arXiv subjects

Jun-ichi Miyachi

Publications and source records attributed to Jun-ichi Miyachi.

7 recordsLinked to original sources

arXiv2020

Derived categories of NDG categories

We study $N$-differential graded ($NDG$) categories and their the derived categories. First, we introduce $N$-differential modules over an $NDG$ category $\mathcal{A}$. Then we show that the category $\mathsf{C}_{Ndg}(\mathcal{A})$ of $N$-differential $\mathcal{A}$-modules is a Frobenius category, and that its homotopy category $\mathsf{K}_{Ndg}(\mathcal{a})$ is a triangulated category. Second, we study the properties of the derived category $\mathsf{D}_{Ndg}(\mathcal{A})$ and give triangle equivalences of Morita type between derived categories of $NDG$ categories. Finally, we show $\mathsf{D}_{Ndg}(\mathcal{A})$ is triangle equivalent to the derived category of some ordinary DG category.

math.CT↗

arXiv2017

Derived categories of $N$-complexes

We study the homotopy category $\mathsf{K}_{N}(\mathcal{B})$ of $N$-complexes of an additive category $\mathcal{B}$ and the derived category $\mathsf{D}_{N}(\mathcal{A})$ of an abelian category $\mathcal{A}$. First we show that both $\mathsf{K}_N(\mathcal{B})$ and $\mathsf{D}_N(\mathcal{A})$ 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 $\mathcal{A}$, we show that $\mathsf{D}_{N}(\mathcal{A})$ is triangle equivalent to the ordinary derived category $\mathsf{D}(\mathsf{Morph}_{N-2}(\mathcal{A}))$ where $\mathsf{Morph}_{N-2}(\mathcal{A})$ is the category of sequential $N-2$ morphisms of $\mathcal{A}$.

math.CT↗

arXiv2016

Polygon of recollements and $N$-complexes

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 $\mathsf{K}_{N}(\mathcal{B})$.

math.CT↗

arXiv2010

Recollement of homotopy categories and Cohen-Macaulay modules

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 $\opn{T}_2(R)$-modules.

math.RA↗

arXiv2002

Recollement and Tilting Complexes

First, we study recollement of a derived category of unbounded complexes of modules induced by a partial tilting complex. Second, we give equivalent conditions for P^{centerdot} to be a recollement tilting complex, that is, a tilting complex which induces an equivalence between recollements $\{\cat{D}_{A/AeA}(A), \cat{D}(A), \cat{D}(eAe)}$ and $\{\cat{D}_{B/BfB}(B), \cat{D}(B), \cat{D}(fBf)}$, where e, f are idempotents of A, B, respectively. In this case, there is an unbounded bimodule complex $\varDelta^{\centerdot}_{T}$ which induces an equivalence between $\cat{D}_{A/AeA}(A)$ and $\cat{D}_{B/BfB}(B)$. Third, we apply the above to a symmetric algebra A. We show that a partial tilting complex $P^{\centerdot}$ for A of length 2 extends to a tilting complex, and that $P^{\centerdot}$ is a tilting complex if and only if the number of indecomposable types of $P^{\centerdot}$ is one of A. Finally, we show that for an idempotent e of A, a tilting complex for eAe extends to a recollement tilting complex for A, and that its standard equivalence induces an equivalence between $\cat{Mod}A/AeA$ and $\cat{Mod}B/BfB$.

math.RA↗

arXiv2000

Derived Picard Groups of Finite Dimensional Hereditary Algebras

Let A be an algebra over a field k, and denote by D^b(Mod A) the bounded derived category of left A-modules. The derived Picard group DPic_k(A) is the group of triangle auto-equivalences of D^b(Mod A) induced by tilting complexes. We study the group DPic_k(A) when A = k Δis the path algebra of a finite quiver Δ. We obtain general results on the structure of DPic_k(A), as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPic_k(A) on a certain infinite quiver. This representation is faithful whenΔis a tree, and then DPic_k(A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPic_k(A). When A is hereditary, DPic_k(A) coincides with the full group of k-linear triangle auto-equivalences of D^b(Mod} A). Hence we can calculate the group of such auto-equivalences for any triangulated category D equivalent to D^b(Mod A). These include the derived categories of certain noncommutative spaces introduced by Kontsevich-Rosenberg.

math.RA↗

arXiv2000

On t-structures and Torsion Theories Induced by Compact Objects

First, we show that a compact object $C$ in a triangulated category, which satisfies suitable conditions, induces a $t$-structure. Second, in an abelian category we show that a complex $P^{\centerdot}$ of small projective objects of term length two, which satisfies suitable conditions, induces a torsion theory. In the case of module categories, using a torsion theory, we give equivalent conditions for $P^{\centerdot}$ to be a tilting complex. Finally, in the case of artin algebras, we give one to one correspondence between tilting complexes of term length two and torsion theories with certain conditions.

math.RA↗