Proof (1) It suffices to prove $^{\perp }({}_{A}T)\cap {\mathcal {P}^{\leqslant n}(A)}\subseteq {\mathrm {cogen}^{*}({}_{A}T)}$ . Let $M\in {^{\perp }({}_{A}T)}\cap {\mathcal {P}^{\leqslant n}(A)}$ . Then there exists an exact sequence in A-mod

$$ \begin{align*}0\to P_{n}\to\cdots\to P_1\to P_0\to M\to 0,\end{align*} $$

where $P_{i}$ is projective for all $0\leqslant i\leqslant n$ . This yields the following exact sequence in $B^{op}\mbox {-mod}$ :

$$ \begin{align*}0\to \mbox{Hom}_{A}(M,T)\to \mbox{Hom}_{A}(P_{0},T)\to \mbox{Hom}_{A}(P_{1},T)\to \cdots \to \mbox{Hom}_{A}(P_{n},T)\to 0.\end{align*} $$

Since $\mbox {Hom}_{A}(P_{i},T)\in {\textrm {add}(T_{B})}\subseteq {^{\perp }(T_{B})}$ for all $0\leqslant i\leqslant n$ and since ${^{\perp }(T_{B})}$ is a resolving subcategory of $B^{op}\mbox {-mod}$ , we have $\mbox {Hom}_{A}(M,T)\in {^{\perp }(T_{B})}$ . It follows that there is a commutative diagram with exact rows in A-mod:

Since $\sigma _{P_{i}}$ is an isomorphism for all $0\leqslant i\leqslant n$ , so is $\sigma _{M}$ . Thus $M\in {\mathrm {cogen}^{*}({}_{A}T)}$ by Lemma 3.1.

(2) Assume that $_{A}T\in {\mathcal {GP}^{<\infty }(A)}$ and $\mathcal {GP}(A)\subseteq {\mathcal {W}{({}_{A}T)}}$ . Since $_{A}T\in {\mathcal {GP}^{<\infty }(A)}$ , we see from [Reference Christensen, Frankild and Holm15, Lemma 2.17] that there is an exact sequence $0\to T\to H\to G\to 0 $ in A-mod, where $H\in {\mathcal {P}^{<\infty }(A)}$ and $G\in {\mathcal {GP}(A)}$ . As $\mathcal {GP}(A)\subseteq {\mathcal {W}{({}_{A}T)}}$ , the exact sequence $0\to T\to H\to G\to 0$ splits. Thus $_{A}T$ is isomorphic to a direct summand of H, which forces $T\in {\mathcal {P}^{<\infty }(A)}$ .

Conversely, assume $_{A}T\in {\mathcal {P}^{<\infty }(A)}$ . Then $\mathcal {GP}(A)\subseteq {^{\perp }({}_{A}T)}$ . Let M be an A-module in $\mathcal {GP}(A)$ . It is enough to prove that M is in $\mathcal {W}{({}_{A}T)}$ . There exists a complete projective resolution over A

$$ \begin{align*}P^{\bullet}:\dots \to P^{-1}\stackrel{d^{-1}}\to P^{0}\stackrel{d^{0}}\to P^{1}\stackrel{d^{1}}\to\dots\end{align*} $$

such that $M\cong \textrm {Im}(d^{-1})$ . Set $K^{i}:=\textrm {Im}(d^{i})$ for all $i\in {\mathbb {Z}}$ . Then $K^{i}\in {\mathcal {GP}(A)}$ . As $\mathcal {GP}(A)\subseteq {^{\perp }({}_{A}T)}$ , the complex $\mbox {Hom}_{A}(P^{\bullet },T)$ is again exact. Hence we have the following commutative diagrams:

and

This implies that both $\sigma _{K^{1}}$ and $\sigma _{M}$ are injective, and therefore $\sigma _{M}$ is an isomorphism by the snake lemma. Similarly, we can show that $\sigma _{K^{i}}:K^{i}\to ((K^{i},T),T)$ is an isomorphism for any $i\in {\mathbb {Z}}$ . Thus $\mbox {Ext}^{i}_{B}(\mbox {Hom}_{A}(M,T),T)=0$ for all $i\geqslant 1$ . By Lemma 3.1, $M\in {\mathrm {cogen}^{*}({}_{A}T)}$ . Since $M\in {{^{\perp }({}_{A}T)}}$ , we have $M\in {\mathcal {W}{({}_{A}T)}}$ .

(3) It suffices to show ${^{\perp }({}_{A}T)}\cap {\mathcal {GP}^{\leqslant n }(A)}\subseteq \mathcal {W}{({}_{A}T)}$ . Let $M\in {^{\perp }({}_{A}T)}\cap {\mathcal {GP}^{\leqslant n }(A)}$ . By [Reference Christensen, Frankild and Holm15, Lemma 2.17], there exists an exact sequence $0\to M\to H\to G\to 0$ in A-mod with proj.dim $({}_{A}H)\leqslant n$ and $G\in {\mathcal {GP}(A)}$ . As $_{A}T\in {\mathcal {P}^{<\infty }(A)}$ , we have $G\in {^{\perp }({}_{A}T)}$ . This implies $H\in {^{\perp }({}_{A}T)}$ . Hence $H\in {\mathcal {W}{({}_{A}T)}}$ by $(1)$ . Since $G\in {\mathcal {W}{({}_{A}T)}}$ by (2), $M\in {\mathcal {W}{({}_{A}T)}}$ by Lemma 3.2.

(4) Let $M\in \mathcal {W}{({}_{A}T)}$ . There exists an exact sequence $0\to M\to X_{0}\to X_{1}\to \dots $ in A-mod with $X_{i}\in {\textrm {add}({}_{A}T)}$ for all $i\geqslant 0$ such that it stays exact after applying $\mbox {Hom}_{A}(-,T)$ . Since proj.dim $({}_{A}T)=\ell $ , we see from the horseshoe lemma that there exists an exact sequence in $A\mbox {-mod}$

$$ \begin{align*}0\to \Omega^{\ell}_{A}(M)\to Q_{0}\to Q_{1}\to \dots,\end{align*} $$

where $Q_{i}$ is projective for each $i\geqslant 0$ . Recall that there exists an exact sequence of A-modules $0\to {{}_{A}A}\to T_{0}\to \dots \to T_{\ell }\to 0$ with $T_{i}\in {\textrm {add}({}_{A}T)}$ for all $0\leqslant i\leqslant \ell $ . Since ${M\in {^{\perp }({}_{A}T)}}$ , there are isomorphisms $\mbox {Ext}_{A}^{i}(\Omega ^{\ell }_{A}(M),A)\cong \mbox {Ext}_{A}^{\ell +i}(M,A)\cong \mbox {Ext}_{A}^{i}(M,T_{\ell })=0$ for all $i\geqslant 1$ . This implies $M\in {\mathcal {GP}^{\leqslant \ell }(A)}$ .

Proof of Theorem 1.2(1)

Recall from [Reference Anderson and Fuller1, Theorem 23.5] that, given inverse (not necessarily resolving) dualities $F:\mathcal {C}\to \mathcal {D}$ and $G:\mathcal {D}\to \mathcal {C}$ with ${{}_A}A\in \mathcal {C}$ and $B_B\in \mathcal {D}$ , there exists a faithfully balanced bimodule $_{A}T_{B}$ with $T_{B}\cong {F({}_{A}A)}\in {\mathcal {D}}$ and ${_{A}T\cong {G(B_{B})}\in {\mathcal {C}}}$ such that $F\cong \mbox {Hom}_{A}(-,T)|_{\mathcal {C}}$ and $G\cong \mbox {Hom}_{B^{op}}(-,T)|_{\mathcal {D}}$ . Now, assume that F and G are resolving dualities. They are exact functors between exact categories $\mathcal {C}$ and $\mathcal {D}$ . Let $\delta : 0\to T\to X\to M\to 0$ be an exact sequence in A-mod with $M\in {\mathcal {C}}$ . Since $\mathcal {C}$ is closed under extensions in A-mod, X belongs to $\mathcal {C}$ . As F is exact, the sequence $F(\delta ): 0\to F(M)\to F(X)\to F(T)\to 0$ is exact. It follows from $F(T)\cong {FG(B_{B})}\cong {B_{B}}$ that $F(\delta )$ splits in $B^{op}\mbox {-mod}$ . Note that $GF$ is naturally isomorphic to the identity functor of $\mathcal {C}$ . Thus $\delta $ splits in A-mod. This implies that $\mbox {Ext}_{A}^{1}(M,T)=0$ for all $M\in {\mathcal {C}}$ . Since $\mathcal {C}\subseteq A\mbox {-mod}$ is a resolving subcategory, $\Omega ^{i}_{A}(M)\in {\mathcal {C}}$ for all $M\in {\mathcal {C}}$ and $i\geqslant 1$ . Consequently, $\mbox {Ext}_{A}^{i}(M,T)\cong {\mbox {Ext}_{A}^{1}(\Omega ^{i-1}_{A}(M),T)=0}$ . This yields $\mathcal {C}\subseteq {^{\perp }({}_{A}T)}$ . In particular, $_AT\in {^{\perp }({}_{A}T)}$ . Similarly, one can prove $\mathcal {D}\subseteq {^{\perp }(T_{B})}$ , which gives rise to ${T_B\in {^{\perp }(T_{B})}}$ . Since ${{}_A}T{{}_B}$ is faithfully balanced, it is Wakamatsu tilting.

To prove (c), it suffices to show the inclusion $\mathcal {C}\subseteq {\mathcal {W}{({}_{A}T)}}$ , while the inclusion ${\mathcal {D}\subseteq {\mathcal {W}{(T_{B})}}}$ can be shown dually.

In fact, as $\mathcal {C}\subseteq {^{\perp }({}_{A}T)}$ , we only need to show $\mathcal {C}\subseteq $ cogen $^{*}({}_{A}T)$ . For each $X\in \mathcal {C}$ , let

$$ \begin{align*}\cdots\to Q_{1}\stackrel{g_{1}}\to Q_{0}\stackrel{g_{0}}\to F(X)\to 0\end{align*} $$

be a projective resolution of $F(X)$ in $B^{op}\mbox {-mod}$ . Clearly, $F(X)\in {\mathcal {D}}$ and $Q_{i}\in {\mathcal {D}}$ for all $i\geqslant 0$ . Since $\mathcal {D}$ is a resolving subcategory of $B^{op}$ -mod, $\textrm {Ker}(g_{i})\in {\mathcal {D}}$ for all $i\geqslant 0$ . By the exactness of G, we obtain an exact sequence of A-modules $0\to X\to G(Q_{0})\to G(Q_{1})\to \dots $ in which $G(Q_{i})\in {\textrm {add}(G(B_{B}))=\textrm {add}({}_{A}T)}$ for all $i\geqslant 0$ . Thus the exactness of F implies $X\in $ cogen $^{*}({}_{A}T)$ . This finishes the proof of Theorem 1.2(1).

Proof (1) The necessity of $(1)$ is clear since $\mathcal {U}\subseteq \mathcal {U}^{\leqslant n}(A)$ and $\mathcal {V}\subseteq \mathcal {V}^{\leqslant m}(B^{op})$ . It suffices to show the sufficiency of $(1)$ . Assume that the condition $(\ast )$ holds. Let $M\in \mathcal {C}\cap \, \mathcal {U}^{\leqslant n}(A)$ . We need to show $F(M)\in {\mathcal {D}\cap \mathcal {V}^{\leqslant m}(B^{op})}$ .

Let $\cdots \to P_1\to P_0\to M\to 0$ be a projective resolution of $_{A}M$ and let $s:=\textrm {max}\{m,n\}$ . Since $\mathcal {C}$ and $\mathcal {U}$ are resolving subcategories of $A\mbox {-mod}$ , $\Omega ^{s}_{A}(M)\in {\mathcal {C}\cap \mathcal {U}}$ by Lemma 2.4. It follows from $F(\mathcal {C}\cap \mathcal {U})\subseteq {\mathcal {D}\cap \mathcal {V}^{\leqslant m}(B^{op})}$ that $F(\Omega ^{s}_{A}(M))\in {\mathcal {D}\cap \mathcal {V}^{\leqslant m}(B^{op})}$ . Since $M\in \mathcal {C}\subseteq {^{\perp }({}_{A}T)}$ by Theorem 1.2(1), there is an exact sequence in $B^{\mathrm {op}}\mbox {-mod}$ :

$$ \begin{align*}0\to F(M)\to F(P_0)\to F(P_1)\to\cdots \to F(P_{s-1})\to F(\Omega^{s}_{A}(M))\to 0.\end{align*} $$

Observe that both $\mathcal {D}$ and $\mathcal {V}^{\leqslant m}(B^{op})$ are resolving subcategories of $B^{op}\mbox {-mod}$ , and so is their intersection ${\mathcal {D}\cap \mathcal {V}^{\leqslant m}(B^{op})}$ . Since $F(P_i)\in F(\mbox {add}({}_{A}A))\subseteq F(\mathcal {C}\cap \mathcal {U})$ for all $0\leqslant i\leqslant s-1$ , we have $F(M)\in {\mathcal {D}\cap \mathcal {V}^{\leqslant m}(B^{op})}$ .

(2) It is enough to prove (a) and (c) since (b) is dual to (a).

(a) To prove $\mathcal {C}\cap {\mathcal {U}^{<\infty }(A)}=\mathcal {C}\cap {\mathcal {U}^{\leqslant n}(A)}$ , it suffices to prove $\mathcal {C}\cap {\mathcal {U}^{<\infty }(A)}\subseteq \mathcal {C}\,\cap {\mathcal {U}^{\leqslant n}(A)}$ . Let $X\in {\mathcal {C}\cap {\mathcal {U}^{<\infty }(A)}}$ . Then there is a natural number t such that $\Omega ^{t}_{A}(X)\in {\mathcal {C}\cap \mathcal {U}}$ by Lemma 2.4. It follows from the proof of the sufficiency of $(1)$ that $F(X)\in {{\mathcal {D}\cap \mathcal {V}^{\leqslant m}(B^{op})}}$ . Since $X\cong {GF(X)}$ , we have $X\in {\mathcal {C}\cap {\mathcal {U}^{\leqslant n}(A)}}$ .

Suppose $\mathcal {U}\subseteq {\mathcal {C}}$ . By Theorem 1.2(1), we only need to check ${^{\perp }({}_{A}T)\cap {\mathcal {U}^{<\infty }(A)}}\subseteq {\mathcal {C}\cap {\mathcal {U}^{<\infty }(A)}}$ . Let $Y\in {^{\perp }({}_{A}T)\cap {\mathcal {U}^{<\infty }(A)}}$ . Then there exists an exact sequence in A-mod:

$$ \begin{align*}0\to U_{t}\to U_{t-1}\to \cdots\to U_1\to U_0\to Y\to 0,\end{align*} $$

where $U_i\in {\mathcal {U}}$ for all $0\leqslant i\leqslant {t}\in \mathbb {N}$ . Note that $\mathcal {C}\subseteq {\mathcal {W}{({}_{A}T)}}\subseteq {^{\perp }({}_{A}T)}$ by Theorem 1.2(1). Since $Y\in {^{\perp }({}_{A}T)}$ and ${\mathcal {U}}\subseteq {\mathcal {C}}$ , applying $F:=\mbox {Hom}_A(-,T)$ to the above exact sequence yields an exact sequence in $B^{op}\mbox {-mod}$ :

$$ \begin{align*}0\to F(Y)\to F(U_0)\to F(U_1)\to \cdots\to F(U_{t-1})\to F(U_{t})\to 0.\end{align*} $$

As $\mathcal {U}\subseteq {\mathcal {C}}$ , we see from the condition $(\ast )$ that $F(U_i)\in F(\mathcal {U})\subseteq {{\mathcal {D}\cap \mathcal {V}^{\leqslant m}(B^{op})}}$ for all $0\leqslant i\leqslant {t}$ . Note that $\mathcal {D}\cap \mathcal {V}^{\leqslant m}(B^{op})$ is a resolving subcategory of $B^{op}\mbox {-mod}$ . Thus $F(Y)\in {{\mathcal {D}\cap \mathcal {V}^{\leqslant m}(B^{op})}}$ . By the exactness of G, the following sequence

$$ \begin{align*}0\to GF(U_{t})\to GF(U_{t-1})\to \cdots\to GF(U_1)\to GF(U_0)\to GF(Y)\to 0\end{align*} $$

is exact in $A\mbox {-mod}$ . Recall that $GF$ is naturally isomorphic to the identity functor of $\mathcal {C}$ . Since $U_{i}\in \mathcal {U}\subseteq \mathcal {C}$ for all $0\leqslant i\leqslant {t}$ , we have $Y\cong GF(Y)\in G(\mathcal {D})\subseteq {{\mathcal {C}}}$ .

(c) Since $_{A}T\cong G(B_{B})$ by Theorem 1.2(1), we obtain $_{A}T\in {\mathcal {C}\cap \mathcal {U}^{\leqslant {s}}(A)}$ . As $\mathcal {C}$ and $\mathcal {U}$ are resolving subcategories of $A\mbox {-mod}$ , it follows from Lemma 2.4 that $\Omega ^{i}_{A}(T)\in {\mathcal {C}\cap \mathcal {U}}$ for all $i\geqslant s$ . In particular, $\Omega ^{s+1}_{A}(T)\in {\mathcal {C}\cap \mathcal {U}}$ . By $(\ast )$ , $F(\Omega ^{s+1}_{A}(T))\in {\mathcal {D}\cap \mathcal {V}^{\leqslant s}(B^{op})}$ . Moreover, from the minimal projective resolution $\cdots \to P_2\to P_1\to P_0\to {{}_A}T\to 0$ of the module ${{}_A}T$ , we obtain an exact sequence in $B^{op}\mbox {-mod}$

$$ \begin{align*}0\to B\to F(P_{0})\to F(P_{1})\to F(P_2)\to \dots,\end{align*} $$

where $F(P_{i})\cong \mbox {Hom}_{A}(P_{i},T)\in {\textrm {add}(T_{B})}$ for all $i\geqslant 0$ . Since $\mathcal {D}\subseteq {^{\perp }(T_{B})}$ by Theorem 1.2(1)(c) and $F(\Omega ^{s+1}_{A}(T))\in \mathcal {D}$ , it is clear that $\mbox {Ext}_{B^{op}}^{j}(F(\Omega ^{s+1}_{A}(T)),F(P_{i}))=0$ for all $j\geqslant 1$ and $i\geqslant 0$ . Consequently, there are isomorphisms

$$ \begin{align*}&\mbox{Ext}_{B^{op}}^{1}\big(F(\Omega^{s+1}_{A}(T)),F(\Omega^{s}_{A}(T))\big) \cong \mbox{Ext}_{B^{op}}^{s+1}\big(F(\Omega^{s+1}_{A}(T)),B\big)\\&\quad\cong\mbox{Ext}_{B^{op}}^1\big(\Omega_{B^{op}}^s(F(\Omega^{s+1}_{A}(T))),B\big).\end{align*} $$

By Lemma 2.4, $\Omega _{B^{op}}^s(F(\Omega ^{s+1}_{A}(T)))\in \mathcal {V}$ . Since $\mathcal {V}\subseteq {\mathcal {SGP}(B^{op})}={^{\perp }(B_B)}$ by assumption, we have $\mbox {Ext}_{B^{op}}^1\big (\Omega _{B^{op}}^s(F(\Omega ^{s+1}_{A}(T))),B\big )=0,$ and hence $\mbox {Ext}_{B^{op}}^{1}\big (F(\Omega ^{s+1}_{A}(T)), F(\Omega ^{s}_{A}(T))\big )=0$ . This implies that the sequence

$$ \begin{align*}0\to F(\Omega^{s}_{A}(T))\to F(P_s)\to F(\Omega^{s+1}_{A}(T))\to 0\end{align*} $$

splits. Thus $F(\Omega ^{s}_{A}(T))$ is a direct summand of $F(P_s)$ , and belongs to $\textrm {add}(T_{B})$ . Since $\Omega ^{s}_{A}(T)\in \mathcal {C}$ , we have $\Omega ^{s}_{A}(T)\cong GF(\Omega ^{s}_{A}(T))\in \mbox {add}({}_{A}A)$ . This forces ${\mathrm {proj.dim}({}_AT)\leqslant s}$ . Similarly, we can show $\mathrm { proj.dim}(T_{B})\leqslant s$ . Thus ${{}_A}T{{}_B}$ is tilting.

Proof of Theorem 1.2(2)

The sufficiency of $(2)$ is a direct consequence of Lemma 3.5. Now, assume that $_{A}T_{B}$ is a tilting bimodule with $\ell =\mathrm {proj.dim}({}_AT)$ . By Theorem 1.2(1) and Corollary 3.4(3), $\mathcal {C}\subseteq \mathcal {W}({{}_A}T)\subseteq \mathcal {GP}^{\leqslant \ell }(A)$ and $\mathcal {D}\subseteq \mathcal {W}(T_B)\subseteq \mathcal {GP}^{\leqslant \ell }(B^{op})$ . It follows that $\mathcal {C}\cap \mathcal {GP}^{\leqslant \ell }(A)=\mathcal {C}$ and $\mathcal {D}\cap \mathcal {GP}^{\leqslant \ell }(B^{op}) =\mathcal {D}$ . Since $\mathcal {GP}^{\leqslant \ell }(A)\subseteq \mathcal {SGP}^{\leqslant \ell }(A)$ , we also have $\mathcal {C}\cap \mathcal {SGP}^{\leqslant \ell }(A)=\mathcal {C}$ and $\mathcal {D}\cap \mathcal {SGP}^{\leqslant \ell }(B^{op})=\mathcal {D}$ . Thus the assertions on (semi-)Gorenstein-projective modules in the necessity of $(2)$ automatically hold.

Now, we apply Lemma 3.5 to show the assertion on projective modules in the necessity of $(2)$ .

Let $\mathcal {U}:=\mbox {add}({}_AA)$ and $\mathcal {V}:=\mbox {add}(B_B)$ . Since $\mathcal {C}\subseteq A\mbox {-mod}$ and $\mathcal {D}\subseteq B^{op}\mbox {-mod}$ are resolving subcategories, $\mathcal {U}={\mathcal {U}\cap \mathcal {C}}$ and $\mathcal {V}={\mathcal {V}\cap \mathcal {D}}$ . Clearly, we have $F(M)=\mbox {Hom}_A(M,T)\in \textrm {add}(T_B)$ for any $M\in \mathcal {U}$ . Since $T_B=F(A)\in {\mathcal {D}\cap \mathcal {V}^{\leqslant \ell }}(B^{op})$ , it is clear that $F(M)\in {\mathcal {D}\cap \mathcal {V}^{\leqslant \ell }}(B^{op})$ . Dually, $G(N)\in {\mathcal {C}\cap \mathcal {U}^{\leqslant \ell }}(A)$ for any $N\in \mathcal {V}$ . By Lemma 3.5(1), F and G can be restricted to inverse resolving dualities $\mathcal {C}\cap {\mathcal {P}^{\leqslant \ell }(A)}\simeq \mathcal {D}\cap {\mathcal {P}^{\leqslant \ell }(B^{op})}$ . Further, by Lemma 3.5(2), we have $\mathcal {C}\cap {\mathcal {P}^{<\infty }(A)}=\mathcal {C}\cap {\mathcal {P}^{\leqslant \ell }(A)}$ and $\mathcal {D}\cap {\mathcal {P}^{\leqslant \ell }(B^{op})}= \mathcal {D}\cap {\mathcal {P}^{<\infty }(B^{op})}$ . This finishes the proof of $(2)$ .

Proof of Corollary 1.3

(1) follows from Corollary 3.4(3) and the resolving dualities $\mathcal {W}{({}_{A}T)}\simeq \mathcal {W}{(T_{B})}$ by [Reference Wakamatsu42, Theorem 4.2].

(2) Suppose $\mathcal {C}\subseteq {\mathcal {GP}^{\leqslant n }(A)}$ and $\mathcal {D}\subseteq {\mathcal {GP}^{\leqslant m}(B^{op})}$ for some $n, m\in \mathbb {N}$ . Then we have

$$ \begin{align*} \mathcal{C}\cap\mathcal{GP}^{\leqslant n}(A)=\mathcal{C} \text{ and } \mathcal{D}\cap\mathcal{GP}^{\leqslant m}(B^{op}) =\mathcal{D}. \end{align*} $$

It follows from Theorem 1.2(2) that ${{}_A}T_{B}$ is a tilting bimodule. In Lemma 3.5, we set $\mathcal {U}:=\mathcal {GP}(A)$ and $\mathcal {V}:=\mathcal {GP}(B^{op})$ , and then the condition $(\ast )$ is satisfied. Further, assume $\mathcal {GP}(A)\subseteq \mathcal {C}$ and $\mathcal {GP}(B^{op})\subseteq \mathcal {D}$ . Then we have $\mathcal {C}={^{\perp }({}_{A}T)\cap {\mathcal {GP}^{<\infty }(A)}}$ and $\mathcal {D}={^{\perp }({}_{A}T)\cap {\mathcal {GP}^{<\infty }(B^{op})}}$ by Lemma 3.5(2). Since T is tilting, we have $\mathcal {C}={^{\perp }({}_{A}T)}\cap {\mathcal {GP}^{\leqslant \ell }(A)}$ and $\mathcal {D}={^{\perp }(T_{B})}\cap {\mathcal {GP}^{\leqslant \ell }(B^{op})}$ by Corollary 3.4(3).

Proof Assume that $(a)$ holds. Let X be the direct sum of all non-isomorphic, indecomposable, finitely generated Gorenstein-projective A-modules. Since A is CM-finite, X is a finite direct sum and $\textrm {add}({}_{A}X)=\mathcal {GP}(A)$ . As $\mathcal {GP}(A)\subseteq \mathcal {C}$ and ${\mathcal {D}\subseteq {\mathcal {GP}^{<\infty }(B^{op})}}$ , we have $F(X)\in {\mathcal {GP}^{<\infty }(B^{op})}$ . This implies that there exists a natural number m such that $F(X)\in \mathcal {GP}^{\leqslant m}(B^{op})$ , and thus $F(\mathcal {C}\cap {\mathcal {GP}(A)})=F(\mathcal {GP}(A))\subseteq {\mathcal {D}\cap \mathcal {GP}^{\leqslant m}(B^{op})}$ . Since $\mbox {Hom}_B(-,B):\mathcal {GP}(B)\to \mathcal {GP}(B^{op})$ is a duality of exact categories and B is CM-finite, $B^{op}$ is also CM-finite. In a similar way, we can show that $G(\mathcal {D}\cap {\mathcal {GP}(B^{op})})=G(\mathcal {GP}(B^{op}))\subseteq {\mathcal {C}\cap \mathcal {GP}^{\leqslant n}(A)}$ for some natural number n. Now, Corollary 3.6 follows from Lemma 3.5.

Assume that $(b)$ holds. Thanks to [Reference Xi43, Lemma 4.4], we have

$$ \begin{align*} \mathcal{GP}^{<\infty}(A)=\mathcal{GP}^{\leqslant s}(A) \text{ and } \mathcal{GP}^{<\infty}(B^{op})=\mathcal{GP}^{\leqslant t}(B^{op}), \end{align*} $$

where $s=\operatorname {findim}(A) $ and $t=\operatorname {findim}(B^{op})$ . Thus Corollary 3.6 follows from Corollary 1.3(2).

Proof Recall that the functors $F:=\mbox {Hom}_{A}(-,T)$ and $G:=\mbox {Hom}_{B^{op}}(-,T)$ between $A\mbox {-mod}$ and $B^{op}\mbox {-mod}$ can be restricted to inverse resolving dualities $\mathcal {W}{({}_{A}T)}\simeq \mathcal {W}{(T_{B})}$ by [Reference Wakamatsu42, Theorem 4.2]. So, the necessity of Corollary 3.7 follows from Theorem 1.2(2) and Lemma 3.3(1)(3). As to the sufficiency of Corollary 3.7, all the cases except $(4)$ are direct consequences of Theorem 1.2(2) and Lemma 3.3(1). It remains to show the case $(4)$ . Suppose that F and G are restricted to the inverse resolving dualities $^{\perp }({}_{A}T)\cap {\mathcal {GP}^{\leqslant n }(A)}\simeq {^{\perp }(T_{B})\cap {\mathcal {GP}^{\leqslant m}(B^{op})}}$ for some $n, m\in \mathbb {N}$ . By Theorem 1.2(1), $^{\perp }({}_{A}T)\cap {\mathcal {GP}^{\leqslant n }(A)}\subseteq {\mathcal {W}{({}_{A}T)}}$ and ${^{\perp }(T_{B})\cap {\mathcal {GP}^{\leqslant m}(B^{op})}}\subseteq {\mathcal {W}{(T_{B})}}$ . This implies that $^{\perp }({}_{A}T)\cap {\mathcal {GP}^{\leqslant n }(A)}=\mathcal {W}({}_{A}T)\cap {\mathcal {GP}^{\leqslant n}(A)}$ and $^{\perp }(T_{B})\cap {\mathcal {GP}^{\leqslant n }(B^{op})}=\mathcal {W}(T_{B})\cap {\mathcal {GP}^{\leqslant n}(B^{op})}$ , and then we return to the case $(3)$ .

Proof Since F induces an equivalence $\mathcal {A}_{1}\stackrel {\simeq }\longrightarrow {\mathcal {A}_{2}}^{op}$ of small exact categories, it follows that $K_{n}(\mathcal {A}_{1})\simeq {K_{n}({\mathcal {A}_{2}}^{op})}$ for each $n\in \mathbb {N}$ . Lemma 4.2 follows from $K_{n}(\mathcal {A}_{2})\simeq {K_{n}({\mathcal {A}_{2}}^{op})}$ (see [Reference Quillen37, Section 2]).

Proof of Corollary 1.4(1)(2)

$(1)$ Let $\ell :=\mathrm {proj.dim}({}_AT)$ . Set $\mathcal {A}_{1}:={^{\perp }({}_{A}T)}\cap {\mathcal {GP}^{\leqslant \ell }(A)}$ and $\mathcal {A}_{2}:={^{\perp }(T_{B})}\cap {\mathcal {GP}^{\leqslant \ell }(B^{op})}$ . By Corollary 1.3(1), $F:=\mbox {Hom}_{A}(-,T):\mathcal {A}_{1}\to \mathcal {A}_{2}$ is a duality of small exact categories. Let $G:=\mbox {Hom}_{B^{op}}(-,B):\mathcal {GP}(B^{op})\to \mathcal {GP}(B)$ . Then G is also a duality of small exact categories. Since $\mathcal {GP}(A)\subseteq \mathcal {A}_{1}$ and $\mathcal {GP}(B^{op})\subseteq \mathcal {A}_{2}$ , there is a diagram of exact categories:

(4.1)

Clearly, $\mathcal {A}_{1}\subseteq A\mbox {-mod}$ and $\mathcal {A}_{2}\subseteq B^{op}\mbox {-mod}$ are resolving subcategories. Now, we apply Lemma 2.1 to the inclusions of exact categories in (4.1), and then obtain the following diagram of triangle equivalences of unbounded derived categories of exact categories:

This implies a triangle equivalence $\mathscr {D}(\mathcal {GP}(A))\simeq {\mathscr {D}(\mathcal {GP}(B))}$ , which can be restricted to an equivalence $\mathscr {D}^{*}(\mathcal {GP}(A))\simeq \mathscr {D}^{*}(\mathcal {GP}(B))$ for any $*\in \{+,-,b\}$ by Lemma 2.1.

$(2)$ By Lemma 4.1, $K_{n}(\mathcal {GP}(A))\simeq {K_{n}(\mathcal {A}_{1})}$ and $K_{n}(\mathcal {GP}(B^{op}))\simeq {K_{n}(\mathcal {A}_{2})}$ . By Lemma 4.2, $K_{n}(\mathcal {A}_{1})\simeq {K_{n}(\mathcal {A}_{2})}$ and $K_{n}(\mathcal {GP}(B))\simeq {K_{n}(\mathcal {GP}(B^{op}))}$ . Thus $K_{n}(\mathcal {GP}(A))\simeq {K_{n}(\mathcal {GP}(B))}$ .

Proof Recall that a strictly exact complex $X\in {\mathscr {C}(\mathcal {P}(A))}$ is exact in $\mathscr {C}(A\mbox {-mod})$ with all of its boundaries in $\mathcal {P}(A)$ . Then X is contractible and thus zero in $\mathscr {K}(\mathcal {P}(A))$ . By the construction of derived categories in Section 2.2, $\mathscr {K}^{*}(\mathcal {P}(A))= {\mathscr {D}^{*}(\mathcal {P}(A))}$ for any $*\in \{\emptyset ,+,-,b\}$ . Set

$$ \begin{align*}\ell:=\mathrm{proj.dim}({}_AT),\;\; \mathcal{B}_{1}:={^{\perp}({}_{A}T)}\cap{\mathcal{P}^{\leqslant\ell}(A)},\; \;\mathcal{B}_{2}:={^{\perp}(T_{B})}\cap{\mathcal{P}^{\leqslant\ell}(B^{op})}.\end{align*} $$

Then $\mathcal {B}_{1}={^{\perp }({}_{A}T)}\cap {\mathcal {P}^{<\infty }(A)}$ and $\mathcal {B}_{2}={^{\perp }(T_{B})}\cap {\mathcal {P}^{<\infty }(B^{op})}$ by Theorem 2.6(c), and $F:\mathcal {B}_{1}\to \mathcal {B}_{2}$ is a duality of small exact categories by Theorem 2.6 (see also Corollary 3.7(2)). Now, in the proof of Corollary 1.4(1)(2), we replace $\mathcal {A}_{1}$ , $\mathcal {A}_{2}$ , $\mathcal {GP}(A)$ and $\mathcal {GP}(B^{op})$ with $\mathcal {B}_{1}$ and $\mathcal {B}_{2}$ , $\mathcal {P}(A)$ and $\mathcal {P}(B^{op})$ , respectively, and then show Corollary 4.3 similarly.

Proof By Corollary 3.4(3), $\mathcal {A}=\mathcal {W}({}_{A}T)$ . Since $\mathcal {A}$ is a resolving subcategory of $A\mbox {-mod}$ by Lemma 3.2(1), it is a small exact category of which the projective objects are exactly projective A-modules. By Lemma 3.2(3), add $({}_{A}T)$ equals the full subcategory of $\mathcal {A}$ consisting of injective objects. So, $\mathcal {A}$ as an exact category has enough projective objects and injective objects. Clearly, $\mathcal {P}^{<\infty }(\mathcal {A})=\mathcal {A}\cap \mathcal {P}^{<\infty }(A)={^{\bot }({}_{A}T)}\cap \mathcal {P}^{\leqslant 1}(A)$ . Further, since $\mathrm { proj.dim}({}_AT)\leqslant 1$ , it follows from Theorem 2.6(c) that ${^{\bot }({}_{A}T)}\cap \mathcal {P}^{\leqslant 1}(A)={^{\bot }({}_{A}T)}\cap \mathcal {P}^{<\infty }(A)$ which consists of those A-modules having finite $\mbox {add}({{}_A}T)$ -coresolutions. Thus $\mathcal {P}^{<\infty }(\mathcal {A})=\mathcal {I}^{<\infty }(\mathcal {A})$ , that is, $\mathcal {A}$ is weakly Gorenstein. Since $\mathcal {P}^{<\infty }(\mathcal {A})\subseteq \mathcal {P}^{\leqslant 1}(A)$ , we have $\mathcal {P}^{<\infty }(\mathcal {A})=\mathcal {P}^{\leqslant 1}(\mathcal {A})$ . Moreover, by the resolving duality $\mbox {Hom}_A(-,T): {^{\bot }({}_{A}T)}\cap \mathcal {P}^{\leqslant 1}(A)\to {^{\bot }(T_B)}\cap \mathcal {P}^{\leqslant 1}(B^{op})$ in Theorem 2.5, each object of ${^{\bot }({}_{A}T)}\cap \mathcal {P}^{\leqslant 1}(A)$ has an $\mbox {add}({{}_A}T)$ -coresolution of length at most $1$ . This implies $\mathcal {I}^{<\infty }(\mathcal {A})=\mathcal {I}^{\leqslant 1}(\mathcal {A})$ , and therefore $\mathcal {A}$ is weakly 1-Gorenstein. Since $\mathcal {A}\subseteq A\mbox {-mod}$ is a resolving subcategory and k is a finite field, it can be checked that $\mathcal {A}$ satisfies (E-a)–(E-d).

Proof (1) Let $H:=\mathcal {H}(A)$ , $I:=I(\mathcal {A})$ and $J:=J(\mathcal {A})$ . For any $K\in {{\mathcal {P}^{\leqslant 1}(\mathcal {A})}}$ and $M\in {\mathcal {A}}$ , it follows from [Reference Lu and Wang29, Lemma A4] that $[M]\diamond [K]=q^{-\langle M,K\rangle }[M\oplus K]$ in $H/I$ . Dually, $[K]\diamond [M]=q^{-\langle K,M\rangle }[M\oplus K]$ in $H/J$ . Thus $q^{\langle M,K\rangle }[M]\diamond [K]=q^{\langle K,M\rangle }[K]\diamond [M]$ in $H/(I+J)$ . This implies that $\Phi _{\mathcal {A}}$ is a left and right Ore subset of the algebra $H/(I+J)$ . By a similar argument as in the proof of [Reference Lu and Wang29, Proposition A5], one can further show that $\Phi _{\mathcal {A}}$ is a left and right denominator subset of $H/(I+J)$ .

(2) By $(1)$ , there is an isomorphism of algebras ${(H/(I+J))[\Phi ^{-1}_{\mathcal {A}}]}\cong {[\Phi ^{-1}_{\mathcal {A}}](H/(I+J))}.$ By Lemma 4.6, it is enough to show the algebra isomorphism

$$ \begin{align*}\mathcal{SDH}(\mathcal{A})\cong{(H/(I+J))[\Phi^{-1}_{\mathcal{A}}]}.\end{align*} $$

The algebra isomorphism ${[\Phi ^{-1}_{\mathcal {A}}](H/(I+J))}\cong {[\mathcal {R}^{-1}_{\mathcal {A}}](H/J)}$ can be proved dually.

Let $\lambda _{1}:H/I\to \mathcal {SDH}(\mathcal {A})$ and $\lambda :H/(I+J)\to (H/(I+J))[\Phi ^{-1}_{\mathcal {A}}]$ be the localizations and let $\pi _{1}:H/I\to H/(I+J)$ be the canonical surjection. Since $\pi _{1}(\mathcal {S}_{\mathcal {A}})=\Phi _{\mathcal {A}}$ , there is a unique homomorphism of algebras

$$ \begin{align*}\sigma:\mathcal{SDH}(\mathcal{A}) \to{(H/(I+J))[\Phi^{-1}_{\mathcal{A}}]}\end{align*} $$

such that $\sigma \lambda _{1}=\lambda \pi _{1}$ . By the statements following [Reference Lu and Wang29, Lemma A8], we have $\lambda _{1}(J)=0$ in $\mathcal {SDH}(\mathcal {A})$ . Then there exists a unique homomorphism of algebras

$$ \begin{align*}\widetilde{\lambda_{1}}:{(H/(I+J))[\Phi^{-1}_{\mathcal{A}}]}\to \mathcal{SDH}(\mathcal{A})\end{align*} $$

such that $\lambda _{1}=\widetilde {\lambda _{1}}\pi _{1}$ , and hence $\widetilde {\lambda _{1}}([K])$ is invertible in $\mathcal {SDH}(\mathcal {A})$ for any $[K]\in {\Phi _{\mathcal {A}}}$ . This implies that $\lambda $ induces a unique homomorphism of algebras

$$ \begin{align*}\rho:{(H/(I+J))[\Phi^{-1}_{\mathcal{A}}]}\to \mathcal{SDH}(\mathcal{A})\end{align*} $$

such that $\rho \lambda =\widetilde {\lambda _{1}}$ . So we have following commutative diagram:

Since $\rho \sigma \lambda _{1}=\rho \lambda \pi _{1}=\widetilde {\lambda _{1}}\pi _{1}=\lambda _{1}$ , it follows from the universal property of $\lambda _{1}$ that $\rho \sigma =\textrm {Id}$ . On the other hand, $\sigma \rho \lambda \pi _{1}=\sigma \widetilde {\lambda _{1}}\pi _{1}=\sigma \lambda _{1}=\lambda \pi _{1}$ . Since $\pi _{1}$ is surjective, $\sigma \rho \lambda =\lambda $ . By the universal property of $\lambda $ , we have $\sigma \rho =\textrm {Id}$ . Thus $\sigma $ and $\rho $ are isomorphisms of algebras.

Proof Note that F induces an equivalence $\mathcal {A}_{1}\stackrel {\simeq }\longrightarrow {\mathcal {A}_{2}}^{op}$ of weakly $1$ -Gorenstein exact categories. Then F induces an isomorphism of algebras: $\mathcal {SDH}(\mathcal {A}_{1})\stackrel {\simeq }\longrightarrow {\mathcal {SDH}({\mathcal {A}_{2}}^{op})}.$ By Lemma 4.7(2), ${\mathcal {SDH}({\mathcal {A}_{2}}^{op})}\simeq (\mathcal {SDH}(\mathcal {A}_2))^{op}$ . Thus Proposition 4.8 holds.

Proof Clearly, $\mathcal {B}$ is weakly $1$ -Gorenstein satisfying (E-a)–(E-d). By Lemma 4.4 and its proof, $\mathcal {A}$ is also weakly $1$ -Gorenstein satisfying (E-a)–(E-d) and ${\mathcal {P}^{{\leqslant 1}}(\mathcal {A})}={^{\bot }({}_{A}T)}\cap \mathcal {P}^{\leqslant 1}(A)$ . This means that $\mathcal {SDH}(\mathcal {B})$ and $\mathcal {SDH}(\mathcal {A})$ are well defined. Moreover, since each object $M\in \mathcal {A}$ has Gorenstein dimension at most $1$ , the exact sequence in Proposition 4.10 always exists. By a similar proof of [Reference Lu and Wang29, Theorem C(1)], one can check that $\widetilde {\phi }$ is a surjective homomorphism of algebras and the map

$$ \begin{align*}\psi:\mathcal{H}(\mathcal{A})\to\mathcal{SDH}(\mathcal{B}),\;\;[M]\mapsto{q^{-\langle M,H_{M}\rangle}[G_{M}]\diamond[H_{M}]^{-1}}\end{align*} $$

is well defined. Next, we claim that $\psi $ is a homomorphism of algebras. It suffices to show

(4.2) $$ \begin{align} \psi([M]\diamond[N])=\psi([M])\diamond\psi([N]) \end{align} $$

for all $M,N\in {\mathcal {A}}$ .

For this aim, we fix two exact sequences in A-mod:

(4.3) $$ \begin{align} 0\to H_{1}\to G_{1}\to M\to0, \end{align} $$

(4.4) $$ \begin{align} 0\to H_{2}\to G_{2}\to N\to0, \end{align} $$

where $H_{1},H_{2}\in {\mbox {add}({}_{A}A)}$ and $G_{1},G_{2}\in {\mathcal {B}}$ . Applying $\mbox {Hom}_{A}(-,N)$ and $ \mbox {Hom}_{A}(G_1,-)$ to the sequences (4.3) and (4.4), respectively, we obtain the following diagram:

(4.5)

Now, let $0\to N\to L\to M\to 0$ be an exact sequence in $\mathcal {A}$ . By (4.5), there is a commutative diagram with exact rows and columns:

(4.6)

Note that $\langle L,H_{L}\rangle ={\langle M+N,H_{1}+H_{2}\rangle }={\langle M,H_{1}\rangle }+{\langle M,H_{2}\rangle }+{\langle N,H_{1}\rangle }+{\langle N,H_{2}\rangle }$ . Since $\mathcal {A}$ and $\mathcal {B}$ are fully exact subcategories of $A\mbox {-mod}$ , there are equalities

$$ \begin{align*} |\mbox{Ext}_{\mathcal{A}}^{1}(X_1,X_2)|&=|\mbox{Ext}_{A}^{1}(X_1,X_2)|, \quad |\mbox{Hom}_{\mathcal{A}}(X_1,X_2)|=|\mbox{Hom}_{A}(X_1,X_2)|,\\ |\mbox{Ext}_{\mathcal{B}}^{1}(Y_1,Y_2)|&=|\mbox{Ext}_{A}^{1}(Y_1,Y_2)|, \quad |\mbox{Hom}_{\mathcal{B}}(Y_1,Y_2)|=|\mbox{Hom}_{A}(Y_1,Y_2)|, \end{align*} $$

for all $X_1,X_2\in {\mathcal {A}}$ and $Y_1,Y_2\in {\mathcal {B}}$ . Thus

$$ \begin{align*} \begin{aligned} \psi([M]\diamond[N])&=\psi(\sum_{[L]\in{\mathrm{Iso}(\mathcal{A})}}\displaystyle\frac{|\mbox{Ext}_{A}^{1}(M,N)_{L}|}{|\mbox{Hom}_{A}(M,N)|}[L])\\ &=\sum_{[L]\in{\mathrm{Iso}(\mathcal{A})}}\displaystyle\frac{|\mbox{Ext}_{A}^{1}(M,N)_{L}|}{|\mbox{Hom}_{A}(M,N)|}\displaystyle q^{-\langle L,H_{L}\rangle}[G_{L}]\diamond[H_{L}]^{-1}\\ &=(\sum_{[L]\in{\mathrm{Iso}(\mathcal{A})}}\displaystyle\frac{|\mbox{Ext}_{A}^{1}(M,N)_{L}|}{|\mbox{Hom}_{A}(M,N)|}q^{-\langle L,H_{L}\rangle}[G_{L}])\diamond[H_{1}\oplus H_{2}]^{-1}\\ &=(\sum_{[L]\in{\mathrm{Iso}(\mathcal{A})}}\displaystyle\frac{|\mbox{Ext}_{A}^{1}(M,N)_{L}|}{|\mbox{Hom}_{A}(M,N)|}q^{-{\langle M,H_{1}\rangle}-{\langle M,H_{2}\rangle}-{\langle N,H_{1}\rangle}-{\langle N,H_{2}\rangle}}[G_{L}])\\&\quad\diamond[H_{1}\oplus H_{2}]^{-1}. \end{aligned} \end{align*} $$

Since $q^{\langle H_1,G_{2}\rangle }[H_1]\diamond [G_2]=[H_1\oplus G_2]=q^{\langle G_2,H_{1}\rangle }[G_2]\diamond [H_1]$ , it follows that

$$ \begin{align*} \begin{aligned} \psi([M]) \diamond \psi([N])&=q^{-{\langle M,H_{1}\rangle}-{\langle N,H_{2}\rangle}}[G_1]\diamond{[H_1]^{-1}}\diamond[G_2]\diamond{[H_2]^{-1}}\\ &=q^{-{\langle M,H_{1}\rangle}-{\langle N,H_{2}\rangle}+{\langle H_{1},G_{2}\rangle}-{\langle G_{2},H_{1}\rangle}}[G_1]\diamond[G_2]\diamond{[H_1]^{-1}}\diamond{[H_2]^{-1}}\\ &=q^{-{\langle M,H_{1}\rangle}-{\langle N,H_{2}\rangle}+{\langle H_{1},G_{2}\rangle}-{\langle G_{2},H_{1}\rangle}}[G_1]\diamond[G_2]\diamond([H_2]\diamond[H_1])^{-1}\\ &=q^{-{\langle M,H_{1}\rangle}-{\langle N,H_{2}\rangle}+{\langle H_{1},G_{2}\rangle}-{\langle G_{2},H_{1}\rangle}+{\langle H_{2},H_{1}\rangle}}[G_1]\kern1.2pt{\diamond}[G_2]\kern1.2pt{\diamond}\kern1.2pt[H_2\kern1.2pt{\oplus}\kern1.2pt H_1]^{-1}. \end{aligned} \end{align*} $$

Consequently, to prove (4.2), we only need to check

(4.7) $$ \begin{align} q^{-{\langle M,H_{2}\rangle}-{\langle N,H_{1}\rangle}}\sum_{[L]\in{\mathrm{ Iso}(\mathcal{A})}}\displaystyle\frac{|\mbox{Ext}_{A}^{1}(M,N)_{L}|}{|\mbox{Hom}_{A}(M,N)|}[G_{L}]& =q^{{\langle H_{1},G_{2}\rangle}-{\langle G_{2},H_{1}\rangle}+{\langle H_{2},H_{1}\rangle}}[G_1]\diamond[G_2]. \end{align} $$

Now, we set $K:={\rm Ker} (\Theta )$ in the diagram (4.5). Then there exists an exact sequence

$$ \begin{align*}0\to \mbox{Hom}_{A}(M,N)\to \mbox{Hom}_{A}(G_1,N) \to \mbox{Hom}_{A}(H_1,N)\to K\to0.\end{align*} $$

Thus

(4.8) $$ \begin{align} |K|=q^{\mathrm{dim}_{k}\scriptstyle\mathrm{Hom}_{A}(H_1,N)-\mathrm{dim}_{k}\scriptstyle\mathrm{Hom}_{A}(G_1,N)+\mathrm{dim}_{k}\scriptstyle\mathrm{ Hom}_{A}(M,N)}. \end{align} $$

For any $\delta ':0\to G_2\to W\to G_1\to 0$ in the set $\mbox {Ext}_{A}^{1}(G_1,G_2)_{W}$ , one can show

$$ \begin{align*}|\{\delta\in{\mbox{Ext}_{A}^{1}(M,N)}\ | \ \Theta(\delta)=\delta'\}|=|K|.\end{align*} $$

This leads to

$$ \begin{align*}|\{\delta\in{\mbox{Ext}_{A}^{1}(M,N)}\ | \ \Theta(\delta)\in{\mbox{Ext}_{A}^{1}(G_1,G_2)_{W}}\}|=|K||\mbox{Ext}_{A}^{1}(G_1,G_2)_{W}|\end{align*} $$

and therefore

(4.9) $$ \begin{align} \sum_{[L]\in{\mathrm{Iso}(\mathcal{A})}}|\mbox{Ext}_{A}^{1}(M,N)_{L}|[G_{L}]= |K|\sum_{[W]\in{\mathrm{Iso}(\mathcal{A})}}|{\mbox{Ext}_{A}^{1}(G_1,G_2)_{W}}|[W]. \end{align} $$

Since $G_1\in \mathcal {B}$ and $H_{2}\in {\mbox {add}({}_{A}A)}$ , the equality $\mbox {Ext}_A^1(G_1, H_2)=0$ holds. This implies that the sequence $0\to \mbox {Hom}_{A}(G_1,H_{2})\to \mbox {Hom}_{A}(G_1,G_{2}) \to \mbox {Hom}_{A}(G_1,N)\to 0$ is exact. Consequently,

(4.10) $$ \begin{align} \small \displaystyle\mathrm{dim}_{k}\mbox{Hom}_{A}(H_1,N)&-\displaystyle\mathrm{dim}_{k}\mbox{Hom}_{A}(G_1,N)= {\langle H_{1},N\rangle}+\displaystyle\mathrm{dim}_{k}\mbox{Hom}_{A}(G_1,H_{2})\\&-\displaystyle\mathrm{dim}_{k}\mbox{Hom}_{A}(G_1,G_2).\nonumber \end{align} $$

So we have

$$ \begin{align*} \begin{aligned} q^{-{\langle M,H_{2}\rangle}-{\langle N,H_{1}\rangle}}\sum_{[L]\in{\mathrm{ Iso}(\mathcal{A})}}\displaystyle\frac{|\mbox{Ext}_{A}^{1}(M,N)_{L}|}{|\mbox{Hom}_{A}(M,N)|}[G_{L}]\\ &=q^{-{\langle M,H_{2}\rangle}-{\langle N,H_{1}\rangle}+\mathrm{dim}_{k}\scriptstyle\mathrm{Hom}_{A}(H_1,N)-\mathrm{dim}_{k}\scriptstyle\mathrm{ Hom}_{A}(G_1,N)}\\&\quad\times\sum\limits_{[W]\in{\mathrm{Iso}(\mathcal{B})}}|\mbox{Ext}_{A}^{1}(G_1,G_2)_{W}|[W]\quad(\mbox{by}\ (4.8),\;(4.9))\\ &=q^{-{\langle M,H_{2}\rangle}-{\langle N,H_{1}\rangle}+{\langle H_{1},N\rangle}+\mathrm{dim}_{k}\scriptstyle\mathrm{Hom}(G_1,H_{2})}\sum\limits_{[W]\in{\mathrm{ Iso}(\mathcal{B})}}\displaystyle\frac{|\mbox{Ext}_{A}^{1}(G_1,G_2)_{W}|}{\mbox{Hom}_{A}(G_1,G_2)}[W] \quad (\mbox{by}\ (4.10))\\ &=q^{-{\langle M,H_{2}\rangle}-{\langle N,H_{1}\rangle}+{\langle H_{1},G_2\rangle}-{\langle H_{1},H_2\rangle}+\mathrm{dim}_{k}\scriptstyle\mathrm{ Hom}_{A}(G_1,H_{2})}[G_1]\diamond[G_2] \quad(\mbox {since}\ {\langle H_{1},G_2\rangle}\\&={\langle H_{1},H_{2}+N\rangle})\\ &=q^{{\langle H_{1},G_2\rangle}-{\langle N,H_{1}\rangle}}q^{-({\langle M,H_{2}\rangle}+{\langle H_{1},H_2\rangle}-\mathrm{dim}_{k}\scriptstyle\mathrm{ Hom}_{A}(G_1,H_{2}))}[G_1]\diamond[G_2] \\ &=q^{{\langle H_{1},G_2\rangle}-{\langle N,H_{1}\rangle}}[G_1]\diamond[G_2] \quad(\mbox {since}\ {\langle G_{1},H_2\rangle}={\langle M+H_{1},H_{2}\rangle})\\ &=q^{{\langle H_{1},G_2\rangle}-{\langle G_{2},H_{1}\rangle}+{\langle H_{2},H_{1}\rangle}}[G_1]\diamond[G_2] \quad(\mbox {since}\ {\langle G_{2},H_1\rangle}={\langle H_{2}+N,H_{1}\rangle}). \end{aligned} \end{align*} $$

This shows that (4.7) is true, and thus (4.2) is also true. So, $\psi $ is a homomorphism of algebras.

Finally, we show that $\psi $ factorizes through the canonical surjection $\mathcal {H}(\mathcal {A})\to \mathcal {H}(\mathcal {A})/I(\mathcal {A})$ .

Suppose $N\in {\mathcal {P}^{{\leqslant 1}}(\mathcal {A})}$ . Since $H_2\in \mbox {add}({}_{A}A)$ , we have $G_{2}\in {\mathcal {P}^{{\leqslant 1}}(A)}$ (see the first column in the diagram (4.6)). As $G_{2}$ lies in $\mathcal {B}$ , it is projective. This implies ${G_{L}\cong {G_{1}\oplus G_{2}}}$ , and therefore

$$ \begin{align*}\psi([L])={q^{-\langle L,H_{L}\rangle}[G_{L}]\diamond[H_{L}]^{-1}}= {q^{-\langle M\oplus N,H_{1}\oplus H_{2}\rangle}[G_{1}\oplus G_{2}]\diamond[H_{1}\oplus H_{2}]^{-1}}=\psi([N\oplus M]).\end{align*} $$

Consequently, $\psi $ induces a homomorphism of algebras $\psi ':\mathcal {H}(\mathcal {A})/I(\mathcal {A})\to \mathcal {SDH}(\mathcal {B}).$ Since $\psi ([K])$ is invertible in $\mathcal {SDH}(\mathcal {B})$ for any $K\in {\mathcal {P}^{{\leqslant 1}}(\mathcal {A})}$ , $\psi '$ induces a unique homomorphism of algebras $\widetilde {\psi }:\mathcal {SDH}(\mathcal {A})\to \mathcal {SDH}(\mathcal {B}).$ Clearly, $\widetilde {\psi }\widetilde {\phi }=\mathrm {Id}$ , which means that $\widetilde {\phi }$ is injective. Since $\widetilde {\phi }$ is surjective, it is an isomorphism of algebras.

Proof By Proposition 4.10, it suffices to show ${^{\perp }({}_{A}T)}\cap {\mathcal {GP}^{\leqslant 1}(A)}={\mathcal {GP}^{\leqslant 1}(A)}$ . Let ${M\in {\mathcal {GP}^{\leqslant 1}(A)}}$ . By [Reference Christensen, Frankild and Holm15, Lemma 2.17], there exists an exact sequence $0\to M\to H\to G\to 0$ in A-mod with proj.dim $({}_{A}H)\leqslant 1$ and $G\in {\mathcal {GP}(A)}$ . It follows from Lemma 3.3(2) that $G\in {\mathcal {W}{({}_{A}T)}}$ . Since $_{A}T$ is a strong tiling module with $\mathrm {proj.dim}({}_AT)\leqslant 1$ , we have $H\in {^{\perp }({}_{A}T)}$ . Hence $M\in {^{\perp }({}_{A}T)}$ . This implies ${\mathcal {GP}^{\leqslant 1}(A)}\subseteq {{^{\perp }({}_{A}T)}}$ .

Proof of Theorem 4.9

Let $\mathcal {A}_{1}:={^{\perp }({}_{A}T)}\cap {\mathcal {GP}^{\leqslant 1}(A)}$ and $\mathcal {A}_{2}:={^{\perp }(T_{B})}\cap {\mathcal {GP}^{\leqslant 1}(B^{op})}$ . It follows from Corollary 1.3(1) that $F:=\mbox {Hom}_{A}(-,T):\mathcal {A}_{1}\to \mathcal {A}_{2}$ is a duality of weakly $1$ -Gorenstein exact categories. Moreover, $G:=\mbox {Hom}_{B^{op}}(-,B):\mathcal {GP}(B^{op})\to \mathcal {GP}(B)$ is also a duality of weakly $1$ -Gorenstein exact categories. By Propositions 4.10 and 4.8, we obtain the following diagram of isomorphisms of algebras:

Define

$$ \begin{align*}\Xi:={(\Upsilon_{G})^{op}}{(\widetilde{\psi}_{B^{op}})^{op}}{\Upsilon_{F}}{\widetilde{\phi}_{A}}: \;\mathcal{SDH}(\mathcal{GP}(A)) \to \mathcal{SDH}(\mathcal{GP}(B)).\end{align*} $$

Then $\Xi $ is an isomorphism of algebras. It remains to show that, for any $G\in {\mathcal {GP}(A)}$ ,

$$ \begin{align*}\Xi([G])=q^{-\langle L,G\rangle}[F(L)]^{-1}\diamond[F(Z)],\end{align*} $$

where L and Z are given in Theorem 4.9.

In fact, since $G\in \mathcal {GP}(A)$ , there is an exact sequence $0\to K\to P\to G\to 0$ in A-mod such that $P\in \mbox {add}({}_AA)$ and $K\in \mathcal {GP}(A)$ . As $_{A}T$ is $1$ -tilting, there is an exact sequence $0\to P\to T_{0}\to T_{1}\to 0$ in A-mod with $T_{0},T_{1}\in {\mbox {add}({}_{A}T)}$ . Consider the following pushout diagram $(\sharp )$ :

Since T is a $1$ -tilting module, $({}_{A}T)^{\perp }=\mathrm {Gen}({}_AT)$ , the smallest full subcategory of $A\mbox {-mod}$ containing $_AT$ and being closed under direct sums and quotients. This forces $X\in {({}_{A}T)^{\perp }}$ . Since $T_1\in \mbox {add}({}_AT)\subseteq ({}_{A}T)^{\perp }$ , the third column in the diagram $(\sharp )$ implies that the map $G\to X$ is a left $({}_{A}T)^{\perp }$ -approximation of G. Now, let $f:G\to Z$ be a minimal left $({}_{A}T)^{\perp }$ -approximation with $L:=\mathrm { Coker}(f)$ . Then f is injective; Z and L are isomorphic to direct summands of X and $T_1$ , respectively. In particular, $Z\in \mathrm {Gen}({}_AT)$ and $L\in \mbox {add}({}_AT)$ . Since $\mathcal {GP}(A)\cup \mbox {add}({}_AT)\subseteq \mathcal {A}_1$ and $\mathcal {A}_1$ is closed under extensions in $A\mbox {-mod}$ , we have $Z\in \mathcal {A}_1$ . As $Z\in \mathrm {Gen}({}_AT)$ , there exists an exact sequence ${0\to N\to H\to Z\to 0}$ of A-modules with $H\in {\mbox {add}({}_{A}T)}$ . Since $\mathcal {A}_1\subseteq A\mbox {-mod}$ is a resolving subcategory, $N\in \mathcal {A}_{1}$ . So, the sequence is exact in $\mathcal {A}_1$ . Applying F to the sequence yields an exact sequence $0\to F(Z)\to F(H)\to F(N)\to 0$ in $\mathcal {A}_2$ . Since $F(H)\in \mbox {add}(B_B)$ and $F(N)\in {\mathcal {GP}^{\leqslant 1}(B^{op})}$ , it follows that $F(Z)\in {\mathcal {GP}(B^{op})}$ . Hence, we have an exact sequence in $\mathcal {A}_2$

(4.11) $$ \begin{align} 0\longrightarrow F(L)\longrightarrow F(Z)\longrightarrow F(G)\longrightarrow 0 \end{align} $$

such that $F(Z)$ is Gorenstein projective and $F(L)$ is projective. Moreover, there are equalities

(4.12) $$ \begin{align} \begin{aligned} \langle F(G),F(L)\rangle&=\displaystyle\textrm{dim}_{k}\mbox{Hom}_{B^{op}}(F(G),F(L))- \displaystyle\textrm{dim}_{k}\mbox{Ext}^{1}_{B}(F(G),F(L))\\ &=\displaystyle\textrm{dim}_{k}\mbox{Hom}_{A}(L,G)- \displaystyle\textrm{dim}_{k}\mbox{Ext}^{1}_{B}(L,G)\quad(\mbox{by}\ \mbox{Lemma} \ 3.1)\\ &=\langle L,G\rangle. \end{aligned} \end{align} $$

Thus

$$ \begin{align*} \begin{aligned} \Xi([G]) &= ((\Upsilon_{G})^{op} (\widetilde{\psi}_{B^{op}})^{op} \Upsilon_{F} \widetilde{\phi}_{A})([G])\\ &={(\Upsilon_{G})^{op}}{(\widetilde{\psi}_{B^{op}})^{op}}([F(G)])\\ &={(\Upsilon_{G})^{op}}(q^{-\langle F(G),F(L)\rangle}[F(L)]^{-1}\diamond[F(Z)])\quad(\mbox{by}\ (4.11))\\ &=q^{-\langle F(G),F(L)\rangle}[GF(L)]^{-1}\diamond[GF(Z)]\\ &=q^{-\langle F(G),F(L)\rangle}[\mbox{Hom}_{A}(T,L)]^{-1}\diamond[\mbox{Hom}_{A}(T,Z)]\quad(\mbox{by}\ \mbox{Lemma} \ 3.1)\\ &=q^{-\langle L,G\rangle}[\mbox{Hom}_{A}(T,L)]^{-1}\diamond[\mbox{Hom}_{A}(T,Z)]\quad(\mbox{by}\ (4.12)). \end{aligned} \end{align*} $$

This finishes the proof of Theorem 4.9.

Proof Let A be a finite-dimensional 1-Gorenstein k-algebra. Then A-mod is a weakly 1-Gorenstein algebra satisfying (E-a)–(E-d). Recall that $\mathcal {SDH}(A):=\mathcal {SDH}(A\mbox {-mod})$ is the semi-derived Ringel–Hall algebra of A. Set $_{A}T:=\mbox {Hom}_{k}(A,k)$ , the ordinary injective cogenerator for $A\mbox {-mod}$ . Then $_{A}T$ is strong tilting and $\mathrm {proj.dim}({}_AT)\leqslant 1$ . Moreover, $A\mbox {-mod}={\mathcal {GP}^{\leqslant 1}(A)}={^{\perp }({}_{A}T)}\cap {\mathcal {GP}^{\leqslant 1}(A)}$ . Now, $(1)$ is a direct consequence of Corollary 4.11 and (2) follows from $(1)$ and Theorem 4.9.