arXiv subjects

Riku Fushimi

Publications and source records attributed to Riku Fushimi.

6 recordsLinked to original sources

arXiv2026

Cluster Morita theorem for negative cluster categories

Fix an integer $d\leq-2$. We characterize Hom-finite algebraic triangulated categories admitting a $(-d)$-simple-minded system as stable categories $\underline{\mathrm{CM}}(B)$ of proper $(-d)$-self-injective non-positive dg algebras; equivalently, each admits a $d$-stable locally finite strictly positive dg model whose cosingular dg quotient recovers the chosen enhancement. Under a $d$-Calabi--Yau hypothesis, we give a characterization theorem for acyclic negative cluster categories in terms of the finite graded extension algebra of a simple-minded system. At chain level, Hochschild and reduced cyclic localization identify right $(d+1)$-Calabi--Yau structures on the finite-dimensional part with normalized right $d$-Calabi--Yau structures on the cosingular quotient. Finally, when the Koszul dual is proper, the Brav--Dyckerhoff evaluation morphism is a quasi-isomorphism of mixed complexes, yielding left--right Calabi--Yau symmetry.

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arXiv2026

A classification of derived-discrete graded algebras

A finite-dimensional algebra is derived-discrete, in the sense of Vossieck, precisely when it is a piecewise hereditary algebra of Dynkin type or a gentle one-cycle algebra not satisfying the clock condition. Building on the discrete triangulated categories of Broomhead, Pauksztello, and Ploog, we study derived-discreteness for locally finite non-positively graded algebras, regarded as connective locally finite dg algebras with trivial differential. Our main result extends Vossieck's classification to this setting: such a graded algebra is derived-discrete if and only if it is graded Morita equivalent to a piecewise hereditary algebra of Dynkin type, or it is a graded gentle one-cycle algebra not satisfying the graded clock condition. Along the way, using surface models we prove the conjecture of Kalck and Yang that the graded clock condition is invariant under derived equivalence. We also establish a restriction on semi-orthogonal decompositions of the bounded derived category of a path algebra of Dynkin type $D$.

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arXiv2026

Silting-discrete graded path algebras

We classify connected finite acyclic graded quivers $Q$ for which the graded path algebra $kQ$, regarded as a formal dg algebra, is silting-discrete. We prove that $kQ$ is silting-discrete if and only if it is derived-discrete, and that both conditions are equivalent to the underlying graph of $Q$ being of type ADE, or of type $\widetilde{A}$ with unequal clockwise and counter-clockwise total degrees. The key ingredient is an explicit construction of an infinite pre-simple-minded collection in $\\text{pvd } kQ$ in the non-discrete case.

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arXiv2026

Silting theory and derived base change

For finite-dimensional algebras over a field, Koenig and Yang established a bijection between silting complexes and simple-minded collections in the bounded derived category, with further contributions by many authors in various settings. In this paper, we work over a commutative complete local noetherian ring $(R,\m,k)$ rather than over a field and establish a bijection in this more general setting. As an application of this generalization, we construct a bijection between silting complexes over a noetherian $R$-algebra $\Lambda$ and silting complexes over $\Lambda\ten^\LL_RS$ for any morphism of commutative complete local noetherian rings $(R,\m,k)\to(S,\n,k)$. This result generalizes some known results on silting complexes over noetherian algebras.

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arXiv2024

Contravariant Koszul duality between non-positive and positive dg algebras

The Koszul dual of locally finite non-positive dg algebra is locally finite positive dg algebra. However, the Koszul dual of locally finite positive dg algebra is not necessary locally finite. We characterize locally finite positive dg algebras whose Koszul dual is locally finite. Moreover, we show that the Koszul dual functor induces contravariant equivalences between the perfect derived category and the perfectly valued derived category. As an application of Koszul dualities, we establish an ST-correspondence. We also show that, under some assumption, every covariantly finite bounded heart is a length heart, and the triangulated analogy of Smal{\o}'s symmetry holds.

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arXiv2023

The correspondence between silting objects and $t$-structures for non-positive dg algebras

We establish a bijective correspondence between isomorphism classes of basic silting objects of $\mathsf{per}(A)$ and algebraic $t$-structures of $\mathsf{D}_{\rm fd}(A)$ for locally finite non-positive dg algebra $A$ over a field $k$ (more generally, we work in the setting of ST-pair inside an algebraic triangulated category). For a non-positive (topologically) homologically smooth dg $k$-algebra $A$ whose zeroth cohomology is finite-dimensional, or for a non-positive proper dg $k$-algebra $A$, the one-to-one correspondence between isomorphism classes of basic silting objects of $\mathsf{per}(A)$ and algebraic $t$-structures on $\mathsf{D}_{\rm fd}(A)$ was already known. The main result of this paper generalizes the above two results to locally finite non-positive dg $k$-algebras.

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