Journal ArticleDOI
Generators and Dimensions of Derived Categories of Modules
Takuma Aihara,Ryo Takahashi +1 more
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In this paper, the dimension of the bounded derived category of finitely generated modules over a commutative Noetherian ring has been studied, and it is shown that it is finite over a complete local ring containing a field with perfect residue field.Abstract:
Several years ago, Bondal, Rouquier, and Van den Bergh introduced the notion of the dimension of a triangulated category, and Rouquier proved that the bounded derived category of coherent sheaves on a separated scheme of finite type over a perfect field has finite dimension. In this article, we study the dimension of the bounded derived category of finitely generated modules over a commutative Noetherian ring. The main result of this article asserts that it is finite over a complete local ring containing a field with perfect residue field. Our methods also give a ring-theoretic proof of the affine case of Rouquier's theorem.read more
Citations
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Annihilation of Cohomology and Strong Generation of Module Categories
TL;DR: In this paper, the cohomology annihilator of a noetherian ring that is finitely generated as a module over its center is introduced, and results are established linking the existence of nontrivial cohomorphology annihilators and strong generators for the category of finite generated modules.
Journal ArticleDOI
Ulrich ideals and almost Gorenstein rings
TL;DR: In this article, the structure of the complex RHomR(R/I,R) is explored for an Ulrich ideal I in a Cohen-Macaulay local ring R. As a consequence, it is proved that in a onedimensional almost Gorenstein but non-Gorenstein local ring, the only possible UIL ideal is the maximal ideal.
Posted Content
The dimension of a subcategory of modules
Hailong Dao,Ryo Takahashi +1 more
TL;DR: In this paper, the dimension of a subcategory of finitely generated R-modules has been shown to have small dimensions when R is a commutative noetherian local ring.
Journal ArticleDOI
Upper bounds for dimensions of singularity categories
TL;DR: In this paper, the dimension of the singularity category of a Cohen-Macaulay local ring with an isolated singularity was shown to be upper bounded by an upper bound given by Ballard, Favero and Katzarkov.
Journal ArticleDOI
The dimension of a subcategory of modules
Hailong Dao,Ryo Takahashi +1 more
TL;DR: In this article, the dimension of a subcategory of finitely generated R-modules has been shown to have small dimensions when R is a commutative noetherian local ring.
References
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Book
Commutative Algebra: with a View Toward Algebraic Geometry
TL;DR: In this article, the authors define basic constructions and dimension theory, and apply them to the problem of homological methods for combinatorial problem solving in the context of homology.
Book
Commutative Ring Theory
Hideyuki Matsumura,Miles Reid +1 more
TL;DR: In this article, the authors introduce the notion of complete local rings and apply it to a wide range of applications, including: I-smoothness, I-flatness revisited, and valuation rings.
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Introduction to Homological Algebra
TL;DR: The landscape of homological algebra has evolved over the last half-century into a fundamental tool for the working mathematician as discussed by the authors, which is suitable for second or third year graduate students.
Book
An introduction to homological algebra
TL;DR: In this paper, the authors propose a theory of homology and cohomology theories of groups and moniods, and derive derived functors from homology functors, including Tensor products, groups of homomorphisms, and projective and injective modules.
Commutative Algebra I
TL;DR: A compilation of two sets of notes at the University of Kansas was published in the Spring of 2002 by?? and the other in the spring of 2007 by Branden Stone.