Endomorphism algebras of silting complexes

Throughout this paper, let $A$ be a finite dimensional algebra over a field $k$. We always assume that all modules are right modules. A composition $g\circ f$ of morphisms $f$ and $g$ means first $f$ then $g$. But a composition $\alpha\beta$ of arrows $\alpha$ and $\beta$ means that first $\alpha$ then $\beta$. The category of all right $A$-modules is denoted by $\mathrm{Mod}A$, and the subcategory of finitely presented $A$-modules is denoted by $\mathrm{mod}A$. For a given $A$-module $X$, we denote by $\mathrm{pd}X$ (resp. $\mathrm{id}X$) the projective (resp. injective) dimension of $X$.

Let $\mathcal{M}\subset\mathrm{Mod}A$ be a class of modules. $\mathrm{Add}\mathcal{M}$ (resp. $\mathrm{add}\mathcal{M}$) denotes the class consisting of all modules isomorphic to direct summands of (finite) direct sums of elements in $\mathcal{M}$, while $\mathrm{Gen}\mathcal{M}$ (resp. $\mathrm{gen}\mathcal{M}$) is the class of epimorphic images of (finite) direct sums of elements in $\mathcal{M}$. Dually, we define $\mathrm{Cogen}\mathcal{M}$ (resp. $\mathrm{cogen}\mathcal{M}$) as the class of all submodules of (finite) direct sums of elements in $\mathcal{M}$.

Denote by $\mathrm{ind}A$ the subcategory of $\mathrm{mod}A$ formed by the indecomposable $A$-modules. Given $X,Y\in\mathrm{ind}A$, a path from $X$ to $Y$ in $\mathrm{ind}A$ is a sequence of non-zero morphisms $X=X_{0}\rightarrow X_{1}\rightarrow\cdots\rightarrow X_{t-1}\rightarrow X_{t}=Y\ (t\geq 1)$, where $X_{i}\in\mathrm{ind}A$ for all $i$. We say that $X$ is a predecessor of $Y$ and $Y$ is a successor of $X$.

Let $\mathcal{C}$ be a subcategory of $\mathrm{ind}A$. Recall that $\mathcal{C}$ is closed under predecessors if, whenever there is a path from $X$ to $Y$ in $\mathrm{ind}A$, with $Y\in\mathcal{C}$, then $X\in\mathcal{C}$. An example is the left part $\mathcal{L}_{A}$ of $\mathrm{mod}A$, defined in [18], which is a full subcategory of $\mathrm{ind}A$ with object class

Dually, we define subcategories closed under successors. An example of such categories is the right part $\mathcal{R}_{A}$ of $\mathrm{ind}A$

Let $\mathcal{X}$ be a subcategory of an additive category $\mathcal{A}$. For an object $M$ in $\mathcal{A}$, a right $\mathcal{X}$-approximation of $M$ is a morphism $f:X\longrightarrow M$ with $X\in\mathcal{X}$ such that any morphism $f^{\prime}:X^{\prime}\longrightarrow M$ with $X^{\prime}\in\mathcal{X}$ factors through $f$. If every object in $\mathcal{A}$ has a right $\mathcal{X}$-approximation, we call $\mathcal{X}$ contravariantly finite in $\mathcal{A}$. The notions of a left $\mathcal{X}$-approximation and a covariantly finite subcategory are defined dually. We say $\mathcal{X}$ is functorially finite if it is both contravariantly finite and covariantly finite.

A torsion pair fulfilling the equivalent properties in the above proposition is a functorially finite torsion pair.

Let $\mathcal{T}$ be a triangulated category with shift functor $[1]$. For a full subcategory $\mathcal{V}$ of $\mathcal{T}$ and a subset $I\subseteq\mathbb{Z}$ (which is usually expressed by symbols such as $\geq n,\leq n$, or just $n$), we set

For instance,

A full subcategory $\mathcal{V}$ of $\mathcal{T}$, closed under direct summands, is said to be suspended if it is closed under positive shifts and extensions, that is, if $X\in\mathcal{V}$, then $X[i]\in\mathcal{V}$ for all integers $i>0$, and if $X\rightarrow Y\rightarrow Z\rightarrow X[1]$ is a triangle in $\mathcal{T}$ with $X,Z\in\mathcal{V}$ then $Y\in\mathcal{V}$. Dually, one can define cosuspended subcategories.

A suspended subcategory $\mathcal{V}$ of $\mathcal{T}$ is called an aisle if the inclusion functor $\mathcal{V}\hookrightarrow\mathcal{T}$ has a right adjoint $u:\mathcal{T}\rightarrow\mathcal{V}$. Similarly, a cosuspended subcategory $\mathcal{W}$ of $\mathcal{T}$ is called a coaisle if the inclusion functor $\mathcal{W}\hookrightarrow\mathcal{T}$ has a left adjoint $v:\mathcal{T}\rightarrow\mathcal{W}$. It is shown in [20] that the following conditions are equivalent for a suspended subcategory $\mathcal{V}$ of $\mathcal{T}$:

$(i)$ $\mathcal{V}$ is an aisle.

$(ii)$ $(\mathcal{V},\mathcal{V}^{\bot_{0}})$ is a t-structure.

$(iii)$ For any $X$ in $\mathcal{T}$, there is a triangle $V\rightarrow X\rightarrow W\rightarrow V[1]$ with $V\in\mathcal{V}$ and $W\in\mathcal{V}^{\bot_{0}}$.

$(iv)$ $\mathcal{V}$ is contravariantly finite in $\mathcal{T}$.

Recall that the heart of the t-structure $(\mathcal{V},\mathcal{W})$ is the subcategory $\mathcal{H}=\mathcal{V}\cap\mathcal{W}[1]$. It is an abelian category by [10]. We denote by $H^{0}:\mathcal{T}\rightarrow\mathcal{H}$ the associated cohomological functor given by $H^{0}(X)=u(v(X)[1])$.

The following Lemma will be useful later.

Silting complexes were introduced in [20] to study t-structures in the derived category of a hereditary algebra.

It is shown in [5, Proposition 4.2] that a bounded complex of finitely generated projective $A$-modules $T$ in $K^{b}(\mathrm{proj}A)$ is a silting object if and only if it satisfies the following conditions:

$(S1)$ $\mathrm{Hom}_{D(A)}(T,T[i])=0$ for all $i>0$;

$(S2)$ $T$ is a generator of $D(A)$, i.e. $T^{\bot_{\mathbb{Z}}}=0$.

We call $T$ a bounded silting complex.

An $n$-term silting complex is a bounded silting complex with $n$ non-zero terms, which we always assume to be concentrated in degrees $0,\dots,-n+1$. If $T$ is a $2$-term silting complex in $K^{b}(\mathrm{proj}A)$, then its cohomology in degree zero $H^{0}(T)$ is called a silting module [5]. Over a finite dimensional algebra, the finite dimensional silting modules are precisely the support $\tau$-tilting modules from [1]. Every silting module generates a torsion class, called silting class.

In particular, if $A$ is a hereditary ring, then $\lambda:A\rightarrow B$ is a homological ring epimorphism if and only if it is pseudoflat, which is equivalent to being a universal localization of $A$ by [23, Theorem 6.1]. This shows that universal localization provides a powerful tool to construct homological ring epimorphisms for hereditary rings.

Moreover, every pseudoflat ring epimorphism $\lambda:A\rightarrow B$ starting in a hereditary ring $A$ induces a silting module $T=B\oplus\mathrm{Coker}\lambda$, see for example [6].

The partial order on bireflective subcategories given by inclusion corresponds, under the bijection in Theorem 2.7, to a partial order on the epiclasses of $A$ defined by setting $\lambda_{1}\leq\lambda_{2}$ whenever $\lambda_{1}:A\longrightarrow B_{1}$ factors through $\lambda_{2}:A\longrightarrow B_{2}$ via a ring homomorphism $\mu:B_{2}\longrightarrow B_{1}$, that is, $\lambda_{1}=\mu\circ\lambda_{2}$.

It is shown in [4, Section 5] that every chain

of homological ring epimorphisms $\lambda_{n}:A\longrightarrow B_{n}$ induces a t-structure in $D(A)$. More precisely, consider the corresponding bireflective subcategories $\mathcal{X}_{n}$ of Mod$A$, which are all extension closed, together with the silting classes

induced by the silting $A$-modules $T_{n}=B_{n}\bigoplus\mathrm{Coker}(\lambda_{n})$. We set

for all $n\in\mathbb{Z}$. Then there is a t-structure $(\mathcal{V},\mathcal{W})$ in $D(A)$ with aisle

The following proposition shows that the t-structure is induced by a silting complex under suitable hypotheses.

Every bounded silting complex over a hereditary algebra $A$ has this form, as explained in the following theorem.

Let now $T$ be an $n$-term silting complex in $K^{b}(\mathrm{proj}A)$. According to Theorem 3.2, $T$ arises from a chain of finite dimensional homological ring epimorphisms

with $\lambda_{i}:A\longrightarrow B_{i}$ for integers $0\leq i\leq n-2$. Consider the corresponding chains of extension-closed bireflective subcategories

and of silting classes

We construct a chain

with

and

These classes induce a t-structure $(\mathcal{V},\mathcal{W})$ in $D(A)$, where the aisle

consists of the complexes $X$ with cohomologies concentrated in degrees $\leq 0$ satisfying $H^{-i}(X)\in\mathrm{Gen}B_{i}\cap\mathrm{Mod}B_{i+1}\ \text{for}\ i=0,1,\dots,n-3$ and $H^{-(n-2)}(X)\in\mathrm{Gen}B_{n-2}$.

Now we compute the coaisle $\mathcal{W}$. Since $A$ is hereditary, we know from [4, Section 3.3] that $\mathcal{W}$ is determined by its cohomologies, that is,

where $\mathcal{W}_{j}=H^{j}(\mathcal{W})$ satisfies $\mathcal{W}_{j}={\mathcal{V}_{-j}}^{\bot_{0}}\cap{\mathcal{V}_{-(j+1)}}^{\bot_{1}}$.

We compute the heart of the t-structure $(\mathcal{V},\mathcal{W})$. Since $\mathcal{W}[1]$ consists of the complexes $X$ with $H^{j}(X)\in\mathcal{W}_{j+1}$ for all $j\in\mathbb{Z}$, we have

Hence a complex $X$ lies in $\mathcal{H}$ if and only if

The objects in $\mathcal{H}$, being directs sums of their cohomologies, decompose as direct sums of stalk complexes in $\mathcal{V}_{0}[0],(\mathcal{V}_{j}\cap\mathcal{W}_{-(j-1)})[j]$ for $j=1,2,\dots,n-2$ and $\mathcal{W}_{-(n-2)}[n-1]$. We will denote these classes by $\mathcal{H}_{j}$, $0\leq j<n$. We have the following result.

We introduce the definition of an $n$-section in $\mathrm{ind}A$.

In particular, for $n=2$ we obtain a split torsion pair, and for $n=3$ a trisection in the sense of [2, 18].

Motivated by [25, p.120], we say that an $n$-section $(\mathcal{A}_{1},\cdots,\mathcal{A}_{n})$ is separated if, for any three adjacent subcategories $(\mathcal{A}_{i-1},\mathcal{A}_{i},\mathcal{A}_{i+1})\ (2\leq i\leq n-1)$ of $\mathrm{ind}A$, any morphism $A_{i-1}\longrightarrow A_{i+1}$ with $A_{i-1}\in\mathcal{A}_{i-1}$ and $A_{i+1}\in\mathcal{A}_{i+1}$ factors through $\mathrm{add}\mathcal{A}_{i}$. Moreover, an $n$-section $(\mathcal{A}_{1},\cdots,\mathcal{A}_{n})$ is called functorially finite if every $\mathrm{add}\mathcal{A}_{i}$ is a functorially finite subcategory of $\mathrm{mod}A$.

Let $A$ be a finite dimensional hereditary algebra, and let $T$ be an $n$-term silting complex in $K^{b}(\mathrm{proj}A)$ with endomorphism ring $B=\mathrm{End}_{D(A)}T$. Following [21, 24], the heart of the t-structure $(\mathcal{V},\mathcal{W})$ induced by $T$ is equivalent to the module category $\mathrm{Mod}B$ via the functor $F=\mathrm{Hom}_{D(A)}(T,-)\arrowvert_{\mathcal{H}}:\mathcal{H}\longrightarrow\mathrm{Mod}B$. Moreover, the t-structure $(\mathcal{V},\mathcal{W})$ restricts to a bounded t-structure $(\mathcal{V}^{\prime},\mathcal{W}^{\prime})$ in $D^{b}(\mathrm{mod}A)$ and $F$ restricts to an equivalence between the heart $\mathcal{H}^{\prime}=\mathcal{H}\cap D^{b}(\mathrm{mod}A)$ and $\mathrm{mod}B$. Put

Then $\mathcal{H}^{\prime}$ is a Krull-Schmidt $k$-category, even a length category, and