra and $P$ a DG $\mathscr{A}$-module such that its underlying graded module $P^{\#}$ is a finitely generated $\mathscr{A}^{\#}$-module. We show that $P$ is semi-free if it is semi-projective and it is categorically free if it is categorically projective. It can be considered as a generalization of the well-known Quillen-Suslin Theorem in DG context. As an application, we show that the ghost length and the cone length of a compact DG module coincide.

math.RA↗
arXiv2022

Local cohomology for Gorenstein homologically smooth DG algebras

In this paper, we introduce the theory of local cohomology and local duality to Notherian connected cochain DG algebras. We show that the notion of local cohomology functor can be used to detect the Gorensteinness of a homologically smooth DG algebra. For any Gorenstein homologically smooth locally finite DG algebra $\mathcal{A}$, we define a group homomorphism $\mathrm{Hdet}: \mathrm{Aut}_{dg}(\mathcal{A})\to k^{\times},$ called the homological determinant. As applications, we present a sufficient condition for the invariant DG subalgebra $\mathcal{A}^G$ to be Gorensten, where $\mathcal{A}$ is a homologically smooth DG algebra such that $H(\mathcal{A})$ is a Noetherian AS-Gorenstein graded algebra and $G$ is a finite subgroup of $\mathrm{Aut}_{dg}(\mathcal{A})$. Especially, we can apply this result to DG down-up algebras and non-trivial DG free algebras generated in two degree-one elements.

math.RA↗
arXiv2022

Cohomology algebras of a family of cochain DG skew polynomial algebras

Let $\mathcal{A}$ be a connected cochain DG algebra such that its underlying graded algebra $\mathcal{A}^{\#}$ is the graded skew polynomial algebra $$k\langle x_1,x_2, x_3\rangle/\left(\begin{array}{ccc} x_1x_2+x_2x_1\\ x_2x_3+x_3x_2\\ x_3x_1+x_1x_3 \end {array}\right), |x_1|=|x_2|=|x_3|=1.$$ From \cite{MWZ} or \cite{MWYZ}, one sees that the differential $\partial_{\mathcal{A}}$ is determined by \begin{align*} \left( \begin{array}{c} \partial_{\mathcal{A}}(x_1) \partial_{\mathcal{A}}(x_2) \partial_{\mathcal{A}}(x_3) \end{array} \right)=M\left( \begin{array}{c} x_1^2 x_2^2 x_3^2 \end{array} \right), \end{align*} for some $M\in M_3(k)$. For the case $1\le r(M)\le 3$, we compute $H(\mathcal{A})$ case by case. The computational results in this paper give substantial support for \cite{MWZ}, where the various homological properties of such DG algebras are systematically studied. We find some examples, which indicate that the cohomology graded algebra of a Koszul Calabi-Yau DG algebra may be not left (right) Gorenstein.

math.KT↗
arXiv2021

Homological properties of $3$-dimensional DG Sklyanin algebras

In this paper, we introduce the notion of DG Sklyanin algebras, which are connected cochain DG algebras whose underlying graded algebras are Sklyanin algebras. Let $\mathcal{A}$ be a $3$-dimensional DG Sklyanin algebra with $\mathcal{A}^{\#}=S_{a,b,c}$, where $(a,b,c)\in \Bbb{P}_k^2-\mathfrak{D}$ and $$\mathfrak{D}=\{(1,0,0), (0,1,0),(0,0,1)\}\sqcup\{(a,b,c)|a^3=b^3=c^3\}.$$ We systematically study its differential structures and various homological properties. Especially, we figure out the conditions for $\mathcal{A}$ to be Calabi-Yau, Koszul, Gorenstein and homologically smooth, respectively.

math.RA↗
arXiv2021

DG Algebra structures on the quantum affine $n$-space $\mathcal{O}_{-1}(k^n)$

Let $\mathcal{A}$ be a connected cochain DG algebra, whose underlying graded algebra $\mathcal{A}^{\#}$ is the quantum affine $n$-space $\mathcal{O}_{-1}(k^n)$. We compute all possible differential structures of $\mathcal{A}$ and show that there exists a one-to-one correspondence between $$\{\text{cochain DG algebra}\,\,\mathcal{A}\,|\,\mathcal{A}^{\#}=\mathcal{O}_{-1}(k^n)\}$$ and the $n\times n$ matrices $M_n(k)$. For any $M\in M_n(k)$, we write $\mathcal{A}_{\mathcal{O}_{-1}(k^3)}(M)$ for the DG algebra corresponding to it. We also study the isomorphism problems of these non-commutative DG algebras. For the cases $n\le 3$, we check their homological properties. Unlike the case of $n=2$, we discover that not all of them are Calabi-Yau when $n=3$. In spite of this, we recognize those Calabi-Yau ones case by case. In brief, we solve the problem on how to judge whether a given such DG algebra $\mathcal{A}_{\mathcal{O}_{-1}(k^3)}(M)$ is Calabi-Yau.

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arXiv2019

Derived Picard groups of homologically smooth Koszul DG algebras

In this paper, we show that the derived Picard group of a homologically smooth Koszul connected cochain DG algebra is isomorphic to the opposite group of the derived Picard group of its finite dimensional local Ext-algebra. As applications, we compute the derived Picard groups of some important DG algebras such as trivial DG polynomial algebras, trivial DG free algebras and several non-trivial DG down-up algebras and DG free algebras.

math.RA↗
arXiv2019

Some dimensions of DG polynomial algebras

Assume that $\mathcal{A}$ is a cochain DG polynomial algebra such that its underlying graded algebra $\mathcal{A}^{#}$ is a polynomial algebra generated by $n$ degree $1$ elements. We determine the DG Krull dimension, the global dimension, the ghost dimension and the Rouquier dimension of $\mathcal{A}$.

math.RA↗
arXiv2017

A note on homologically smooth connected cochain DG algebras

In this paper, we obtain two interesting results on homologically smooth connected cochain DG algebras. More precisely, we show that any Koszul DG module in $\mathrm{D_{fg}}(A)$ is compact, when $A$ is a homologically smooth connected cochain DG algebra with a Noetherian cohomology graded algebra $H(A)$. And we prove that the homologically smoothness of $A$ is equivalent to $$\mathrm{D_{fg}}(A)=\mathrm{D}^c(A),$$ if $A$ is a Koszul connected cochain DG algebra such that $H(A)$ is a Noetherian graded algebra with a balanced dualizing complex.

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