自入射代数的平凡扩张与斜群代数
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摘要
这是一篇关于自入射代数的平凡扩张与斜群代数的博士论文,主要包含以下三个方面的内容.
1.分次自入射Koszul代数Λ的平凡扩张T(Λ)的Koszul'性,在本文第三章中,我们定义了两类T(Λ-)模并且构造了T(Λ)上这两类模的投射覆盖.接着我们讨论了自入射代数Λ中函子F=D(A)(?)A-的性质,得到若Λ-模Λ0有极小投射分解…→P2→P2P1→P1P0→P0A0→0,则FΛ0作为Λ-模有极小投射分解…→FP2→FP2FP1→FP2FP0→FP0FA0→0.最后通过构造平凡扩张代数上分次单模的极小投射分解,我们得到连通的有限维分次自入射Koszul代数的平凡扩张代数亦是分次自入射Koszul代数.
2.外代数上的斜群代数及其平凡扩张的稳定范畴的三角维数和他们的表示维数.在本文第四章中,我们首先考虑有限维自入射代数与其上的斜群代数的稳定范畴的三角维数之间的关系,根据三角范畴的定义,利用代数与其上斜群代数上的模之间的关系我们得到两者相等.因此利用Rouquier关于n维空间上外代数稳定范畴三角维数的结果及自入射代数稳定范畴三角维数与表示维数之间的关系,我们找到了斜群代数表示维数的下界,另一方面利用自入射代数的表示维数与根长的关系,得到n维空间上外代数上的斜群代数的表示维数为n+1.最后我们讨论n维空间上外代数的斜群代数的平凡扩张的表示维数,我们首先得到n维空间上外代数的斜群代数的基本代数的Gabriel箭图和关系,利用该结果我们证明了n维空间上的外代数的斜群代数的平凡扩张的表示维数为n+2.
3.有限维代数与其上的斜群代数中的n-丛倾斜子范畴之间的关系以及2维空间上外代数中n-丛倾斜子范畴的存在性.在本文第五章中我们首先从有限维代数上的斜群代数中的n-丛倾斜子范畴构造了有限维代数中的n-丛倾斜子范畴.接着我们从有限维代数中的n-丛倾斜子范畴构造了其上的斜群代数中的n-丛倾斜子范畴.最后我们具体地讨论了2维空间上外代数中的n-丛倾斜子范畴的存在性,通过具体的计算,我们发现2维空间上外代数中不存在n-丛倾斜子范畴
This thesis for Ph.D degree is on the trivial extension and skew group algebra of selfinjective algebras, it mainly consists of the following three parts.
1. the Koszulity of the trivial extension T(A) of the graded selfinjective algebra A. In Chapter 3, firstly we define two classes of modules over the trivial extension of selfinjective algebra and construct their projective covers. Secondly we discuss the properties of the functor F=DA(?)A-in selfinjective algebras A and obtain that the minimal projective resolution of FA0 as a A-module is…→FP2→FP2FP1→FP1FP0→FP0FA0→00, if A0 as a A-module has the minimal projective resolution…→P2→P2P1→P1P0→P0A0→0. At last by constructing the minimal projective resolutions of graded simple modules over the trivial extension of selfinjective algebra, we obtain if A is a connected selfinjective Koszul algebra, then the trivial extension algebra is also a selfinjective Koszul algebra.
2. The dimension of stable module categories of skew group algebra over exterior algebra and its trivial extension and the representation dimensions of them. In chapter 4, we discuss the relations of dimensions of stable mod-ule categories between selfinjective algebra and skew group algebra over it, by the definition of dimension of triangulated category and the relations between A—modules and A*G—modules, we show both of them are equal. So we find the lower bound for the skew group algebra over exterior algebra by Rouquier's result on the dimension of stable module category of selfinjective algebra and the relations between the dimension of stable module category and the represen-tation dimension of selfinjective algebra, on the other hand, using the relations between the Loewy length and the representation dimension of selfinjective al-gebra, we obtain the representation dimension of the skew group algebra of exterior algebra over n-dimensional vector space is n+1. Lastly we investigate the representation dimension of the trivial extension algebra of skew group al-gebra over exterior algebra, and obtain the quiver and relations of its basic algebra, so we show that the representation dimension of the trivial extension algebra of skew group algebra over exterior algebra on n-dimensional vector space is n+2.
3. The relations between n-cluster tilting subcategories of finite dimen-sional algebra and those of its skew group algebra and the existence of n-cluster tilting subcategories of exterior algebra of 2-dimensional vector space. In chap-ter 5, we construct the n-cluster tilting subcategories of finite-dimensional alge-bra from the n-cluster tilting subcategories of its skew group algebra. Then we construct the n-cluster tilting subcategories of skew group algebra over the finite-dimensional algebra from the n-cluster tilting subcategories of finite-dimensional algebra. And we investigate the existence of n-cluster tilting sub-categories of exterior algebra of 2-dimensional vector space finally, we find there are no n-cluster tilting subcategories in exterior algebra of 2-dimensional vector space by specific computation.
1.分次自入射Koszul代数Λ的平凡扩张T(Λ)的Koszul'性,在本文第三章中,我们定义了两类T(Λ-)模并且构造了T(Λ)上这两类模的投射覆盖.接着我们讨论了自入射代数Λ中函子F=D(A)(?)A-的性质,得到若Λ-模Λ0有极小投射分解…→P2→P2P1→P1P0→P0A0→0,则FΛ0作为Λ-模有极小投射分解…→FP2→FP2FP1→FP2FP0→FP0FA0→0.最后通过构造平凡扩张代数上分次单模的极小投射分解,我们得到连通的有限维分次自入射Koszul代数的平凡扩张代数亦是分次自入射Koszul代数.
2.外代数上的斜群代数及其平凡扩张的稳定范畴的三角维数和他们的表示维数.在本文第四章中,我们首先考虑有限维自入射代数与其上的斜群代数的稳定范畴的三角维数之间的关系,根据三角范畴的定义,利用代数与其上斜群代数上的模之间的关系我们得到两者相等.因此利用Rouquier关于n维空间上外代数稳定范畴三角维数的结果及自入射代数稳定范畴三角维数与表示维数之间的关系,我们找到了斜群代数表示维数的下界,另一方面利用自入射代数的表示维数与根长的关系,得到n维空间上外代数上的斜群代数的表示维数为n+1.最后我们讨论n维空间上外代数的斜群代数的平凡扩张的表示维数,我们首先得到n维空间上外代数的斜群代数的基本代数的Gabriel箭图和关系,利用该结果我们证明了n维空间上的外代数的斜群代数的平凡扩张的表示维数为n+2.
3.有限维代数与其上的斜群代数中的n-丛倾斜子范畴之间的关系以及2维空间上外代数中n-丛倾斜子范畴的存在性.在本文第五章中我们首先从有限维代数上的斜群代数中的n-丛倾斜子范畴构造了有限维代数中的n-丛倾斜子范畴.接着我们从有限维代数中的n-丛倾斜子范畴构造了其上的斜群代数中的n-丛倾斜子范畴.最后我们具体地讨论了2维空间上外代数中的n-丛倾斜子范畴的存在性,通过具体的计算,我们发现2维空间上外代数中不存在n-丛倾斜子范畴
This thesis for Ph.D degree is on the trivial extension and skew group algebra of selfinjective algebras, it mainly consists of the following three parts.
1. the Koszulity of the trivial extension T(A) of the graded selfinjective algebra A. In Chapter 3, firstly we define two classes of modules over the trivial extension of selfinjective algebra and construct their projective covers. Secondly we discuss the properties of the functor F=DA(?)A-in selfinjective algebras A and obtain that the minimal projective resolution of FA0 as a A-module is…→FP2→FP2FP1→FP1FP0→FP0FA0→00, if A0 as a A-module has the minimal projective resolution…→P2→P2P1→P1P0→P0A0→0. At last by constructing the minimal projective resolutions of graded simple modules over the trivial extension of selfinjective algebra, we obtain if A is a connected selfinjective Koszul algebra, then the trivial extension algebra is also a selfinjective Koszul algebra.
2. The dimension of stable module categories of skew group algebra over exterior algebra and its trivial extension and the representation dimensions of them. In chapter 4, we discuss the relations of dimensions of stable mod-ule categories between selfinjective algebra and skew group algebra over it, by the definition of dimension of triangulated category and the relations between A—modules and A*G—modules, we show both of them are equal. So we find the lower bound for the skew group algebra over exterior algebra by Rouquier's result on the dimension of stable module category of selfinjective algebra and the relations between the dimension of stable module category and the represen-tation dimension of selfinjective algebra, on the other hand, using the relations between the Loewy length and the representation dimension of selfinjective al-gebra, we obtain the representation dimension of the skew group algebra of exterior algebra over n-dimensional vector space is n+1. Lastly we investigate the representation dimension of the trivial extension algebra of skew group al-gebra over exterior algebra, and obtain the quiver and relations of its basic algebra, so we show that the representation dimension of the trivial extension algebra of skew group algebra over exterior algebra on n-dimensional vector space is n+2.
3. The relations between n-cluster tilting subcategories of finite dimen-sional algebra and those of its skew group algebra and the existence of n-cluster tilting subcategories of exterior algebra of 2-dimensional vector space. In chap-ter 5, we construct the n-cluster tilting subcategories of finite-dimensional alge-bra from the n-cluster tilting subcategories of its skew group algebra. Then we construct the n-cluster tilting subcategories of skew group algebra over the finite-dimensional algebra from the n-cluster tilting subcategories of finite-dimensional algebra. And we investigate the existence of n-cluster tilting sub-categories of exterior algebra of 2-dimensional vector space finally, we find there are no n-cluster tilting subcategories in exterior algebra of 2-dimensional vector space by specific computation.
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