论文标题
在三角类别上排名函数的功能方法
A functorial approach to rank functions on triangulated categories
论文作者
论文摘要
我们通过其Abelianisation $ \ operatatorName {mod} \ Mathcal {c} $在三角形类别$ \ MATHCAL {C} $上研究排名函数。我们证明，$ \ Mathcal {C} $上的每个排名函数都可以解释为$ \ perivatorName {mod} \ Mathcal {C} $上的加法功能。结果，每个积分等级函数都具有不可还原的级别分解。此外，我们将积分等级函数与函数类别中的许多重要概念相关联$ \ pereratatorName {mod} \ Mathcal {c} $。我们研究了等级函数与函子之间的连接，从$ \ Mathcal {C} $到本地有限的三角形类别，以及Chuang和Lazarev的概括结果。在特殊情况下，对于紧凑的三角形类别$ \ MATHCAL {T} $，该连接变得特别好，在$ \ nathcal {c} $ smashing Bocalisation of $ \ nathcal的$ \ Mathcal {在这种情况下，可以使用$ \ Mathcal {t} $中某些内福对象的组合长度来描述任何积分级别函数。最后，如果$ \ Mathcal {c} = \ propatorAtorName {per}（a）$对于差异分级代数$ a $ a $，我们将同源表达式$ a \ $ a \ a \ b $用$ \ propatateOrnAme {per pertatorNearname {per}（per}（per}（b）$ local locally local locally等级函数，我们呼吁我们呼叫idempotent。
We study rank functions on a triangulated category $\mathcal{C}$ via its abelianisation $\operatorname{mod}\mathcal{C}$. We prove that every rank function on $\mathcal{C}$ can be interpreted as an additive function on $\operatorname{mod}\mathcal{C}$. As a consequence, every integral rank function has a unique decomposition into irreducible ones. Furthermore, we relate integral rank functions to a number of important concepts in the functor category $\operatorname{Mod}\mathcal{C}$. We study the connection between rank functions and functors from $\mathcal{C}$ to locally finite triangulated categories, generalising results by Chuang and Lazarev. In the special case $\mathcal{C}=\mathcal{T}^c$ for a compactly generated triangulated category $\mathcal{T}$, this connection becomes particularly nice, providing a link between rank functions on $\mathcal{C}$ and smashing localisations of $\mathcal{T}$. In this context, any integral rank function can be described using the composition length with respect to certain endofinite objects in $\mathcal{T}$. Finally, if $\mathcal{C}=\operatorname{per}(A)$ for a differential graded algebra $A$, we classify homological epimorphisms $A\to B$ with $\operatorname{per}(B)$ locally finite via special rank functions which we call idempotent.
