January 01, 1970
For a tilting complex over a finite-dimensional symmetric algebra over an algebraically closed field, whose distinct graded components have no indecomposable projective direct summands in common, we explicitly (re)construct a two-sided tilting complex that corresponds to the tilting complex, starting from a bimodule inducing a stable equivalence of Morita type.
Let \(k\) be an algebraically closed field and \(A\) and \(B\) finite dimensional symmetric \(k\)-algebras. One of the central topics in the representation theory of finite dimensional algebras is to determine whether two algebras \(A\) and \(B\) are derived equivalent or not. In [1], Rickard showed that the algebras \(A\) and \(B\) are derived equivalent if and only if there exists a tilting complex \(T\) of \(A\)-modules whose endomorphism algebra is isomorphic to the algebra \(B\).
On the other hand, we say that a complex \(C\) of \((B,A)\)-bimodules is a two-sided tilting complex if the functor \({-} \otimes_{B}^{\mathbb{L}} C: \mathop{D^b}{(B)}\to\mathop{D^b}{(A)}\) induces a triangulated equivalence. In [2], [3], Rickard and Keller showed that for the tilting complex \(T\), there theoretically exists a two-sided tilting complex \(C\) of \((B, A)\)-bimodules whose restriction to \(A\) is isomorphic to the tilting complex \(T\) in the derived category of \(A\)-modules. Such equivalences sometimes make it easier to track the correspondences of complexes in derived categories. This can be one of the motivations to construct two-sided tilting complexes from one-sided tilting complexes.
Stable equivalences of Morita type were introduced by Broué in [4]. They are defined by the existence of certain bimodules whose tensor functors induce equivalences between stable categories. Rickard showed in [2] that derived equivalences induce stable equivalences of Morita type by taking minimal projective resolutions of two-sided tilting complexes. Consequently, for symmetric algebras, stable equivalences of Morita type are weaker than derived equivalences.
A natural problem is how to explicitly construct a two-sided tilting complex realizing a given stable equivalence of Morita type for concrete symmetric algebras. Although one-sided tilting complexes are often constructed explicitly in practice, analogous explicit constructions for two-sided tilting complexes are sometimes much less understood. Developing such constructions enables us to give a concrete description of the induced derived equivalences and to investigate derived equivalences between algebras.
There have been some studies on constructing two-term two-sided tilting complexes from bimodules inducing stable equivalences of Morita type by taking projective covers of bimodules, although their primary goal is not necessarily to obtain explicit two-sided tilting complexes. For example, Rouquier constructed a two-term two-sided tilting complex for blocks of group algebras with cyclic defect groups in order to give a character correspondence in [5].
Moreover, in [6], Okuyama showed that tensoring successively with the constructed two-sided tilting complexes corresponds to the derived equivalences obtained by Okuyama’s method described in [7].
There are studies aimed at constructing explicit two-sided tilting complexes for specific classes of algebras, such as Brauer tree algebras [8], [9], and generalized Brauer tree algebras [10], which are determined by combinatorial graph structures. These studies respectively start from specific (not necessarily two-term) tilting complexes constructed by Rickard for Brauer tree algebras [11], by Rickard–Schaps for Brauer tree algebras [12], and by Membrillo-Hernández for generalized Brauer tree algebras [13], together with bimodules inducing stable equivalences of Morita type.
Motivated by these results, we aim to give an explicit description of two-sided tilting complexes corresponding to given tilting complexes under suitable assumptions. Our approach is to construct such complexes by adding appropriate projective bimodules to each graded component of a stalk complex inducing a stable equivalence of Morita type.
Then, in this paper, we start with two algebras \(A\) and \(B\) that are derived equivalent via a tilting complex \(T\) of \(A\)-modules. Rather than asking whether the two algebras are derived equivalent, we investigate how this derived equivalence can be realized by giving an explicit construction of a two-sided tilting complex whose unique graded component induces a stable equivalence of Morita type. We show that if the tilting complex \(T\) has no indecomposable projective summands in common among its different degrees, then we can construct a two-sided tilting complex by deleting certain direct summands from a minimal projective resolution of a bimodule inducing a specific stable equivalence of Morita type (see 4). It is required that the images of simple modules under tensoring with the bimodule coincide with those images from the derived equivalence induced by the tilting complex \(T\). Since each graded component of the two-sided tilting complex is projective on both sides, the tensor functor directly induces a derived equivalence. We remark that any two-term tilting complex, Rickard tree-to-star tilting complex, Rickard–Schaps tree-to-star tilting complex, Membrillo-Hernández tree-to-star tilting complex has no indecomposable projective summands in common among its different degrees. We also prove that the restriction of the constructed two-sided tilting complex to \(A\) is isomorphic to the tilting complex \(T\) with the \(A\)-action twisted by an automorphism of the algebra \(A\) (see 2).
Throughout this paper, \(\Gamma\) and \(\Lambda\) mean finite dimensional indecomposable symmetric algebras over an algebraically closed field \(k\). Let \(\Gamma^{^\mathrm{op}}\) denote the opposite algebra of \(\Gamma\). All \(\Gamma\)-modules are finitely generated right \(\Gamma\)-modules unless otherwise stated. We identify \((\Lambda,\Gamma)\)-bimodules with \(\Lambda^{^\mathrm{op}}\otimes_k \Gamma\)-modules, which restricts to \(\Lambda^{^\mathrm{op}}\)-modules or \(\Gamma\)-modules. Given a \(\Gamma\)-module \(U\), we denote by \(P(U)\) a projective cover of \(U\), by \(\if\relax\detokenize{}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle }} \fi U\) the kernel of a projective cover of \(U\), and by \(\if\relax\detokenize{-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -1}} \fi U\) the cokernel of an injective hull of \(U\). We define inductively \(\if\relax\detokenize{n+1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle n+1}} \fi U= \if\relax\detokenize{}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle }} \fi ( \if\relax\detokenize{n}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle n}} \fi U)\) and \(\if\relax\detokenize{-n-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -n-1}} \fi U= \if\relax\detokenize{-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -1}} \fi ( \if\relax\detokenize{-n}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -n}} \fi U)\) for each integer \(n\ge 1\) and a \(\Gamma\)-module \(U\). We denote by \(U^{\ast}:=\mathop{\mathrm{Hom}}\nolimits_{\Gamma}(U,\Gamma)\) the dual module of a \(\Gamma\)-module \(U\). This is a \(\Gamma^{^\mathrm{op}}\)-module defined by \((af)(u)=f(ua)\) for all \(a\in \Gamma^{^\mathrm{op}}\), \(f\in U^{\ast}\) and \(u\in U\). We note that \(V^{\ast}\otimes_k U\) is a \((\Lambda,\Gamma)\)-bimodule for a \(\Gamma\)-module \(U\) and a \(\Lambda\)-module \(V\). For automorphisms \(\alpha, \beta\) of the algebra \(\Gamma\), we denote \({}_{\alpha}\Gamma_{\beta}\) by \(\Gamma^{^\mathrm{op}}\otimes_k \Gamma\)-module defined by \(x(a\otimes_k b)=\alpha(a)x\beta(b)\) for \(x\in \Gamma\) and \(a\otimes_k b \in \Gamma^{^\mathrm{op}}\otimes_k\Gamma\). We note that \({({}_{\alpha}\Gamma_{\beta})}^{\ast}\) is isomorphic to \({}_{\beta}\Gamma_{\alpha}\). Let \(\{C^n\}_{n\in\mathbb{Z}}\) be a family of \(\Gamma\)-modules indexed by \(\mathbb{Z}\). Let \(d^n\) be a \(\Gamma\)-homomorphism from \(C^n\) to \(C^{n+1}\), which satisfies \(d^{n+1}\circ d^n=0\) for each integer \(n\). Then \(C=(C^n, d^n)_{n\in \mathbb{Z}}\) is a complex of \(\Gamma\)-modules. We define \(C^{\ast}\) to be \(({(C^{-n})}^{\ast}, \mathop{\mathrm{Hom}}\nolimits_{\Gamma}(d^{-n-1},\Gamma):{(C^{-n})}^{\ast}\to {(C^{-n-1})}^{\ast})_{n\in \mathbb{Z}}\). The dual complex of \(C\) is a complex of \(\Gamma^{^\mathrm{op}}\)-modules. We denote by \(C[n]\) the \(n\)-shifted complex of \(C\) and by \(H^n(C)\) the \(n\)th cohomology of \(C\) for each \(n\).
We denote the category of finitely generated right \(\Gamma\)-modules by \(\mathop{\mathrm{mod}}\nolimits\Gamma\), the full subcategory of \(\mathop{\mathrm{mod}}\nolimits\Gamma\) whose objects are finitely generated right projective \(\Gamma\)-modules by \(\mathop{\mathrm{proj}}\Gamma\), the bounded homotopy category of \(\mathop{\mathrm{proj}}\Gamma\) by \(\mathop{K^b}(\mathop{\mathrm{proj}}\Gamma)\), the bounded homotopy category of \(\mathop{\mathrm{mod}}\nolimits\Gamma\) by \(\mathop{K^b}(\mathop{\mathrm{mod}}\nolimits\Gamma)\), the bounded derived category of \(\mathop{\mathrm{mod}}\nolimits\Gamma\) by \(\mathop{D^b}(\Gamma)\), and the stable module category of \(\Gamma\) by \(\mathop{\underline{\mathrm{mod}}}\nolimits{\Gamma}\). We denote by \(\mathop{\mathrm{add}}(T)\) the smallest full subcategory of \(\mathop{K^b}(\mathop{\mathrm{proj}}\Gamma)\) containing a complex \(T\) closed under isomorphisms, direct sums and direct summands.
Given a set \(X\), we denote by \(\delta_{ij}\) the Kronecker delta for \(i, j \in X\) given by \[\delta_{ij}=\begin{cases} 1 & \text{if i=j,} \\ 0 & \text{if i\neq j}. \end{cases}\]
Tilting complexes are important to consider derived equivalences. We describe them in this subsection. Let \(k\) be an algebraically closed field and \(\Gamma\) and \(\Lambda\) two finite dimensional indecomposable symmetric \(k\)-algebras. We say that \(\Gamma\) and \(\Lambda\) are derived equivalent if \(\mathop{D^b}(\Gamma)\) and \(\mathop{D^b}(\Lambda)\) are equivalent as triangulated categories.
Definition 1. We call a bounded complex \(T\) of projective \(\Gamma\)-modules a tilting complex* if the following conditions are satisfied:*
\(\mathop{\mathrm{Hom}}\nolimits_{\mathop{K^b}(\mathop{\mathrm{proj}}\Gamma)}(T, T[n])=0\) for any non-zero integer \(n\).
The subcategory \(\mathop{\mathrm{add}}(T)\) generates \(\mathop{K^b}(\mathop{\mathrm{proj}}\Gamma)\) as a triangulated category.
The second condition means that \(\Gamma\) is obtained by applying a finite sequence of operations for \(T\), including taking direct sums, direct summands, mapping cones, and shifts. We recall the definition of two-sided tilting complexes.
Definition 2 ([2]). We call a bounded complex \(C\) of \(\Lambda^{^\mathrm{op}}\otimes_k \Gamma\)-modules a two-sided tilting complex* if \({-} \otimes_\Lambda^{\mathbb{L}} C\) is a triangulated functor \(\mathop{D^b}(\Lambda)\to \mathop{D^b}(\Gamma)\) which induces an equivalence.*
We note that \({-} \otimes_\Lambda^{\mathbb{L}} C\) is the left derived functor for \({-} \otimes_\Lambda C\). This definition of \(C\) is equivalent to saying that \(C^{\ast}\otimes_{\Lambda}^{\mathbb{L}}C\) is isomorphic to \(\Gamma\) in \(\mathop{D^b}{(\Gamma^{^\mathrm{op}}\otimes_k \Gamma)}\) and \(C\otimes_{\Gamma}^{\mathbb{L}}C^{\ast}\) is isomorphic to \(\Lambda\) in \(\mathop{D^b}{(\Lambda^{^\mathrm{op}}\otimes_k \Lambda)}\). The following proposition holds.
Proposition 3 ([14]). Let \(C\) be a two-sided tilting complex of \(\Lambda^{^\mathrm{op}}\otimes_k \Gamma\)-modules which are projective when seen as \(\Lambda^{^\mathrm{op}}\)-modules and \(\Gamma\)-modules. Then \({-} \otimes_\Lambda^{\mathbb{L}} C\) is equivalent to \({-} \otimes_\Lambda C\) as a functor.
The following proposition establishes a relationship between tilting complexes, two-sided tilting complexes, and derived equivalences.
Proposition 4 ([1], [2], [3]). The following conditions are equivalent:
The \(k\)-algebra \(\Gamma\) is derived equivalent to \(\Lambda\).
There exists a tilting complex \(T\) of \(\Gamma\)-modules such that \(\mathop{\mathrm{End}}\nolimits_{\mathop{K^b}(\mathop{\mathrm{proj}}{\Gamma})}(T)\) is isomorphic to \(\Lambda\) as a \(k\)-algebra.
There exists a two-sided tilting complex \(C\) of \(\Lambda^{^\mathrm{op}}\otimes_k \Gamma\)-modules.
Let \(T\) be a tilting complex of \(\Gamma\)-modules and set \(\Lambda= \mathop{\mathrm{End}}\nolimits_{\mathop{K^b}(\mathop{\mathrm{proj}}{\Gamma})}(T)\). Then an equivalence from \(\mathop{D^b}{(\Gamma)}\) to \(\mathop{D^b}{(\Lambda)}\) is obtained by taking a \(T\)-resolution of an object in \(\mathop{D^b}{(\Gamma)}\) and then applying the functor \(\mathop{\mathrm{Hom}}\nolimits_{\mathop{K^b}(\mathop{\mathrm{proj}}{\Gamma})}(T, -)\). We say that this equivalence is induced by the tilting complex \(T\).
Proposition 5 ([2]). Let \(T\) be a tilting complex of \(\Gamma\)-modules and \(F:\mathop{D^b}{(\Gamma)}\to\mathop{D^b}{(\Lambda)}\) a derived equivalence induced by \(T\). Then, there exists a two-sided tilting complex \(C\) which restricts to \(T\) in \(\mathop{D^b}(\Gamma)\) and to \(F(\Gamma)^{\ast}\) in \(\mathop{D^b}(\Lambda^{^\mathrm{op}})\).
We say that a two-sided tilting complex \(C\) corresponds to \(T\) if it satisfies the condition in 5.
Proposition 6 ([7]). Let \(T\) be a tilting complex of \(\Gamma\)-modules. Let \(F:\mathop{D^b}{(\Gamma)}\to\mathop{D^b}{(\Lambda)}\) be a derived equivalence induced by \(T\). We fix \(X\in \mathop{D^b}{(\Gamma)}\) and \(\ell_0 \in \mathbb{Z}\). If \(\mathop{\mathrm{Hom}}\nolimits_{\mathop{K^b}{(\mathop{\mathrm{mod}}\nolimits\Gamma)}}(T, X[\ell])=0\) for all \(\ell \in \mathbb{Z}\) with \(\ell \neq \ell_0\), then \[H^{\ell}(F(X))\cong \begin{cases} \mathop{\mathrm{Hom}}\nolimits_{\mathop{K^b}{(\mathop{\mathrm{mod}}\nolimits\Gamma)}}(T, X[\ell]) & (\text{for \ell = \ell_0}), \\ 0 & (\text{for all \ell \neq \ell_0}), \end{cases}\] as \(\Lambda\)-modules.
For \(X, Y \in \mathop{D^b}{(\Gamma)}\), we write \(\mathbb{R}\mathop{\mathrm{Hom}}\nolimits_{\Gamma}(X,Y)\) for the derived Hom complex.
Proposition 7 ([15], [16]). Let \(X, Y\in \mathop{D^b}{(\Gamma)}\). Then for any integer \(\ell\), we have \[H^{\ell}(\mathbb{R}\mathop{\mathrm{Hom}}\nolimits_{\Gamma} (X,Y)) \cong \mathop{\mathrm{Hom}}\nolimits_{\mathop{K^b}{(\mathop{\mathrm{mod}}\nolimits\Gamma)}}(X, Y[\ell]).\]
Throughout this subsection, let \(T=(T^{\ell}, d^{\ell})_{\ell \in \mathbb{Z}}=\bigoplus_{i=1}^n T_i\) be a basic tilting complex of \(\Gamma\)-modules, where each complex \(T_i=(T_i^{\ell}, d_i^{\ell})_{\ell\in \mathbb{Z}}\) is indecomposable and not homotopy equivalent to \(0\). Thus, \(d_i^{\ell}\) is a radical homomorphism. We denote a triangulated equivalence from \(\mathop{D^b}(\Gamma)\) to \(\mathop{D^b}(\Lambda)\) induced by the tilting complex \(T\) by \(F\). We denote the pair-wise non-isomorphic simple \(\Gamma\)-modules by \(S_1, \dots, S_n\).
Proposition 8. The following are equivalent for each \(i \in \{1, \dots, n\}\).
\(P(S_i)\) appears as a direct summand of a unique graded component of \(T\).
The complex \(F(S_i)\) is isomorphic to a shift of a \(\Lambda\)-module.
Moreover, if \(P(S_i)\) is a summand of the unique graded component \(T^{n_i}\), then \(F(S_i)\) is isomorphic to \(\mathop{\mathrm{Hom}}\nolimits_{K^b(\mathop{\mathrm{mod}}\nolimits\Gamma)}(T,S_i[-n_i])[n_i]\) and vice versa.
Proof. ([eachtrivcap] \(\Rightarrow\) [eachmodshiftB]). We have \(\mathop{\mathrm{Hom}}\nolimits_\Gamma(T^{\ell}, S_i)=0\) for all \(\ell\neq n_i\) by the assumption. Thus we have \(\mathop{\mathrm{Hom}}\nolimits_{\mathop{K^b}(\mathop{\mathrm{mod}}\nolimits\Gamma)}(T, S_i[-\ell])=0\) for all \(\ell\neq n_i\). By 6, \(F(S_i)\cong W_i[n_i]\), where we set a \(\Lambda\)-module \(W_i=\mathop{\mathrm{Hom}}\nolimits_{\mathop{K^b}(\mathop{\mathrm{mod}}\nolimits\Gamma)}(T, S_i[-n_i])\).
([eachmodshiftB] \(\Rightarrow\) [eachtrivcap]). Since \(F(S_i)\cong W_i[n_i]\) by the assumption, we have \[H^{\ell}(F(S_i)) \cong \begin{cases} W_i & (\text{for \ell=-n_i}), \\ 0 & (\text{for \ell\neq -n_i}). \end{cases}\] By 7, \[H^{\ell}(F(S_i))\cong H^{\ell}(\mathbb{R}\mathop{\mathrm{Hom}}\nolimits_{\Gamma} (T,S_i)) \cong \mathop{\mathrm{Hom}}\nolimits_{\mathop{K^b}{(\mathop{\mathrm{mod}}\nolimits\Gamma)}}(T, S_i[\ell]).\] Since \(S_i\) is a simple module, we have \[\mathop{\mathrm{Hom}}\nolimits_{\mathop{K^b}{(\mathop{\mathrm{mod}}\nolimits\Gamma)}}(T, S_i[\ell]) \cong \mathop{\mathrm{Hom}}\nolimits_{\Gamma}(T^{-\ell}, S_i).\] Thus \[\mathop{\mathrm{Hom}}\nolimits_\Gamma(T^{\ell}, S_i)\cong \begin{cases} W_i & (\text{for \ell = n_i}), \\ 0 & (\text{for \ell \neq n_i}). \end{cases}\] Therefore, \(P(S_i)\) is a summand of \(T^{n_i}\) but not a summand of \(T^{\ell}\) for \(\ell \neq n_i\). ◻
Proposition 9. The following are equivalent:
\(\mathop{\mathrm{add}}{T^{\ell}}\cap \mathop{\mathrm{add}}{T^{\ell'}} = 0\) if \(\ell \neq \ell'\).
For each \(i\), there exist an indecomposable \(\Lambda\)-module \(W_i\) and an integer \(n_i\) such that \(F(S_i) = W_i[n_i]\).
Proof. By 8, condition [modshiftB] is equivalent to the following: for each \(i\), there exists an integer \(n_i\) such that \[P(S_i) \in \mathop{\mathrm{add}}{T^{\ell}} \quad \text{if and only if} \quad \ell = n_i.\] Thus, if \(P(S_i)\in \mathop{\mathrm{add}}{T^{\ell}}\cap \mathop{\mathrm{add}}{T^{\ell'}}\), then \(\ell =\ell' = n_i\). By using this fact, we can easily check that the two conditions are equivalent. ◻
In this subsection, we recall the basic results on stable equivalences for symmetric algebras, which are weaker equivalences than derived equivalences. Let \(\Gamma\) and \(\Lambda\) be two finite dimensional indecomposable symmetric \(k\)-algebras. For \(\Gamma\)-modules \(U\) and \(V\), we denote the \(k\)-linear space of all homomorphisms from \(U\) to \(V\) which factor through projective modules by \(\mathop{\mathrm{Hom}^{\mathrm{pr}}}\nolimits(U,V)\). The stable category of \(\Gamma\)-modules denoted by \(\mathop{\underline{\mathrm{mod}}}\nolimits{\Gamma}\) is defined as follows:
The objects are the same as those of \(\mathop{\mathrm{mod}}\nolimits\Gamma\).
For \(\Gamma\)-modules \(U\) and \(V\), the set of morphisms from \(U\) to \(V\) is \(\mathop{\mathrm{Hom}}\nolimits(U,V)/\mathop{\mathrm{Hom}^{\mathrm{pr}}}\nolimits(U,V)\). We denote this by \(\mathop{\underline{\mathrm{Hom}}}\nolimits(U,V)\).
In addition, the category \(\mathop{\underline{\mathrm{mod}}}\nolimits{\Gamma}\) is a triangulated category with the shift functor \(\if\relax\detokenize{-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -1}} \fi\).
Definition 10 ([4]). We say that \(\Gamma\) and \(\Lambda\) are stably equivalent of Morita type* if there exist a \((\Lambda, \Gamma)\)-bimodule \(M\) and a \((\Gamma, \Lambda)\)-bimodule \(N\) satisfying the following conditions.*
The bimodules \(M\) and \(N\) are projective as left modules and right modules.
\(N\otimes_\Lambda M\cong \Gamma\oplus P\) as \((\Gamma,\Gamma)\)-bimodules for some projective \((\Gamma,\Gamma)\)-bimodule \(P\).
\(M\otimes_\Gamma N\cong \Lambda\oplus Q\) as \((\Lambda,\Lambda)\)-bimodules for some projective \((\Lambda,\Lambda)\)-bimodule \(Q\).
Then, we say that \(M\) induces a stable equivalence of Morita type.
We remark that the above \((\Lambda, \Gamma)\)-bimodule \(M\) induces a functor \({-}\otimes_\Lambda M: \mathop{\mathrm{mod}}\nolimits{\Lambda} \to \mathop{\mathrm{mod}}\nolimits{\Gamma}\), which induces a stable equivalence \(\mathop{\underline{\mathrm{mod}}}\nolimits{\Lambda} \to \mathop{\underline{\mathrm{mod}}}\nolimits{\Gamma}\).
Let \(\mathcal{S}'\) denote a complete set of representatives of isomorphism classes of simple \(\Lambda\)-modules.
Proposition 11 ([5]). Let \(M\) be a \(\Lambda^{^\mathrm{op}}\otimes_k \Gamma\)-module, which is projective as a \(\Gamma\)-module and as a \(\Lambda^{^\mathrm{op}}\)-module. A projective cover of \(M\) is isomorphic to \[\bigoplus_{V\in \mathcal{S}'} P(V)^{\ast} \otimes_k P(V\otimes_\Lambda M).\]
Let \(M\) be an indecomposable \(\Lambda^{^\mathrm{op}}\otimes_k \Gamma\)-module inducing a stable equivalence of Morita type between \(\Lambda\) and \(\Gamma\). Let \(P=(P^t, d_M^t)_{t\in \mathbb{Z}}\) be a projective resolution of the \(\Lambda^{^\mathrm{op}}\otimes_k \Gamma\)-module \(M\). Since \(\if\relax\detokenize{\ell}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle \ell}} \fi M\) gives a stable equivalence of Morita type for \(\ell \ge 1\) (see [17] and [8]), the definition of a minimal projective resolution and 11 give the following proposition.
Proposition 12. The \((\Lambda,\Gamma)\)-bimodule \(P^{-t}\) is isomorphic to \(M\) for \(t=0\), to \(0\) for all \(t<0\), and to \[\bigoplus_{V\in \mathcal{S}'} P(V)^{\ast}\otimes_k P(V\otimes_{\Lambda} \if\relax\detokenize{t-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle t-1}} \fi M)\] for all \(t>0\).
The following proposition is useful to consider a minimal projective resolution of the module \(M\).
Proposition 13 ([8]). We have \(V \otimes_\Lambda \if\relax\detokenize{\ell}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle \ell}} \fi M \cong \if\relax\detokenize{\ell}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle \ell}} \fi (V \otimes_\Lambda M )\) for any simple \(\Lambda\)-module \(V\) and \(\ell\ge 0\).
The following proposition is proved similarly to those propositions in [18] and [8]. This is important to show that some complexes are two-sided tilting complexes.
Proposition 14 (see [18] and [8]). Let \(C=(C^{\ell}, d_C^{\ell})_{\ell \in \mathbb{Z}}\) be a bounded complex of \(\Lambda^{^\mathrm{op}}\otimes_k \Gamma\)-modules such that \(C^t=0\) for all \(t>0\), \(C^{0} = M\) and \(C^t\) is projective for all \(t<0\). If \[\mathop{\mathrm{Hom}}\nolimits_{\mathop{D^b}(\Gamma)}(V\otimes_\Lambda C, W\otimes_\Lambda C[-n]) \cong \,\delta_{VW}\delta_{n0}k\] for \(V,W\in \mathcal{S}'\) and \(n\in \mathbb{Z}_{\ge 0}\), then \(C\) is a two-sided tilting complex.
In this section, we explicitly construct a two-sided tilting complex corresponding to a tilting complex without common projective summands. We take a bimodule inducing the stable equivalence of Morita type induced by the tilting complex. Then we take a minimal projective resolution of the bimodule and take a subcomplex of the resolution. We show that the subcomplex is a two-sided tilting complex.
Let \(T=\bigoplus_{i=1}^n T_i\) be a basic tilting complex for an indecomposable symmetric algebra \(A\), where each \(T_i\) is indecomposable and not homotopy equivalent to \(0\). We put \(B=\mathop{\mathrm{End}}\nolimits_{\mathop{K^b}(\mathop{\mathrm{proj}}{A})}(T)\). We denote a triangulated equivalence from \(\mathop{D^b}(A)\) to \(\mathop{D^b}(B)\) induced by the tilting complex \(T\) by \(F\). We denote a simple module corresponding to an indecomposable projective \(B\)-module \(F(T_i)\) by \(V_i\). We make the following assumption on the derived equivalence induced by the tilting complex.
Assumption 1. We assume that image of each \(S_i\) under the equivalence \(F\) is isomorphic to a positive shift of a \(B\)-module.
For \(i\in \{1,\dots, n\}\), we set \(n_i \ge 0\) and a \(B\)-module \(W_i\) satisfying \[F(S_i)\cong W_i[n_i]\] in \(\mathop{D^b}(B)\). In this situation, we remark that all the negative degrees of tilting complex \(T\) are zero modules by 9. By [2], [7], there exists an \(A^{^\mathrm{op}}\otimes_k B\)-module \(M\) inducing a stable equivalence of Morita type satisfying \[S_i\otimes_A M \cong \if\relax\detokenize{-n_i}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -n_i}} \fi W_i \oplus (\text{projective}).\]
By [19], we may assume that \(M\) is an indecomposable \(A^{^\mathrm{op}}\otimes_k B\)-module and we have an isomorphism \[S_i \otimes_A M \cong \if\relax\detokenize{-n_i}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -n_i}} \fi W_i.\]
Let \(P^{\bullet}(M) = (P^j(M), p^j)_{j \in \mathbb{Z}}\) be a minimal projective resolution of the \(A^{^\mathrm{op}}\otimes_k B\)-module \(M\). That is, \[P^{-j}(M) \cong \begin{cases} M & (\text{for j=0}), \\ P( \if\relax\detokenize{j-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle j-1}} \fi M) & (\text{for 0 < j} ), \\ 0 & (\text{for j < 0}). \end{cases}\] By 12 13, it holds that \(P^{-j}(M)= P( \if\relax\detokenize{j-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle j-1}} \fi M)\) is isomorphic to \[\bigoplus_{i=1}^n P(S_i)^{\ast}\otimes_k P( \if\relax\detokenize{j-1-n_i}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle j-1-n_i}} \fi W_i)\] for \(j>0\). Thus, for a simple \(A\)-module \(S_i\) and for \(j>0\), the \(B\)-module \(S_i\otimes_A P^{-j}(M)\) is isomorphic to \(P( \if\relax\detokenize{j-1-n_i}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle j-1-n_i}} \fi W_i)\) because \[S_i\otimes_A P(S_{i'})^{\ast}\cong \mathop{\mathrm{Hom}}\nolimits_A (P(S_{i'}), S_i)\cong \delta_{i'i}k.\] Since \(P^{-j}(M)\) and \(M\) are projective \(A^{^\mathrm{op}}\)-modules, each short exact sequence of \(A^{^\mathrm{op}}\otimes_k B\)-modules \[\begin{tikzcd}[] 0\rar[]& \if\relax\detokenize{j}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle j}} \fi M\rar[]&P( \if\relax\detokenize{j-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle j-1}} \fi M )\rar[]& \if\relax\detokenize{j-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle j-1}} \fi M\rar[]&0 \end{tikzcd}\] splits as a short exact sequence of \(A^{^\mathrm{op}}\)-modules. Therefore, the exact sequence of \(B\)-modules \[\begin{tikzcd}[] S_i\otimes_A \if\relax\detokenize{j}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle j}} \fi M\rar[]&S_i\otimes_A P( \if\relax\detokenize{j-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle j-1}} \fi M )\rar[]&S_i\otimes_A \if\relax\detokenize{j-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle j-1}} \fi M\rar[]&0 \end{tikzcd}\] is a split short exact sequence of \(k\)-modules. Thus \(S_i\otimes_A P^{\bullet}(M)\) is an exact sequence, and hence a projective resolution of \(S_i \otimes_A M\). Moreover, since each graded component in the complex is minimal, this is in fact a minimal projective resolution of \(S_i \otimes_A M\).
We construct a subcomplex \((C^j, p'^{j})_{j \in \mathbb{Z}}\) of \(P^{\bullet}(M)\) as follows: \[C^{-j} = \begin{cases} M & (\text{for j=0}), \\ \bigoplus_{i\in \{1,\dots, n\}, n_i\ge j} P(S_i)^{\ast}\otimes_k P( \if\relax\detokenize{j-1-n_i}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle j-1-n_i}} \fi W_i) & (\text{for 0 < j } ), \\ 0 & (\text{for j < 0}). \end{cases}\] For convenience of notation, we put \(P^{-j}=P^{-j}(M)\). Let \(\iota^{-j}\colon C^{-j} \hookrightarrow P^{-j}\) be the inclusion and \(\pi^{-j}\colon P^{-j} \twoheadrightarrow C^{-j}\) the projection associated with the direct summand \(C^{-j}\) of \(P^{-j}\). We define a chain complex \(C = (C^{j}, p'^{j})_{j \in \mathbb{Z}}\) by setting \[p'^{-j} = \pi^{-j+1} \circ p^{-j} \circ \iota^{-j}.\] The following lemma holds.
Lemma 1. For each \(i\), we have \(S_i\otimes_A C\cong F(S_i)\) in \(\mathop{D^b}{(B)}.\)
Proof. We note that \[S_i\otimes_A C^{-j} \cong \begin{cases} \if\relax\detokenize{-n_i}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -n_i}} \fi W_i & (\text{for j=0}), \\ P( \if\relax\detokenize{j-1-n_i}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle j-1-n_i}} \fi W_i) & (\text{for 0 < j \le n_i} ), \\ 0 & (\text{for j > n_i and j < 0}). \end{cases}\] and we have a commutative diagram \[\begin{tikzcd}[column sep=huge] S_i\otimes_A P^{-j}\rar["S_i\otimes_A p^{-j}"] &S_i\otimes_A P^{-j+1} \dar["S_i\otimes_A \pi^{-j+1}"]\\ S_i\otimes_A C^{-j}\rar["S_i\otimes_A p'^{-j}"]\uar["S_i\otimes_A \iota^{-j}"] &S_i\otimes_A C^{-j+1} \end{tikzcd}.\] Let \(D^{-j}\) be a complement of \(C^{-j}\) in \(P^{-j}\). The functor \(S_i \otimes_A {-}\) sends a split short exact sequence \[\begin{tikzcd}[sep=large] 0\rar[]&C^{-j}\rar["\iota^{-j}"]&P^{-j}\rar[]&D^{-j}\rar[]&0 \end{tikzcd}\] to a split short exact sequence \[\begin{tikzcd}[sep=large] 0\rar[]&S_i\otimes_A C^{-j}\rar["S_i\otimes_A \iota^{-j}"]&S_i\otimes_A P^{-j}\rar[]&S_i\otimes_A D^{-j}\rar[]&0 \end{tikzcd}\] and similarly sends \[\begin{tikzcd}[sep=large] 0\rar[]&D^{-j+1}\rar[]&P^{-j+1}\rar["\pi^{-j+1}"]&C^{-j+1}\rar[]&0 \end{tikzcd}\] to \[\begin{tikzcd}[sep=large] 0\rar[]& S_{i}\otimes_A D^{-j+1}\rar[]&S_i\otimes_A P^{-j+1}\rar["S_i\otimes_A \pi^{-j+1}"]&S_i\otimes_A C^{-j+1}\rar[]&0. \end{tikzcd}\] By the definition of \(D^{-j}\), we have \[D^{-j} = \bigoplus_{i\in \{1,\dots, n\}, n_i< j} P(S_i)^{\ast}\otimes_k P( \if\relax\detokenize{j-1-n_i}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle j-1-n_i}} \fi W_i)\] for \(0< j\). Therefore, if \(0< j \le n_i\), \[S_i\otimes_A D^{-j} \cong 0 \qquad \text{and} \qquad S_i\otimes_A D^{-j+1}\cong 0.\] Hence, the split morphisms \(S_i\otimes_A \iota^{-j}\) and \(S_i\otimes_A \pi^{-j+1}\) are isomorphisms. By the commutative diagram above, the complex \(S_i\otimes_A C\) is a stupid truncated minimal projective resolution of \(S_i\otimes_A M\) at the \(-n_i\)th degree. Therefore, \[S_i\otimes_A C\cong \if\relax\detokenize{n_i}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle n_i}} \fi (S_i\otimes_A M)[n_i] \cong \if\relax\detokenize{n_i}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle n_i}} \fi ( \if\relax\detokenize{-n_i}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -n_i}} \fi W_i)[n_i] \cong W_i[n_i]\cong F(S_i)\] in \(\mathop{D^b}(B)\). ◻
By 1, for \(\ell\le 0\), \[\begin{align} \mathop{\mathrm{Hom}}\nolimits_{\mathop{D^b}(B)}(S_i\otimes_A C, S_j\otimes_A C[\ell]) & \cong \mathop{\mathrm{Hom}}\nolimits_{\mathop{D^b}(B)}(F(S_i), F(S_j)[\ell]) \\ & \cong \mathop{\mathrm{Hom}}\nolimits_{\mathop{D^b}(A)}(S_i, S_j[\ell])\\ & \cong \delta_{ij}\delta_{\ell0}k. \end{align}\] By 14, we have the following proposition.
Theorem 1. The complex \(C\) of \(A^{^\mathrm{op}}\otimes_k B\)-modules is a two-sided tilting complex.
Let \(G={-}\otimes_A C\). By 1, the functor \(G\) is a triangulated equivalence. Since \((F^{-1}\circ G )(S_i)\cong S_i\) by 1, the functor \(F^{-1}\circ G\) sends the projective module \(P(S_i)\) to itself. In particular, \((F^{-1}\circ G)(A)\cong A\). Thus we have \[A\otimes_A C\cong G(A)\cong F(A)\] in \(\mathop{D^b}{(B)}\). Hence, we have the following proposition.
Proposition 15. The complex \(C\) is isomorphic to the tilting complex \(F(A)\) in \(\mathop{D^b}{(B)}\).
Let \(Y\) be a two-sided tilting complex which restricts to \(T^{\ast}\) as \(A^{^\mathrm{op}}\)-modules and to \(F(A)\) as \(B\)-modules. Such \(Y\) exists by 5. By [20], there exists a \(k\)-algebra automorphism \(\alpha\) of \(A\) such that \({}_{\alpha}A_{1}\!\otimes_A C\cong Y\) in \(\mathop{D^b}(A^{op})\). By taking dual complexes, we have \(C^{\ast}\otimes_A {}_{1}A_{\alpha} \cong Y^{\ast}\cong T\) in \(\mathop{D^b}(A)\). Therefore, we have the following theorem.
Theorem 2. There exists a \(k\)-algebra automorphism \(\alpha\) of the algebra \(A\) such that the complex \(C^{\ast}\otimes_{A}{}_{1}\!A_{\alpha}\) is isomorphic to the tilting complex \(T\) in \(\mathop{D^b}{(A)}\).
We remark that the assumption \(n_i \ge 0\) for all \(i\in \{1, \dots, n\}\) is not essential. If there are some \(i\) such that \(n_i<0\), then we denote the minimum of \(n_i\) for \(i\in \{1, \dots, n\}\) by \(N\). Then the bimodule \(\if\relax\detokenize{-N}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -N}} \fi M\) induces a stable equivalence of Morita type by [17]. We can do the same argument for the complex of \(A^{^\mathrm{op}}\otimes_k B\)-modules which of the \(N\)th degree is isomorphic to \(\if\relax\detokenize{-N}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -N}} \fi M\).
The following corollary is obtained by combining 2 with 9.
Corollary 1. For a tilting complex \(T=(T^{\ell}, d^{\ell})_{\ell \in \mathbb{Z}}\) satisfying \(\mathop{\mathrm{add}}{T^{\ell}} \cap \mathop{\mathrm{add}}{T^{\ell'}} = 0\) for all \(\ell \neq \ell'\), we can construct a corresponding two-sided tilting complex by deleting some indecomposable summands of a minimal projective resolution of some bimodule inducing a stable equivalence of Morita type.
Remark 16. For Brauer tree algebras, tilting complexes in [11] and tilting complexes in [20] satisfy the conditions in 9.
For generalized Brauer tree algebras, tilting complexes in [13] satisfy the conditions in 9.
For arbitrary finite dimensional algebras, two-term tilting complexes satisfy the conditions in 9 by [21].
We may regard that we generalize the construction method of two-sided tilting complexes in [8] corresponding to Rickard tree-to-star tilting complexes for Brauer tree algebras, [9] corresponding to Rickard–Schaps tree-to-star tilting complexes for Brauer tree algebras and [10] for Membrillo-Hernandez tree-to-star tilting complexes for generalized Brauer tree algebras.
Example 1. Let \(A\) be the Brauer graph algebra associated with the following Brauer graph, with counterclockwise cyclic order around each vertex and all multiplicities equal to \(1\). \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/udtmoahz.png}\label{vqlarwtd}\end{figure}\qquad{(1)}\] Silting mutation, introduced in [22], provides a way to construct new silting objects from given ones in a triangulated category. We apply the right silting mutation to the tilting complex \(A\), first with respect to \(\{1\}\), and then with respect to \(\{1,2,4\}\). As a result, we have the tilting complex \(T=\bigoplus_{i=1}^4 T_i\) as follows: \[\begin{tikzcd}[ampersand replacement=\&, row sep=small, column sep=small ] \phantom{0}\&[-6pt]\phantom{0}\&[-5pt]\phantom{0} \& 0\text{th} \& 1\text{st} \& 2\text{nd}\\[-5pt] \&\&T_1\rar[":",phantom]\&P_3\oplus P_3\rar[]\&P_2\oplus P_4 \rar[]\& P_1\\ \phantom{0}\&\&T_2\rar[":",phantom]\&P_3\rar[]\&P_2\rar[]\& 0\\ \phantom{0}\&\&T_3\rar[":",phantom]\&P_3\rar[]\&0\rar[]\&0\\ \&\&T_4\rar[":",phantom]\&P_3\rar[]\&P_4\rar[]\&0 \end{tikzcd}\] Generalized Kauer move, introduced in [23], provides a way to calculate the endomorphism algebra of a tilting complex obtained by a silting mutation of a regular module over a Brauer graph algebra. The endomorphism algebra \(B=\mathop{\mathrm{End}}\nolimits_{\mathop{K^b}({\mathop{\mathrm{proj}}A})}(T)\) is the Brauer graph algebra associated with the following Brauer graph: \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/dwzbtlop.png}\label{pxhrwcdt}\end{figure}\qquad{(2)}\] Following 8, the images of simple modules \(S_i\) through the derived equivalence \(F\) induced by the tilting complex \(T\) are as follows: \[F(S_1)\cong W_1[2],\quad F(S_2)\cong W_2[1],\quad F(S_3)\cong W_3,\quad F(S_4)\cong W_4[1].\] Here we put indecomposable modules \(W_1,\dots,W_4\) by writing the composition factors from top to socle vertically as follows: \[W_1= V_1,\quad W_2= \,\begin{matrix} V_2 \\V_3 \end{matrix}\,,\quad W_3= \, \begin{matrix} \begin{matrix} V_1 \\V_4 \end{matrix}\quad \begin{matrix} V_1 \\V_2 \end{matrix} \\V_3 \end{matrix}\, ,\quad W_4= \,\begin{matrix} V_4 \\V_3 \end{matrix}\,,\] where \(V_1,\dots,V_4\) denote simple \(B\)-modules. Let \(M\) be an \(A^{^\mathrm{op}}\otimes_k B\)-module and induce a stable equivalence of Morita type satisfying \[S_1\otimes_A M\cong \if\relax\detokenize{-2}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -2}} \fi W_1,\quad S_2\otimes_A M\cong \if\relax\detokenize{-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -1}} \fi W_2,\quad S_3\otimes_A M\cong W_3,\quad S_4\otimes_A M \cong \if\relax\detokenize{-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -1}} \fi W_4.\] Let \(P^{\bullet}(M)\) be a minimal projective resolution of \(M\). \[\begin{tikzcd}[row sep=0cm] \phantom{\cdots\cdots\cdots\cdots} &-2\text{nd}&-1\text{st}& 0\text{th}\\[-0.1cm] \phantom{\cdots} & {P(S_1)^{\ast}\otimes_k P( \if\relax\detokenize{-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -1}} \fi W_1)}& P(S_1)^{\ast}\otimes_k P( \if\relax\detokenize{-2}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -2}} \fi W_1)& \phantom{0}\\[-0.1cm] &|[opacity=1]|\oplus & \oplus &\\[-0.1cm] \phantom{\cdots}& |[opacity=1]|{P(S_2)^{\ast}\otimes_k P( W_2)}& P(S_2)^{\ast}\otimes_k P( \if\relax\detokenize{-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -1}} \fi W_2)\\[-0.1cm] |[opacity=1]|{\cdots}\rar[shorten=30pt,opacity=1, ] &|[opacity=1]|{\oplus}\rar[shorten=35pt] & \oplus\rar[shorten <= 40pt,] &M\\[-0.1cm] \phantom{\cdots}& |[opacity=1]|{P(S_3)^{\ast}\otimes_k P( \if\relax\detokenize{}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle }} \fi W_3)}& |[opacity=1]|{P(S_3)^{\ast}\otimes_k P( W_3)}& \phantom{0}\\[-0.1cm] &|[opacity=1]|{\oplus} & |[opacity=1]|{\oplus} &\\[-0.1cm] \phantom{\cdots}&|[opacity=1]|{P(S_4)^{\ast}\otimes_k P( W_4)}& P(S_4)^{\ast}\otimes_k P( \if\relax\detokenize{-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -1}} \fi W_4)& \phantom{0} \end{tikzcd}\] Following our construction method of our two-sided tilting complexes, we delete some direct summands. Then we have a two-sided tilting complex \(C\) of \(A^{^\mathrm{op}}\otimes_k B\)-modules. \[\begin{tikzcd}[row sep=0cm] \phantom{\cdots\cdots\cdots\cdots} &-2\text{nd}&-1\text{st}& 0\text{th}\\[-0.1cm] \phantom{\cdots} & {P(S_1)^{\ast}\otimes_k P( \if\relax\detokenize{-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -1}} \fi W_1)}& P(S_1)^{\ast}\otimes_k P( \if\relax\detokenize{-2}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -2}} \fi W_1)& \phantom{0}\\[-0.1cm] &|[opacity=0]|\oplus & \oplus &\\[-0.1cm] \phantom{\cdots}& |[opacity=0]|{P(S_2)^{\ast}\otimes_k P( W_2)}& P(S_2)^{\ast}\otimes_k P( \if\relax\detokenize{-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -1}} \fi W_2)\\[-0.1cm] |[opacity=0]|{\cdots}\rar[shorten=30pt,opacity=0, ] &|[opacity=0]|{\oplus}\rar[shorten=35pt] & \oplus\rar[shorten <= 40pt,] &M\\[-0.1cm] \phantom{\cdots}& |[opacity=0]|{P(S_3)^{\ast}\otimes_k P( \if\relax\detokenize{}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle }} \fi W_3)}& |[opacity=0]|{P(S_3)^{\ast}\otimes_k P( W_3)}& \phantom{0}\\[-0.1cm] &|[opacity=0]|{\oplus} & |[opacity=0]|{\oplus} &\\[-0.1cm] \phantom{\cdots}&|[opacity=0]|{P(S_4)^{\ast}\otimes_k P( W_4)}& P(S_4)^{\ast}\otimes_k P( \if\relax\detokenize{-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -1}} \fi W_4)& \phantom{0} \end{tikzcd}\] The dual complex \(C^{\ast}\) is also a two-sided tilting complex of \(B^{^\mathrm{op}}\otimes_k A\)-modules, which restricts to the tilting complex \(T\otimes_{A}{}_{\alpha}\!A_{1}\) of \(A\)-modules for a \(k\)-algebra automorphism \(\alpha\) of \(A\).
\[\begin{tikzcd}[ row sep = 0 ] & 0\text{th} & 1\text{st} & 2\text{nd}\\ & \phantom{0} & P( \if\relax\detokenize{-2}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -2}} \fi W_1)^{\ast}\otimes_k P(S_1) & P( \if\relax\detokenize{-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -1}} \fi W_1)^{\ast}\otimes_k P(S_1) \\ & & \oplus & |[opacity=0]| \oplus \\ & & P( \if\relax\detokenize{-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -1}} \fi W_2)^{*} \otimes_k P(S_2) & |[opacity=0]| {P(W_2)^{*} \otimes_k P(S_2)} \\ & M^{\ast} \arrow[r, shorten >= 50pt] & \oplus \arrow[r, shorten = 35pt] & |[opacity=0]| {\oplus} \\ & & |[opacity=0]| \oplus & |[opacity=0]| \oplus & \phantom{0} \\ & |[opacity=0]| {P(W_3)^{*} \otimes_k P(S_3)} & |[opacity=0]| {P( \if\relax\detokenize{}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle }} \fi W_3)^{*} \otimes_k P(S_3)} \\ & \phantom{0} & P( \if\relax\detokenize{-1}\relax \mathop{\Omega} \else \mathop{\Omega^{\scriptstyle -1}} \fi W_4)^{*} \otimes_k P(S_4) & |[opacity=0]| {P(W_4)^{*} \otimes_k P(S_4)} \end{tikzcd}\]
The authors would like to thank Professor Naoko Kunugi for her advice. This work was supported by JSPS KAKENHI Grant Number JP25K17238.
Shuji Fujino 1124702@ed.tus.ac.jp
Department of Mathematics, Graduate School of Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan
Yuta Kozakai kozakai@rs.tus.ac.jp
Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan