Tau-tilting theory

In mathematics, specifically the representation theory of finite dimensional algebras, $\tau$-tilting theory is a recent development that can be viewed as a completion of tilting theory from the viewpoint of mutation. The subfield was introduced in the early 2010s by Takahide Adachi, Osamu Iyama and Idun Reiten and is now a very active area of research. The name reflects the combination of Auslander–Reiten theory and tilting theory, where the $\tau$ represents the Auslander-Reiten translation in the module category of an algebra. Many connections of $\tau$-tilting theory with other areas of mathematics have been explored in recent years.

Basic definitions

Let $K$ be an algebraically closed field and $A$ be a basic finite dimensional algebra, that is, an algebra with pairwise non-isomorphic projective modules. Let $\tau$ denote the Auslander-Reiten translation, which is defined as $\tau =D\text{Tr}$, where $D={\text{Hom}}_K(-,K)$ denotes the standard duality and $\text{Tr}$ denotes the transposition. Denote by $\text{mod}A$ the category of finitely presented right modules of $A$ and for a module $M$ denote by $|M|$ the number of non-isomorphic indecomposable direct summands of $M$. The central notion studied in this area is the following.

τ-rigidity

Let $M$ be an object in $\text{mod}A$ and $P$ be a projective module in $\text{mod}A$.

• The module $M$ is called $\tau$-rigid if ${\text{Hom}}_A(M,\tau M)=0$. In other words, there does not exist a non-zero module homomorphism from the module $M$ to its Auslander-Reiten translation $\tau M$.
• A $\tau$-rigid module $M$ is called $\tau$-tilting if $|M|=|A|$. In other words, the number of direct non-isomorphic indecomposable direct summands of $M$ equals that of $A$, where we may express the algebra as a direct sum of all projective modules.
• A pair $(M,P)$ of a module $M$ and a projective module $P$ is called a $\tau$-rigid pair if $M$ is $\tau$-rigid and ${\text{Hom}}_A(P,M)=0$.
• A $\tau$-rigid pair $(M,P)$ is called (support-)$\tau$-tilting if $|M|+|P|=|A|$.

Connections with other areas

Already in the introductory paper a few connections with other areas were given, and more have been discovered over the years.

Silting Theory

A complex $𝑷=(P^i,d^i)$ in the bounded homotopy category $K^b(\text{proj}A)$ of projective $A$-modules is a 2-term-silting complex if

• $𝑷$ is concentrated in degrees $-1$ and $0$, i.e. $P^i=0$ for all $i\ne 0,-1$;
• ${\text{Hom}}_{K^b(\text{proj A})}(𝑷,𝑷[i])=0$ for any $i>0$, where $𝑷[i]$ denotes the $i$-th shift in the triangulated category;
• Each object in $K^b(\text{proj}A)$ can be obtained from $𝑷$ by taking finitely many shifts, mapping cones and direct summands. (Without this property $𝑷$ is called 2-term pre-silting).

There exists a bijection between $\tau$-tilting pairs $(M,P)$ and two-term silting complexes.

Cluster-Tilting Theory

Let $𝒞$ be a $\text{Hom}$-finite Krull–Schmidt 2-Calabi-Yau triangulated category.

• An object $T$ in $𝒞$ is called rigid if ${\text{Hom}}_{𝒞}(T,T[1])=0$.
• An object $T$ in $𝒞$ is called cluster-tilting if $\text{add}(T)=\{X\in 𝒞:{\text{Hom}}_{𝒞}(T,X[1])=0\}$, where $\text{add}(T)$ is the full subcategory consisting of all direct summands of finite direct sums of copies of $T$.

Assume that $𝒞$ has a cluster-tilting object $T$, and let $A={\text{End}}_{𝒞}(T{)}^{\text{op}}$ be the opposite endomorphism algebra of $T$ over $𝒞$. Then there is a bijection between basic rigid objects in $𝒞$ and basic $\tau$-rigid objects in $A$. And there is a bijection between cluster-tilting objects in $𝒞$ and basic support-$\tau$-tilting objects in $A$.

Related notions

support-tau-tilting subcategories, tau-exceptional sequences.

References

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