The Stable Monomorphism Category of a Frobenius category
The Stable Monomorphism Category of a Frobenius category
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DOI：
10.4310/mrl.2011.v18.n1.a9
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发表时间：
2009-11
期刊：
影响因子：
--
通讯作者：
Xiao-Wu Chen
中科院分区：
文献类型：
--
作者：
Xiao-Wu Chen
For a Frobenius abelian category $\mathcal{A}$, we show that the category ${\rm Mon}(\mathcal{A})$ of monomorphisms in $\mathcal{A}$ is a Frobenius exact category; the associated stable category $\underline{\rm Mon}(\mathcal{A})$ modulo projective objects is called the stable monomorphism category of $\mathcal{A}$. We show that a tilting object in the stable category $\underline{\mathcal{A}}$ of $\mathcal{A}$ modulo projective objects induces naturally a tilting object in $\underline{{\rm Mon}}(\mathcal{A})$. We show that if $\mathcal{A}$ is the category of (graded) modules over a (graded) self-injective algebra $A$, then the stable monomorphism category is triangle equivalent to the (graded) singularity category of the (graded) $2\times 2$ upper triangular matrix algebra $T_2(A)$. As an application, we give two characterizations to the stable category of Ringel-Schmidmeier (\cite{RS3}).
For a Frobenius abelian category $\mathcal{A}$, we show that the category ${\rm Mon}(\mathcal{A})$ of monomorphisms in $\mathcal{A}$ is a Frobenius exact category; the associated stable category $\underline{\rm Mon}(\mathcal{A})$ modulo projective objects is called the stable monomorphism category of $\mathcal{A}$. We show that a tilting object in the stable category $\underline{\mathcal{A}}$ of $\mathcal{A}$ modulo projective objects induces naturally a tilting object in $\underline{{\rm Mon}}(\mathcal{A})$. We show that if $\mathcal{A}$ is the category of (graded) modules over a (graded) self-injective algebra $A$, then the stable monomorphism category is triangle equivalent to the (graded) singularity category of the (graded) $2\times 2$ upper triangular matrix algebra $T_2(A)$. As an application, we give two characterizations to the stable category of Ringel-Schmidmeier (\cite{RS3}).