We generalise $\tau$-cluster morphism categories to the setting of triangulated categories containing a silting object. The compatibility of silting reduction with support $\tau$-tilting reduction will be an essential ingredient when linking our definition to that of Buan--Marsh. We also define two-term presilting sequences in the bounded derived category in such a way that they correspond to signed $\tau$-exceptional sequences in the module category.

Two-term silting and $\tau$-cluster morphism categories

Let <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda">
 <mml:semantics>
 <mml:mi mathvariant="normal">Λ</mml:mi>
 <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation>
 </mml:semantics>
</mml:math>
</inline-formula> be a finite dimensional algebra. There exists a natural injective map from the set of wide subcategories in the category of finitely generated <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda">
 <mml:semantics>
 <mml:mi mathvariant="normal">Λ</mml:mi>
 <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation>
 </mml:semantics>
</mml:math>
</inline-formula>-modules to the set of thick subcategories in the bounded derived category of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda">
 <mml:semantics>
 <mml:mi mathvariant="normal">Λ</mml:mi>
 <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation>
 </mml:semantics>
</mml:math>
</inline-formula>.We show, when <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda">
 <mml:semantics>
 <mml:mi mathvariant="normal">Λ</mml:mi>
 <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation>
 </mml:semantics>
</mml:math>
</inline-formula> is elementary, that the natural map is bijective if and only if <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda">
 <mml:semantics>
 <mml:mi mathvariant="normal">Λ</mml:mi>
 <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation>
 </mml:semantics>
</mml:math>
</inline-formula> is hereditary.

A characterization of hereditary algebras via thick subcategories

For a triangulated category with a bounded t-structure, we prove that there is a bijection between wide subcategories of its heart and thick subcategories of the triangulated category which are closed under the corresponding cohomological functor. We prove that a finite-dimensional triangular algebra over an algebraically closed field is hereditary if and only if any thick subcategory of the bounded derived category is closed under the usual cohomological functor. Dedicated to Professor Yingbo Zhang on the occasion of her 70th birthday.

https://www.intlpress.com/site/pub/files/_fulltext/journals/hha/2017/0019/0002/HHA-2017-0019-0002-a008.pdf

A note on thick subcategories and wide subcategories

Let G be a graph and σ:E(G)→{+1,−1} be a mapping. The pair (G,σ), denoted by Gσ, is called a signed graph. A (proper) l-edge coloring γ of Gσ is a mapping from each vertex–edge incidence of Gσ to Mq such that γ(v,e)=−σ(e)γ(w,e) for each edge e=vw, and no two vertex–edge incidences have the same color; that is, γ(v,e)≠γ(v,f). The chromatic index is the minimal number q such that Gσ has a proper q-edge coloring, denoted by χ′(Gσ). In 2020, Behr proved that the chromatic index of a signed graph is its maximum degree or maximum plus one. In this paper, we considered the chromatic index of the signed generalized Petersen graph GP(n,2) and show that its chromatic index is its maximum degree for most cases. In detail, we proved that (1) χ′(GPσ(n,2))=3 if n≡3 mod 6(n≥9); (2) χ′(GPσ(n,2))=3 if n=2p(p≥4).

/pdf/on-the-chromatic-index-of-the-signed-generalized-petersen-2cscqy3i.pdf

On the Chromatic Index of the Signed Generalized Petersen Graph GP(n, 2)

The homomorphism and coloring of the direct product of signed graphs

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A note on thick subcategories and wide subcategories

On the Chromatic Index of the Signed Generalized Petersen Graph GP(n, 2)

The homomorphism and coloring of the direct product of signed graphs