We introduce the homotopy category of unbounded complexes with bounded homologies. We study a rec... more We introduce the homotopy category of unbounded complexes with bounded homologies. We study a recollement of its a quotient by the homotopy category of bounded complexes. This leads to the existence of quotient categories which are equivalent to a homotopy category of acyclic comlpexes, that is a stable derived category. In the case of a coherent ring R of self-injective dimension both sides, we show that the above recollement are triangulated equivalent to a recollement of the stable module category of Cohen-Macaulay R-modules.
In a triangulated category T with a pair of triangulated subcategories X and Y, one may consider ... more In a triangulated category T with a pair of triangulated subcategories X and Y, one may consider the subcategory of extensions X*Y. We give conditions for X*Y to be triangulated and use them to provide tools for constructing stable t-structures. In particular, we show how to construct so-called triangles of recollements, that is, triples of stable t-structures of the form (X,Y), (Y,Z), (Z,X). We easily recover some triangles of recollements known from the literature.
We study the homotopy category of unbounded complexes with bounded homologies and its quotient ca... more We study the homotopy category of unbounded complexes with bounded homologies and its quotient category by the homotopy category of bounded complexes. We show the existence of a recollement of the above quotient category and it has the homotopy category of acyclic complxes as a triangulated subcategory. In the case of the homotopy category of finitely generated projective modules over an Iwanaga-Gorenstein ring, we show that the above quotient category are triangle equivalent to the stable module category of Cohen-Macaulay T_2(R)-modules.
Every homomorphism of modules is projective-stably equivalent to an epimorphism but is not always... more Every homomorphism of modules is projective-stably equivalent to an epimorphism but is not always to a monomorphism. We prove that a map is projective-stably equivalent to a monomorphism if and only if its kernel is torsionless, that is, a first syzygy. If it occurs although, there can be various monomorphisms that are projective-stably equivalent to a given map. But in this case there uniquely exists a "perfect" monomorphism to which a given map is projective-stably equivalent.
We study the homotopy category K_N(B) of N-complexes of an additive category B and the derived ca... more We study the homotopy category K_N(B) of N-complexes of an additive category B and the derived category D_N(A) of an abelian category A. First we show that both K_N(B) and D_N(A) have natural structures of triangulated categories. Then we establish a theory of projective (resp., injective) resolutions and derived functors. Finally, under some conditions of an abelian category A, we show that D_N(A) is triangle equivalent to the ordinary derived category D(Morph_N-2(A)) where Morph_N-2(A) is the category of sequential N-2 morphisms of A.
We study the homotopy category K N (B) of N-complexes of an additive category B and the derived c... more We study the homotopy category K N (B) of N-complexes of an additive category B and the derived category D N (A) of an abelian category A. First we show that both K N (B) and D N (A) have natural structures of triangulated categories. Then we establish a theory of projective (resp., injective) resolutions and derived functors. Finally, under some conditions of an abelian category A, we show that D N (A) is triangle equivalent to the ordinary derived category D(Mor N−2 (A)) where Mor N−2 (A) is the category of sequential N − 2 morphisms of A.
A commutative noetherian ring with a dualizing complex is Gorenstein if and only if every acyclic... more A commutative noetherian ring with a dualizing complex is Gorenstein if and only if every acyclic complex of injective modules is totally acyclic. We extend this characterization, which is due to Iyengar and Krause, to arbitrary commutative noetherian rings, i.e. we remove the assumption about a dualizing complex. In this context Gorenstein, of course, means locally Gorenstein at every prime.
Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology, 2011
We study the homotopy category of unbounded complexes with bounded homologies and its quotient ca... more We study the homotopy category of unbounded complexes with bounded homologies and its quotient category by the homotopy category of bounded complexes. In the case of the homotopy category of finitely generated projective modules over an Iwanaga-Gorenstein ring, we show the existence of a new structure in the above quotient category, which we call a triangle of recollements. Moreover, we show that this quotient category is triangle equivalent to the stable module category of Cohen-Macaulay T2(R)-modules.
We study a structure of subcategories which are called a polygon of recollements in a triangulate... more We study a structure of subcategories which are called a polygon of recollements in a triangulated category. First, we study a 2n-gon of recollements in an (m/n)-Calabi-Yau triangulated category. Second, we show the homotopy category K(Mor_N-1(B)) of complexes of an additive category Mor_N-1(B) of N-1 sequences of split monomorphisms of an additive category B has a 2N-gon of recollments. Third, we show the homotopy category K_N(B) of N-complexes of B has also a 2N-gon of recollments. Finally, we show there is a triangle equivalence between K(Mor_N-1(B)) and K_N(B).
Abstract. We define the symmetric Auslander category As(R) to consist of complexes of projective ... more Abstract. We define the symmetric Auslander category As(R) to consist of complexes of projective modules whose left- and right-tails are equal to the left- and right-tails of totally acyclic com-plexes of projective modules.
In a triangulated category T with a pair of triangulated subcategories X and Y, one may consider ... more In a triangulated category T with a pair of triangulated subcategories X and Y, one may consider the subcategory of extensions X * Y. We give conditions for X * Y to be triangulated and use them to provide tools for constructing stable t-structures. In particular, we show how to construct so-called triangles of recollements, that is, triples of stable t-structures of the form (X, Y), (Y, Z), (Z, X). We easily recover some triangles of recollements known from the literature.
Abstract. We answer a question posed by Auslander and Bridger. Every homomor-phism of modules is ... more Abstract. We answer a question posed by Auslander and Bridger. Every homomor-phism of modules is projective-stably equivalent to an epimorphism but is not always to a monomorphism. We prove that a map is projective-stably equivalent to a monomor-phism if and only if its kernel is torsionless, that is, a first syzygy. If it occurs although, there can be various monomorphisms that are projective-stably equivalent to a given map. But in this case there uniquely exists a ”perfect ” monomorphism to which a given map is projective-stably equivalent. 1
Every homomorphism of modules is projective-stably equivalent to an epimorphism but is not always... more Every homomorphism of modules is projective-stably equivalent to an epimorphism but is not always to a monomorphism. We prove that a map is projectivestably equivalent to a monomorphism if and only if its kernel is torsionless, that is, a first syzygy. If it occurs although, there can be various monomorphisms that are projective-stably equivalent to a given map. But in this case there uniquely exists a ”perfect ” monomorphism to which a given map is projective-stably equivalent. 2000 Mathematics Subjects Classification: 13D02, 13D25, 16D90 1
Abstract. In a triangulated category T with a pair of triangulated subcategories X and Y, one may... more Abstract. In a triangulated category T with a pair of triangulated subcategories X and Y, one may consider the subcategory of extensions X ∗ Y. We give conditions for X∗Y to be triangulated and use them to provide tools for constructing stable t-structures. In particular, we show how to construct so-called triangles of recollements, that is, triples of stable t-structures of the form (X,Y), (Y,Z), (Z,X). We easily recover some triangles of recollements known from the literature.
Auslander and Bridger introduced the notion of projective stabilization modR of a category of fin... more Auslander and Bridger introduced the notion of projective stabilization modR of a category of finite modules. The category modR is known to be non-abelian. But realistically, modR is almost abelian. It fails to be abelian because of the lack of kernel and cokernel. In fact, each morphism has a pseudo-kernel and a pseudo-cokernel (see §3). On the other hand, a pseudo-kernel of a monomorphism does not necessarily vanish. In this paper we focus on how modR is similar or dissimilar to an abelian category (§4). What is a monomorphism? Which object makes monomorphisms split? One reason for similarity is that modR is closely related to the homotopy category of complexes. We discuss the functor from modR to homotopy category(§2). The method we use already produced important results in representation theory on commutative rings [2], [5]. Throughout the paper, R is a commutative semiperfect ring, equivalently a finite direct sum of local rings; that is, each finite module has a projective cov...
We study a structure of subcategories which are called a polygon of recollements in a triangulate... more We study a structure of subcategories which are called a polygon of recollements in a triangulated category. First, we study a $2n$-gon of recollements in an $(m/n)$-Calabi-Yau triangulated category. Second, we show the homotopy category $\mathsf{K}(\mathsf{Mor}_{N-1}(\mathcal{B}))$ of complexes of an additive category $\mathsf{Mor}_{N-1}(\mathcal{B})$ of $N-1$ sequences of split monomorphisms of an additive category $\mathcal{B}$ has a $2N$-gon of recollments. Third, we show the homotopy category $\mathsf{K}_{N}(\mathcal{B})$ of $N$-complexes of $\mathcal{B}$ has also a $2N$-gon of recollments. Finally, we show there is a triangle equivalence between $\mathsf{K}(\mathsf{Mor}_{N-1}(\mathcal{B}))$ and $\mathsf{K}_{N}(\mathcal{B})$.
Let T be a triangulated category with triangulated subcategories X and Y. We show that the subcat... more Let T be a triangulated category with triangulated subcategories X and Y. We show that the subcategory of extensions X ∗ Y is triangulated if and only if Y ∗ X ⊆ X ∗ Y. In this situation, we show the following analogue of the Second Isomorphism Theorem: (X ∗ Y)/X ≃ Y/(X ∩ Y) and (X ∗ Y)/Y ≃ X/(X ∩ Y). This follows from the existence of a stable t-structure ( X X∩Y , Y X∩Y ) in (X ∗Y)/(X∩Y). We use the machinery to give a recipe for constructing triangles of recollements and recover some triangles of recollements from the literature.
Every homomorphism of modules is projective-stably equivalent to an epimorphism but is not always... more Every homomorphism of modules is projective-stably equivalent to an epimorphism but is not always to a monomorphism. We prove that a map is projective-stably equivalent to a monomorphism if and only if its kernel is torsionless, that is, a first syzygy. If it occurs although, there can be various monomorphisms that are projective-stably equivalent to a given map. But in this case there uniquely exists a "perfect" monomorphism to which a given map is projective-stably equivalent.
We introduce the homotopy category of unbounded complexes with bounded homologies. We study a rec... more We introduce the homotopy category of unbounded complexes with bounded homologies. We study a recollement of its a quotient by the homotopy category of bounded complexes. This leads to the existence of quotient categories which are equivalent to a homotopy category of acyclic comlpexes, that is a stable derived category. In the case of a coherent ring R of self-injective dimension both sides, we show that the above recollement are triangulated equivalent to a recollement of the stable module category of Cohen-Macaulay R-modules.
In a triangulated category T with a pair of triangulated subcategories X and Y, one may consider ... more In a triangulated category T with a pair of triangulated subcategories X and Y, one may consider the subcategory of extensions X*Y. We give conditions for X*Y to be triangulated and use them to provide tools for constructing stable t-structures. In particular, we show how to construct so-called triangles of recollements, that is, triples of stable t-structures of the form (X,Y), (Y,Z), (Z,X). We easily recover some triangles of recollements known from the literature.
We study the homotopy category of unbounded complexes with bounded homologies and its quotient ca... more We study the homotopy category of unbounded complexes with bounded homologies and its quotient category by the homotopy category of bounded complexes. We show the existence of a recollement of the above quotient category and it has the homotopy category of acyclic complxes as a triangulated subcategory. In the case of the homotopy category of finitely generated projective modules over an Iwanaga-Gorenstein ring, we show that the above quotient category are triangle equivalent to the stable module category of Cohen-Macaulay T_2(R)-modules.
Every homomorphism of modules is projective-stably equivalent to an epimorphism but is not always... more Every homomorphism of modules is projective-stably equivalent to an epimorphism but is not always to a monomorphism. We prove that a map is projective-stably equivalent to a monomorphism if and only if its kernel is torsionless, that is, a first syzygy. If it occurs although, there can be various monomorphisms that are projective-stably equivalent to a given map. But in this case there uniquely exists a "perfect" monomorphism to which a given map is projective-stably equivalent.
We study the homotopy category K_N(B) of N-complexes of an additive category B and the derived ca... more We study the homotopy category K_N(B) of N-complexes of an additive category B and the derived category D_N(A) of an abelian category A. First we show that both K_N(B) and D_N(A) have natural structures of triangulated categories. Then we establish a theory of projective (resp., injective) resolutions and derived functors. Finally, under some conditions of an abelian category A, we show that D_N(A) is triangle equivalent to the ordinary derived category D(Morph_N-2(A)) where Morph_N-2(A) is the category of sequential N-2 morphisms of A.
We study the homotopy category K N (B) of N-complexes of an additive category B and the derived c... more We study the homotopy category K N (B) of N-complexes of an additive category B and the derived category D N (A) of an abelian category A. First we show that both K N (B) and D N (A) have natural structures of triangulated categories. Then we establish a theory of projective (resp., injective) resolutions and derived functors. Finally, under some conditions of an abelian category A, we show that D N (A) is triangle equivalent to the ordinary derived category D(Mor N−2 (A)) where Mor N−2 (A) is the category of sequential N − 2 morphisms of A.
A commutative noetherian ring with a dualizing complex is Gorenstein if and only if every acyclic... more A commutative noetherian ring with a dualizing complex is Gorenstein if and only if every acyclic complex of injective modules is totally acyclic. We extend this characterization, which is due to Iyengar and Krause, to arbitrary commutative noetherian rings, i.e. we remove the assumption about a dualizing complex. In this context Gorenstein, of course, means locally Gorenstein at every prime.
Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology, 2011
We study the homotopy category of unbounded complexes with bounded homologies and its quotient ca... more We study the homotopy category of unbounded complexes with bounded homologies and its quotient category by the homotopy category of bounded complexes. In the case of the homotopy category of finitely generated projective modules over an Iwanaga-Gorenstein ring, we show the existence of a new structure in the above quotient category, which we call a triangle of recollements. Moreover, we show that this quotient category is triangle equivalent to the stable module category of Cohen-Macaulay T2(R)-modules.
We study a structure of subcategories which are called a polygon of recollements in a triangulate... more We study a structure of subcategories which are called a polygon of recollements in a triangulated category. First, we study a 2n-gon of recollements in an (m/n)-Calabi-Yau triangulated category. Second, we show the homotopy category K(Mor_N-1(B)) of complexes of an additive category Mor_N-1(B) of N-1 sequences of split monomorphisms of an additive category B has a 2N-gon of recollments. Third, we show the homotopy category K_N(B) of N-complexes of B has also a 2N-gon of recollments. Finally, we show there is a triangle equivalence between K(Mor_N-1(B)) and K_N(B).
Abstract. We define the symmetric Auslander category As(R) to consist of complexes of projective ... more Abstract. We define the symmetric Auslander category As(R) to consist of complexes of projective modules whose left- and right-tails are equal to the left- and right-tails of totally acyclic com-plexes of projective modules.
In a triangulated category T with a pair of triangulated subcategories X and Y, one may consider ... more In a triangulated category T with a pair of triangulated subcategories X and Y, one may consider the subcategory of extensions X * Y. We give conditions for X * Y to be triangulated and use them to provide tools for constructing stable t-structures. In particular, we show how to construct so-called triangles of recollements, that is, triples of stable t-structures of the form (X, Y), (Y, Z), (Z, X). We easily recover some triangles of recollements known from the literature.
Abstract. We answer a question posed by Auslander and Bridger. Every homomor-phism of modules is ... more Abstract. We answer a question posed by Auslander and Bridger. Every homomor-phism of modules is projective-stably equivalent to an epimorphism but is not always to a monomorphism. We prove that a map is projective-stably equivalent to a monomor-phism if and only if its kernel is torsionless, that is, a first syzygy. If it occurs although, there can be various monomorphisms that are projective-stably equivalent to a given map. But in this case there uniquely exists a ”perfect ” monomorphism to which a given map is projective-stably equivalent. 1
Every homomorphism of modules is projective-stably equivalent to an epimorphism but is not always... more Every homomorphism of modules is projective-stably equivalent to an epimorphism but is not always to a monomorphism. We prove that a map is projectivestably equivalent to a monomorphism if and only if its kernel is torsionless, that is, a first syzygy. If it occurs although, there can be various monomorphisms that are projective-stably equivalent to a given map. But in this case there uniquely exists a ”perfect ” monomorphism to which a given map is projective-stably equivalent. 2000 Mathematics Subjects Classification: 13D02, 13D25, 16D90 1
Abstract. In a triangulated category T with a pair of triangulated subcategories X and Y, one may... more Abstract. In a triangulated category T with a pair of triangulated subcategories X and Y, one may consider the subcategory of extensions X ∗ Y. We give conditions for X∗Y to be triangulated and use them to provide tools for constructing stable t-structures. In particular, we show how to construct so-called triangles of recollements, that is, triples of stable t-structures of the form (X,Y), (Y,Z), (Z,X). We easily recover some triangles of recollements known from the literature.
Auslander and Bridger introduced the notion of projective stabilization modR of a category of fin... more Auslander and Bridger introduced the notion of projective stabilization modR of a category of finite modules. The category modR is known to be non-abelian. But realistically, modR is almost abelian. It fails to be abelian because of the lack of kernel and cokernel. In fact, each morphism has a pseudo-kernel and a pseudo-cokernel (see §3). On the other hand, a pseudo-kernel of a monomorphism does not necessarily vanish. In this paper we focus on how modR is similar or dissimilar to an abelian category (§4). What is a monomorphism? Which object makes monomorphisms split? One reason for similarity is that modR is closely related to the homotopy category of complexes. We discuss the functor from modR to homotopy category(§2). The method we use already produced important results in representation theory on commutative rings [2], [5]. Throughout the paper, R is a commutative semiperfect ring, equivalently a finite direct sum of local rings; that is, each finite module has a projective cov...
We study a structure of subcategories which are called a polygon of recollements in a triangulate... more We study a structure of subcategories which are called a polygon of recollements in a triangulated category. First, we study a $2n$-gon of recollements in an $(m/n)$-Calabi-Yau triangulated category. Second, we show the homotopy category $\mathsf{K}(\mathsf{Mor}_{N-1}(\mathcal{B}))$ of complexes of an additive category $\mathsf{Mor}_{N-1}(\mathcal{B})$ of $N-1$ sequences of split monomorphisms of an additive category $\mathcal{B}$ has a $2N$-gon of recollments. Third, we show the homotopy category $\mathsf{K}_{N}(\mathcal{B})$ of $N$-complexes of $\mathcal{B}$ has also a $2N$-gon of recollments. Finally, we show there is a triangle equivalence between $\mathsf{K}(\mathsf{Mor}_{N-1}(\mathcal{B}))$ and $\mathsf{K}_{N}(\mathcal{B})$.
Let T be a triangulated category with triangulated subcategories X and Y. We show that the subcat... more Let T be a triangulated category with triangulated subcategories X and Y. We show that the subcategory of extensions X ∗ Y is triangulated if and only if Y ∗ X ⊆ X ∗ Y. In this situation, we show the following analogue of the Second Isomorphism Theorem: (X ∗ Y)/X ≃ Y/(X ∩ Y) and (X ∗ Y)/Y ≃ X/(X ∩ Y). This follows from the existence of a stable t-structure ( X X∩Y , Y X∩Y ) in (X ∗Y)/(X∩Y). We use the machinery to give a recipe for constructing triangles of recollements and recover some triangles of recollements from the literature.
Every homomorphism of modules is projective-stably equivalent to an epimorphism but is not always... more Every homomorphism of modules is projective-stably equivalent to an epimorphism but is not always to a monomorphism. We prove that a map is projective-stably equivalent to a monomorphism if and only if its kernel is torsionless, that is, a first syzygy. If it occurs although, there can be various monomorphisms that are projective-stably equivalent to a given map. But in this case there uniquely exists a "perfect" monomorphism to which a given map is projective-stably equivalent.
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