2.1. The setting

This follows [Reference Cruz10]. Throughout we fix a Noetherian commutative ring $R$ with identity, and $A$ is an $R$-algebra which is finitely generated and projective as an $R$-module. We refer to $A$ as a projective Noetherian $R$-algebra. The set of invertible elements of $R$ is denoted by $R^\times$.

We denote by $A\!\operatorname {-mod}$ the category of finitely generated (left) $A$-modules. Given $M\in A\!\operatorname {-mod}$, we denote by $\!\operatorname {add}_A M$ (or just $\!\operatorname {add} M$) the full subcategory of $A\!\operatorname {-mod}$ whose modules are direct summands of a finite direct sum of copies of $M$. We also denote $\!\operatorname {add} A$ by $A\!\operatorname {-proj}$.

The endomorphism algebra of a module $M\in A\!\operatorname {-mod}$ is denoted by ${\rm End}_A(M)$. We denote by $D_R$ or just $D$ the standard duality ${\rm Hom}_R(-, R): A\!\operatorname {-mod} \to A^{op}\!\operatorname {-mod}$ where $A^{op}$ is the opposite algebra of $A$.

A module $M \in A\!\operatorname {-mod}\cap R\!\operatorname {-proj}$ is said to be $(A, R)$-injective if it belongs to ${\rm add}\, DA$, and we write $(A,R)\!\operatorname {-inj}\cap R\!\operatorname {-proj}$ for the full subcategory of $A\!\operatorname {-mod}\cap R\!\operatorname {-proj}$ whose modules are $(A, R)$-injective.

Furthermore, an exact sequence of $A$-modules which is split as an exact sequence of $R$-modules is said to be $(A, R)$-exact. In particular, an $(A, R)$-monomorphism is a homomorphism $f: M\to N$ that fits into an $(A,R)$-exact sequence $0\to M\stackrel {f}\to N$.

Given a left exact covariant additive functor $G$, we say that $X$ is a $G$-acyclic object if $\operatorname {R}^{i>0}G(X)=0$. An exact sequence $0\rightarrow L\rightarrow X_0\rightarrow X_1\rightarrow \cdots$ is called a $G$-acyclic coresolution of $L$ if all objects $X_0, X_1, \ldots$ are $G$-acyclic. Given $X\in A\!\operatorname {-mod}\cap R\!\operatorname {-proj}$, we denote by $X^\perp$ the full subcategory

\[ {\{M\in A\!\operatorname{-mod}\cap R\!\operatorname{-proj}\colon \operatorname{Ext}_A^{i>0}(Z, M)=0, \forall Z\in \!\operatorname{add} X \}}, \]

and by ${}^\perp X$ the full subcategory

\[\{M\in A\!\operatorname {-mod}\cap R\!\operatorname {-proj}\colon \operatorname {Ext}_A^{i>0}(M, Z)=0, \forall Z\in \!\operatorname {add} X \}.\]

2.2. Approximations

Assume that $A$ is an $R$-algebra as above, and $Q$ is a fixed module in $A\!\operatorname {-mod}\cap R\!\operatorname {-proj}$.

An $A$-homomorphism $f: M\to N$ is a left $\!\operatorname {add}_A Q$-approximation of $M$ provided that $N$ belongs to $\!\operatorname {add}_A Q$, and moreover the induced map

\[ {\rm Hom}_A(N, X) \to {\rm Hom}_A(M, X) \]

is surjective for every $X\in \!\operatorname {add}_A Q$. Dually, one defines right $\!\operatorname {add}_A Q$-approximations. Note that every module $M \in A\!\operatorname {-mod}$ has a left and a right $\!\operatorname {add}_A Q$-approximation.

2.3. Relative (co)dominant dimension with respect to a module

We recall from [Reference Cruz10] the definition of relative (co)dominant dimensions.

Let $Q, X\in A\!\operatorname {-mod}\cap R\!\operatorname {-proj}$. If $X$ does not admit a left $\!\operatorname {add} Q$-approximation which is an $(A, R)$-monomorphism then the relative dominant dimension of $X$ with respect to $Q$ is zero. Otherwise, the relative dominant dimension of $X$ with respect to $Q$, denoted by $Q\!\operatorname {-domdim}_{(A, R)}X$, or $Q\!\operatorname {-domdim}_A X$ when $R$ is a field, is the supremum of all $n\in \mathbb {N}$ such that there is an $(A, R)$-exact sequence:

\[ 0\to X\to Q_1\to Q_2 \to \cdots \to Q_n \]

with all $Q_i\in \!\operatorname {add} Q$, which remains exact under ${\rm Hom}_A(-, Q)$.

Dually, one defines the relative codominant dimension, denoted by $Q{\rm }\hbox{-}{\rm codomdim}_{(A, R)}(X)$ with $Q, X$ as above: if $X$ does not admit a surjective right $\!\operatorname {add} Q$-approximation, then ${Q\!\operatorname {-codomdim}_{(A, R)}(X)} = 0$. Otherwise it is the supremum of all $n\in \mathbb {N}$ such that there is an $(A, R)$-exact sequence:

\[ Q_n\to Q_{n-1}\to\cdots \to Q_1\to X\to 0 \]

with all $Q_i\in \!\operatorname {add} Q$, which remains exact under ${\rm Hom}_A(Q, -)$.

Hence, $Q\!\operatorname {-codomdim}_{(A, R)} X=DQ\!\operatorname {-domdim}_{(A^{op}, R)} DX$. By $Q\!\operatorname {-domdim} {(A, R)}$ we mean the value $Q\!\operatorname {-domdim}_{(A, R)} A$. We will write $Q\!\operatorname {-codomdim}_A X$ to denote $Q\!\operatorname {-codomdim}_{(A, R)} X$ when $R$ is a field.

2.3.1. Results from [10]

We will give criteria towards finding these invariants. These depend essentially on the theorems [Reference Cruz10, Theorems 3.1.1, 3.1.4] and their dual versions. We briefly describe part of it. Assume $Q\in A\!\operatorname {-mod}\cap R\!\operatorname {-proj}$ such that the endomorphism algebra of $Q$, $B=\operatorname {End}_A(Q)$, is a projective Noetherian $R$-algebra. The Schur functor $F=\operatorname {Hom}_A(Q, -)$ has a left adjoint $\mathcal {I}:=Q\otimes _B -$. Let $\chi$ be the associated counit $\chi \colon \mathcal {I}F\rightarrow \operatorname {id}_{A\!\operatorname {-mod}}$. Explicitly, if $M$ belongs to $A\!\operatorname {-mod}\cap R\!\operatorname {-proj}$, then $\chi _M$ is the $A$-homomorphism:

\[ Q\otimes_B \operatorname{Hom}_A(Q, M)\rightarrow M, \quad \chi_M(q\otimes g)=g(q). \]

By Theorem 3.1.1 of [Reference Cruz10], for a module $M\in A\!\operatorname {-mod}\cap R\!\operatorname {-proj}$, $Q\!\operatorname {-codomdim}_{(A, R)} M\geq 2$ if and only if $\chi _M$ is an isomorphism. Theorem 3.1.4 of [Reference Cruz10] states that $Q\!\operatorname {-codomdim}_{(A, R)}\geq n$ for $n\geq 2$ if and only if $\chi _M$ is an isomorphism and $Q\otimes _B -$ is exact on the first part of a projective resolution of $\operatorname {Hom}_A(Q, M)$ as a $B$-module.

Section 3.1 of [Reference Cruz10] describes several consequences; we will use especially [Reference Cruz10, Corollary 3.1.5, Corollary 3.1.8, Proposition 3.1.11 and Corollary 3.1.12].

Lemma 2.1 Assume $M \in A\!\operatorname {-mod}\cap R\!\operatorname {-proj}$, and let $Q_i \in \!\operatorname {add} Q$. An exact sequence

\[ 0\to M \xrightarrow{\alpha_0} Q_0 \stackrel{\alpha_1}\to Q_1\to \cdots \to Q_t \]

remains exact under ${\rm Hom}_A(-, Q)$ if and only if for every factorization $Q_i\to {\rm im}\, \alpha _{i+1} \to Q_{i+1}$ of $\alpha _{i+1}$, the $(A,R)$-monomorphism ${\rm im}\,\alpha _{i+1}\to Q_{i+1}$ and $\alpha _0$ are left $\!\operatorname {add} Q$-approximations.

Proof. See [Reference Cruz10, Lemma 2.1.4].

In addition to the assumptions on $R, A$, and $Q$, in the following, we also assume that $\operatorname {Hom}_A(Q, Q) \in R\!\operatorname {-proj}$.

It is crucial to compare relative dominant dimensions for end terms of a short exact sequence which remains exact under ${\rm Hom}_A(-, Q)$. This is completely described in [Reference Cruz10, Lemma 3.1.7], part of it is as follows.

Lemma 2.2 Let $M\in A\!\operatorname {-mod}\cap R\!\operatorname {-proj}$ and consider an $(A,R)$-exact sequence

\[ 0\to M_1 \to M \to M_2\to 0 \]

which remains exact under ${\rm Hom}_A(-, Q)$. Let $n=Q\!\operatorname {-domdim}_AM$ and $n_i= Q\!\operatorname {-domdim}_AM_i$ for $i=1, 2$, then$:$

1. (a) $n\geq {\rm min}\{ n_1, n_2\}$.

2. (b) If $n=\infty$ and $n_1 < \infty$ then $n_2 = n_1-1$.

Corollary 2.3 Let $M_i$ for $i\in I$ be a finite set of modules in $A\!\operatorname {-mod}\cap R\!\operatorname {-proj}$. Then

\[ Q\!\operatorname{-domdim}_A\left( \bigoplus_{i\in I} M_i\right) = {\rm inf} \{ Q\!\operatorname{-domdim}_AM_i\mid i\in I\}. \]

Proof. See [Reference Cruz10, Corollary 3.1.8].

Recall ${}^{\perp }Q = \{ M \in A\!\operatorname {-mod}\cap R\!\operatorname {-proj} \mid {\rm Ext}_A^{i>0}(M, Q)=0\}$. The following is proved in [Reference Cruz10, Proposition 3.1.11].

Proposition 2.4 Assume ${\rm Ext}^{i>0}_A(Q, Q)=0$, and $M \in {}^{\perp }Q$. An exact sequence

\[ 0\to M\to Q_1\to \cdots \to Q_n \]

yields $Q\!\operatorname {-domdim}_{(A, R)}(M)\geq n$ if and only if $Q_i \in \!\operatorname {add} Q$ and the cokernel of $Q_{n-1}\to Q_n$ belongs to ${}^{\perp }Q$.

The following is an application of lemma 2.2.

Corollary 2.5 Assume $Q\in {}^\perp Q$. Let $M\in A\!\operatorname {-mod}\cap R\!\operatorname {-proj}$, and consider an $(A, R)$-exact sequence:

\[ 0\to M \to Q_1 \to \cdots \to Q_t \to X\to 0 \]

with $Q_i\in \!\operatorname {add} Q$ and $t\in \mathbb {N}$. If ${\rm Ext}_A^i(X, Q)=0$ for $1\leq i\leq t$, then

\[ Q\!\operatorname{-domdim}_{(A, R)}M = t + Q\!\operatorname{-domdim}_{(A, R)}X. \]

Proof. See [Reference Cruz10, Corollary 3.1.12].

2.4. Split quasi-hereditary algebras with duality

For the definition and general properties of split quasi-hereditary algebras we refer to [Reference Cline, Parshall and Scott5, Reference Cruz9–Reference Cruz11, Reference Rouquier49]. In particular, we follow the notation of [Reference Cruz9–Reference Cruz11]. One of the advantages to use such setup stems from the fact that split quasi-hereditary $R$-algebras $(A, \{\Delta (\lambda )_{\lambda \in \Lambda }\})$ are exactly the algebras so that $(S\otimes _R A, \{S\otimes _R \Delta (\lambda )_{\lambda \in \Lambda }\})$ are quasi-hereditary algebras for every commutative Noetherian ring $S$ which is an $R$-algebra. Concerning the terminology, we remark the word split arises from the endomorphism algebra $\operatorname {End}_A(\Delta (\lambda ))$ being isomorphic to the ground ring $R$. As it was observed in [Reference Rouquier49], when $(A, \{\Delta (\lambda )_{\lambda \in \Lambda }\})$ is a split quasi-hereditary $R$-algebra, the objects $T(\lambda )$ satisfying $\!\operatorname {add} \bigoplus _{\lambda \in \Lambda } T(\lambda )=\mathcal {F}(\tilde {\Delta })\cap \mathcal {F}(\tilde {\nabla })$ are no longer unique, in contrast to quasi-hereditary algebras over a field. For this reason, we will say that $T$ is a characteristic tilting module of $A$ if $\!\operatorname {add} T=\mathcal {F}(\tilde {\Delta })\cap \mathcal {F}(\tilde {\nabla })$ and $T$ is the (basic) characteristic tilting module of $A$ if $A$ is a quasi-hereditary algebra over a field and $T=\bigoplus _{\lambda \in \Lambda } T(\lambda )$.

The following prepares the ground for quasi-hereditary covers, constructed from the Ringel dual $R(A)= {\rm End}_A(T)^{op}$ of a quasi-hereditary algebra $A$ with a characteristic tilting module $T$. To see that Ringel duality is well defined in the integral setup, we refer to [Reference Cruz10, Subsection 2.2.3].

Recall that $\operatorname {Hom}_A(T, DA)\simeq DT$ is a characteristic tilting module over $R(A)$. Relative codominant dimension can also be used to observe that a characteristic tilting module behaves like a flat module in $\mathcal {F}(\tilde {\Delta }_{R(A)})$.

Let $(A, \{\Delta (\lambda )_{\lambda \in \Lambda }\})$ be a split quasi-hereditary algebra over a field $k$. The $\nabla$-filtration dimension of $X$, denoted by $\dim _{\mathcal {F}(\nabla )} X$, is the infimum of all $n\geq 0$ such that there exists an exact sequence:

\[ 0\rightarrow X\rightarrow M_0\rightarrow \cdots \rightarrow M_n\rightarrow 0 \]

with $M_0, \ldots, M_n \in \mathcal {F}(\nabla )$. Analogously, the $\Delta$-filtration dimension is defined. The $\nabla$-filtration dimensions first appeared in [Reference Friedlander and Parshall31] in the study of cohomology of algebraic groups.

$\nabla$- and $\Delta$-filtration dimensions play a crucial role in [Reference Erdmann and Parker25, Reference Mazorchuk and Ovsienko45] establishing that the global dimension of a quasi-hereditary algebra having a simple preserving duality is always an even number. For us, they are of importance due to the following result.

2.5. Cover theory

The concept of a cover, and in particular, of a split quasi-hereditary cover was introduced in [Reference Rouquier49] to give an abstract framework to connections in representation theory like Schur–Weyl duality. Given a split quasi-hereditary algebra $(A, \{\Delta (\lambda )_{\lambda \in \Lambda }\})$ over a commutative Noetherian ring $R$ and a finitely generated projective $A$-module $P$, let $B:= {\rm End}_A(P)^{op}$. We say that $(A, P)$ is a split quasi-hereditary cover of $B$ if the restriction of the functor

\[F:=\operatorname {Hom}_A(P, -)\colon A\!\operatorname {-mod}\rightarrow B\!\operatorname {-mod},\]

known as Schur functor, to $A\!\operatorname {-proj}$ is fully faithful. Given, in addition, $i\in \mathbb {N}\cup \{-1, 0, +\infty \}$, following the notation of [Reference Cruz9], we say that $(A, P)$ is an $i$- $\mathcal {F}(\tilde {\Delta })$ (quasi-hereditary) cover of $B$ if the following conditions hold:

• • $(A, P)$ is a split quasi-hereditary cover of $\operatorname {End}_A(P)^{op}$;

• • The restriction of $F$ to $\mathcal {F}(\tilde {\Delta })$ is faithful;

• • The Schur functor $F$ induces bijections $\operatorname {Ext}_A^j(M, N)\simeq \operatorname {Ext}^j_{B}(FM, FN)$, for every $M, N\in \mathcal {F}(\tilde {\Delta })$ and $0\leq j\leq i$.

Here, $\mathcal {F}(\tilde {\Delta })$ denotes the resolving subcategory of $A\!\operatorname {-mod}\cap R\!\operatorname {-proj}$ whose modules admit a finite filtration into direct summands of direct sums of standard modules $\Delta (\lambda )$, $\lambda \in \Lambda$.

The optimal value of the quality of a cover is known as the Hemmer–Nakano dimension. More precisely, if $(A, P)$ is a $(-1)$-$\mathcal {F}(\tilde {\Delta })$ (quasi-hereditary) cover of $B$, the Hemmer–Nakano dimension of $\mathcal {F}(\tilde {\Delta })$ with respect to $F$ is $i\in \mathbb {N}\cup \{-1, 0, +\infty \}$ if $(A, P)$ is an $i$-$\mathcal {F}(\tilde {\Delta })$ (quasi-hereditary) cover of $\operatorname {End}_A(P)^{op}$ but $(A, P)$ is not an $(i+1)$-$\mathcal {F}(\tilde {\Delta })$ (quasi-hereditary) cover of $B$. The Hemmer–Nakano dimension of $\mathcal {F}(\tilde {\Delta })$ is denoted by $\operatorname {HNdim}_{F}(\mathcal {F}(\tilde {\Delta }))$.

Major tools to compute Hemmer–Nakano dimensions are classical dominant dimension and relative dominant dimensions. This can be traced back to [Reference Fang and Koenig29] which was later amplified in several directions in [Reference Cruz9] and in [Reference Cruz10], and it is briefly summarized in the following result proved in [Reference Cruz10, Theorem 5.3.1, Corollary 5.3.4]. Note that ${\rm Hom}_A(T, Q)$ is projective as a $B$-module.

Let $B$ be a projective Noetherian $R$-algebra, $(A, P)$ be an $(-1)$-$\mathcal {F}(\tilde {\Delta })$ (quasi-hereditary) cover of $B$ and $(A', P')$ be an $(-1)$-$\mathcal {F}(\tilde {\Delta }')$ (quasi-hereditary) cover of $B$. We say that $(A, P)$ is equivalent to $(A', P')$ as quasi-hereditary covers if there exists an equivalence $H\colon A\!\operatorname {-mod}\rightarrow A'\!\operatorname {-mod}$ which restricts to an equivalence of categories between $\mathcal {F}(\tilde {\Delta })$ and $\mathcal {F}(\tilde {\Delta }')$ making the following diagram commutative:

for some equivalence of categories $L$. The first application of uniqueness of covers goes back to [Reference Rouquier49]. Split quasi-hereditary covers with higher values of Hemmer–Nakano dimension associated with them are essentially unique. In fact, this is due to the following result which can be found in [Reference Cruz9, Corollary 4.3.6].

For a more detailed exposition on cover theory and Hemmer–Nakano dimensions we refer to [Reference Cruz9, Reference Cruz10].

3.1. Outline of the proof

The proof will be divided in two parts: $\leq$, $\geq$. The easier part is $\geq$, that is, to prove that:

\begin{align*}\notag\\ Q\!\operatorname{-domdim}_A A & \geq 2\cdot Q\!\operatorname{-domdim}_A T \text{ or equivalently } \notag\\ Q\!\operatorname{-codomdim}_A DA& \geq 2\cdot Q\!\operatorname{-codomdim}_A T, \end{align*}

the latter being a particular case of lemma 3.4. For this inequality, there are two main ideas. First, we construct (using Ringel duality) an exact sequence:

(3.2)\begin{equation} 0\rightarrow \overline{C}\rightarrow \overline{X_n}\rightarrow \cdots\rightarrow \overline{X_1}\rightarrow DA\rightarrow 0 \end{equation}

with the following properties:

• • $n=Q\!\operatorname {-domdim}_A T$;

• • $\overline {C}\in \mathcal {F}(\nabla )$ and $\overline {X_i}\in \!\operatorname {add} Q$.

Hence, (3.2) remains exact under $\operatorname {Hom}_A(T, -)$ and under $\operatorname {Hom}_A(Q, -)$. Second, we use the formula:

\[ Q\!\operatorname{-codomdim}_A T= \inf_{M\in \mathcal{F}(\nabla)} Q\!\operatorname{-codomdim}_A M. \]

On the other hand, to establish the part $\leq$, that is, to prove that $Q\!\operatorname {-codomdim}_A DA$ cannot be larger than $2Q\!\operatorname {-codomdim}_A T=2n$, there are two key steps:

1. (1) From any exact sequence $\overline {X_{n+1}}\rightarrow \overline {X_n}\rightarrow \cdots \rightarrow \overline {X_1}\rightarrow DA\rightarrow 0$ that gives ${Q\!\operatorname {-codomdim}_A DA\geq n+1}$ we can infer that the kernel of $\overline {X_{n+1}}\rightarrow \overline {X_n}$, denoted here by $E$ (with $\overline {X_0}:=DA$ if $n=0$) satisfies $\operatorname {Ext}_A^{i\geq n+2}(X, E)=0$ for every $X\in {}^\perp Q$;

2. (2) Proposition 2.9 provides a method to use the simple preserving duality functor to construct a module $X\in {}^\perp Q$ that satisfies $\operatorname {Ext}_A^{n+2}(X, E)\neq 0$ if $Q\!\operatorname {-codomdim}_A DA>2n$.

3.2. Relative injective dimension

Before we start with the proof of theorem A we need to discuss the concept of relative injective dimension.

Definition 3.2 Let $\mathcal {A}$ be a full subcategory of $A\!\operatorname {-mod}$. We define the $\mathcal {A}$-injective dimension of $N\in A\!\operatorname {-mod}$ (or the relative injective dimension of $N$ with respect to $\mathcal {A}$) as the value:

(3.3)\begin{equation} \inf\{n\in \mathbb{N}\cup \{0\}\colon \operatorname{Ext}_A^{i>n}(M, N)=0, \forall M\in \mathcal{A} \}. \end{equation}

We denote by $\operatorname {idim}_\mathcal {A} N$ the $\mathcal {A}$-injective dimension of $N$. Analogously, we define the $\mathcal {A}$-projective dimension of $N\in A\!\operatorname {-mod}$ as the value:

(3.4)\begin{equation} \inf\{n\in \mathbb{N}\cup \{0\}\colon \operatorname{Ext}_A^{i>n}(N, M)=0, \forall M\in \mathcal{A} \}. \end{equation}

3.3. Computing relative dominant dimension of the regular module using a characteristic tilting module

Let $T$ be a characteristic tilting module of a split quasi-hereditary algebra $A$ and let $Q$ be a partial tilting module. By proposition 2.6(ii)–(iii), the relative codominant dimension of $T$ with respect to $Q$ is a lower bound to the relative dominant dimension of $A$ with respect to $Q$. In the following we will see that this lower bound can be sharpened.

Lemma 3.4 Let $(A, \{\Delta (\lambda )_{\lambda \in \Lambda }\})$ be a split quasi-hereditary $R$-algebra with a characteristic tilting module $T$. Denote by $R(A)$ the Ringel dual $\operatorname {End}_A(T)^{op}$ of $A$. Suppose that $Q\in \!\operatorname {add} T$. Then,

(3.5)\begin{equation} Q\!\operatorname{-domdim} (A, R) \geq Q\!\operatorname{-domdim}_{(A, R)} T + Q\!\operatorname{-codomdim}_{(A, R)} T. \end{equation}

Proof. Observe that $DQ\otimes _A Q\in R\!\operatorname {-proj}$ (see e.g. [Reference Cruz9, A.4.3]). By proposition 2.6(ii) and (iii), we obtain that:

(3.6)\begin{equation} Q\!\operatorname{-domdim}_{(A, R)} A=Q\!\operatorname{-codomdim}_{(A, R)} DA\geq Q\!\operatorname{-codomdim}_{(A, R)} T.\end{equation}

If $Q\!\operatorname {-domdim}_A T=0$, then there is nothing more to prove. Assume that $n:= {Q\!\operatorname {-domdim}_{(A, R)} T}\geq 1$. By proposition 2.6(i), $\operatorname {Hom}_A(T, Q)\!\operatorname {-codomdim}_{R(A)} DT=n$. That is, there exists an exact sequence:

(3.7)\begin{equation} 0\rightarrow C\rightarrow X_n\rightarrow \cdots \rightarrow X_1\rightarrow DT\rightarrow 0 \end{equation}

with all $X_i\in \!\operatorname {add} (\operatorname {Hom}_A(T, Q))$, and so they are projective modules over $R(A)$. The modules $DT$ and $X_i$ are in $\mathcal {F}(\tilde {\Delta }_{R(A)})$, which is closed under kernels of epimorphisms and hence also $C$ is in $\mathcal {F}(\tilde {\Delta }_{R(A)})$.

By the Ringel dual equivalence described in proposition 2.7, the functor $T\otimes _{R(A)} -$ takes (3.7) to the exact sequence:

(3.8)\begin{equation} 0\rightarrow \overline{C}\rightarrow \overline{X_n}\rightarrow \cdots\rightarrow \overline{X_1}\rightarrow T\otimes_{R(A)} DT\rightarrow 0, \end{equation}

and $\overline {X_i}\in \!\operatorname {add} Q$ since $T\otimes _{R(A)} \operatorname {Hom}_A(T, Q)=\!\operatorname {add} Q$. Moreover,

\[ T\otimes_{R(A)} DT\simeq T\otimes_{R(A)} \operatorname{Hom}_A(T, DA)\simeq DA \]

and $\overline {C}\in \mathcal {F}(\tilde {\nabla })$. Thus, the exact sequence (3.8) remains exact under $\operatorname {Hom}_A(Q, -)$. By the dual version of corollary 2.5, we obtain that

\[ {Q\!\operatorname{-codomdim}_{(A, R)} DA}=n+Q\!\operatorname{-codomdim}_{(A, R)} \overline{C}. \]

Applying again proposition 2.6(iii) gives that ${Q\!\operatorname {-codomdim}_{(A, R)} \overline {C}}\geq Q\!\operatorname {-codomdim}_{(A, R)} T.$

Surprisingly, theorem 3.1 generalizes [Reference Fang and Koenig29, Theorem 4.3] without using any techniques on symmetric algebras. In particular, for the following proof we do not need to require $A$ to be an endomorphism algebra of a generator over a symmetric algebra.

Proof of theorem 3.1 By proposition 2.6(ii), proposition 2.8, and lemma 3.4, part $\geq$ follows. So, it remains to show that

\[Q\!\operatorname {-codomdim}_A DA\leq 2 \cdot Q\!\operatorname {-codomdim}_A T.\]

If $Q\!\operatorname {-codomdim}_A T=+\infty$, then there is nothing to prove. Denote by $n$ the value $Q\!\operatorname {-domdim}_A T=Q\!\operatorname {-codomdim}_A T$. Assume first that $n>0$. So, we can consider again exact sequences of the form (3.7) and (3.8). Assume, for a contradiction, that $Q\!\operatorname {-codomdim}_A DA>2n$. Hence, also $Q\!\operatorname {-codomdim}_A \overline {C}>n$ according to the dual of corollary 2.5. So, there exists an exact sequence:

(3.9)\begin{equation} 0\rightarrow L\rightarrow \overline{X_{2n+1}}\rightarrow \overline{X_{2n}}\rightarrow \cdots\rightarrow \overline{X_{n+1}}\rightarrow \overline{C}\rightarrow 0 \end{equation}

which remains exact under $\operatorname {Hom}_A(Q, -)$ and $\overline {X_i}\in \!\operatorname {add} Q$, $i=n+1, \ldots, 2n+1$. In particular, $0\rightarrow L\rightarrow \overline {X_{2n+1}}\rightarrow \cdots \rightarrow \overline {X_1}\rightarrow DA\rightarrow 0$ is an $\operatorname {Hom}_A(Q, -)$-acyclic coresolution of $L$, so it can be used to compute $\operatorname {Ext}_A^i(Q, L)$ for all $i$. Since it remains exact under $\operatorname {Hom}_A(Q, -)$ we obtain that $\operatorname {Ext}_A^{i>0}(Q, L)=0$ and so $L\in Q^\perp$ and ${}^\diamond \! L\in {}^\perp Q$. Let $E$ be the kernel of the map $\overline {X_{n+1}}\rightarrow \overline {C}$ and consider the exact sequence:

(3.10)\begin{equation} 0\rightarrow E\rightarrow \overline{X_{n+1}}\rightarrow \overline{X_n}\rightarrow \cdots\rightarrow \overline{X_1}\rightarrow DA\rightarrow 0. \end{equation}

By lemma 3.3(2), $\operatorname {idim}_{{}^\perp Q} E\leq n+1$. Since (3.10) remains exact under $\operatorname {Hom}_A(Q, -)$ we have that $E\in Q^{\perp }$. On the other hand, observe that $E$ cannot belong to $\mathcal {F}(\nabla )$ because otherwise (3.10) would remain exact under $\operatorname {Hom}_A(T, -)$ yielding that $n<\operatorname {Hom}_A(T, Q)\!\operatorname {-codomdim}_{R(A)} DT$ contradicting the definition of $n$.

So, the exact sequence $0\rightarrow E\rightarrow \overline {X_{n+1}}\rightarrow \overline {C}\rightarrow 0$ yields that $\dim _{\mathcal {F}(\nabla )}E=1$. Hence, $\dim _{\mathcal {F}(\Delta )}{}^\diamond \! E=1$. By proposition 2.9, we obtain that $0\neq \operatorname {Ext}_A^2({}^\diamond \! E, {}^\diamond \! {}^\diamond \! E)\simeq \operatorname {Ext}_A^2({}^\diamond \! E, E)$. By lemma 3.3(3) on the exact sequence $0\rightarrow {}^\diamond \! E\rightarrow {}^\diamond \! \overline {X_{n+2}}\rightarrow \cdots \rightarrow {}^\diamond \! \overline {X_{2n+1}}\rightarrow {}^\diamond \! L\rightarrow 0$ we obtain $0\neq \operatorname {Ext}_A^2({}^\diamond \! E, E)\simeq \operatorname {Ext}_A^{n+2}({}^\diamond \! L, E)$. This contradicts $\operatorname {idim}_{{}^\perp Q} E$ being at most $n+1$. We will now treat the case $n=0$. Assume, for the sake of contradiction, that $Q\!\operatorname {-codomdim}_A DA\geq 1$, then there exists an exact sequence $0\rightarrow L\rightarrow X_1\rightarrow DA\rightarrow 0$ which remains exact under $\operatorname {Hom}_A(Q, -)$ and $X_1\in \!\operatorname {add} Q$. Hence, $L\in Q^{\perp }$, ${}^\diamond \! L\in {}^\perp Q$, $\dim _{\mathcal {F}(\nabla )}(L)\leq 1$ and the ${}^\perp Q$-injective dimension of $L$ is at most one. In particular, $\operatorname {Ext}_A^2({}^\diamond \! L, L)= 0$. By proposition 2.9, we must have that $L\in \mathcal {F}(\nabla )$. But, then applying $\operatorname {Hom}_A(T, -)$ to $0\rightarrow L\rightarrow X_1\rightarrow DA\rightarrow 0$ yields that $\operatorname {Hom}_A(T, Q)\!\operatorname {-codomdim}_{R(A)} \operatorname {Hom}_A(T, DA)\geq 1$ which, in turn, implies that $n=Q\!\operatorname {-domdim} T\geq 1$ by proposition 2.6(i).

The following gives a positive answer to the Conjecture 6.2.4 of [Reference Cruz8], which is a special case by taking $A$ to be a Schur algebra and $Q$ the tensor space $V^{\otimes d}$.

Proposition 3.6 Let $(A, \{\Delta (\lambda )_{\lambda \in \Lambda }\})$ be a split quasi-hereditary algebra over a field $k$ with a simple preserving duality. Let $T$ be a characteristic tilting module of $A$. Assume that $Q\in \!\operatorname {add} T$ satisfies $Q\!\operatorname {-domdim}_A A\geq 2$. Then,

(3.11)\begin{equation} Q\!\operatorname{-domdim}_A A=2\cdot Q\!\operatorname{-domdim}_A T\geq 2\operatorname{domdim}_A T=\operatorname{domdim} A. \end{equation}

Proof. Let $P$ be a faithful projective–injective module over $A$. By assumption, $Q\!\operatorname {-domdim}_A A\geq 2$, so by corollary 2.3 it follows that $Q\!\operatorname {-domdim}_A P\geq 2$. Since $P$ is injective we must have that $P\in \!\operatorname {add} Q$. Hence, $\operatorname {Hom}_A(T, P)\in \!\operatorname {add} \operatorname {Hom}_A(T, Q)$.

By proposition 2.6(i),

(3.12)\begin{align} Q\!\operatorname{-domdim}_A T& =\operatorname{Hom}_{R(A)}(T, Q)\!\operatorname{-codomdim}_{R(A)} DT\notag\\ & \geq \operatorname{Hom}_A(T, P)\!\operatorname{-codomdim}_{R(A)} DT \nonumber\\ & =P\!\operatorname{-domdim}_A T=\operatorname{domdim}_A T. \end{align}

Applying theorem 3.1 to the partial tilting modules $P$ and $Q$, the result follows.

Example 3.7 We illustrate the proof of theorem A, using the Schur algebra $S(2, 4)$ over a field of characteristic $2$. Its basic algebra $A$ is isomorphic to $KQ/I$ where $Q$ is the quiver

and $I = \langle \beta \alpha, \gamma \delta, (\alpha \beta )\delta, \gamma (\alpha \beta )\rangle$ (see [Reference Erdmann26, 5.6]). (Here, we label vertices consistent with later sections.) The partial order is $4> 2 > 0$, all standard modules are uniserial, $\Delta (2)$ has composition factors labelled $2, 0$ and $\Delta (4)$ has composition factors labelled $4, 0, 2$. Then, $T(4)$ and $T(2)$ are uniserial, with

\[ 0\to \Delta(4)\to T(4)\to \Delta(2)\to 0,\quad 0\to \Delta(2)\to T(2) \to \Delta(0)\to 0 \]

We take $Q= T(4)\oplus T(2)$, then the characteristic tilting module is $Q\oplus T(0)$ and $T(0)$ is the simple module $L(0)$.

Consider the sequences in the proof of lemma 3.4. First, sequence (3.7) is the start of a minimal projective resolution of $DT$ as a module for $R(A)$. We apply $T\otimes _{R(A)}(-)$ which gives sequence (3.8). We know that it is a resolution of $D(A)$ with terms in ${\rm add}(Q)$ which is exact under ${\rm Hom}(Q, -)$, and from this information we see that it is

\[ 0 \to \overline{C} \simeq T (0)^2 \to \overline{X_2} \to \overline{X_1} \to D(A) \to 0 \]

(where $\overline {X_1}= T(4)^3\oplus T(2)$ and $\overline {X_2}\simeq T(2)^2$). This shows, by the dual of corollary 2.5, that

\[ Q\!\operatorname{-codomdim}_A D(A) = 2 + Q\!\operatorname{-codomdim}_A T(0). \]

In particular, $\operatorname {Hom}_A(T, Q)\!\operatorname {-codomdim}_{R(A)} DT=2$. By proposition 2.6(i), $n=Q\!\operatorname {-domdim}_A T=2$. In this case, $\overline {C}$ is already too simple and we know its relative codominant dimension using the simple preserving duality, thus this is enough already to deduce that $Q\!\operatorname {-codomdim}_A DA=4$.

Assume, for the moment, that at this point we did not know the value of $Q\!\operatorname {-codomdim}_A \overline {C}$. We consider now the proof of theorem 3.1. The exact sequence (3.10) is precisely:

\[ 0\rightarrow \Delta(2)\rightarrow T(2)\rightarrow \overline{X_2}\rightarrow \overline{X_1}\rightarrow DA\rightarrow 0, \]

where the map $T(2)\rightarrow \overline {X_2}$ factors through $T(0)$. Hence, $E= \Delta (2)$. Step (1), that is, (3.10) together with lemma 3.3(2) gives that $\operatorname {idim}_{{}^\perp Q} \Delta (2)\leq 3.$ Indeed, the (absolute) injective dimension $\operatorname {idim}_A \Delta (2)$ is exactly $3$, so $\operatorname {idim}_{{}^\perp Q} \Delta (2)$ must be at most 3. By the proof of theorem 3.1, if $Q\!\operatorname {-codomdim}_A DA>2\cdot 2=4$, then proposition 2.9 through the object ${}^\diamond \! L$ would give that $\operatorname {idim}_{{}^\perp Q} \Delta (2)>3$ which does not happen.

4.1. On the quasi-hereditary structure of $S(2, d)$

Let $S=S(2, d)$, and let $e=\xi _{(d)}$ be the idempotent corresponding to the largest weight (in the notation of [Reference Green33, 3.2]). Then, $SeS$ is an idempotent heredity ideal and $S/SeS$ is isomorphic to $S(2, d-2)$ (for details see e.g. [Reference Erdmann26, 1.3]). Since $SeS$ is a heredity ideal corresponding to $(d)$, factoring it out is compatible with the quasi-hereditary structure. Furthermore, as it is proved in the appendix of [Reference Dlab and Ringel22] computing ${\rm Ext}^i$'s for $S/SeS$-modules is the same whether in $S$ or in $S/SeS$. In particular, $S(2, d)\!\operatorname {-mod}$ is the full subcategory of $S(2, d+2)\!\operatorname {-mod}$ consisting of modules whose composition factors are different from those appearing in the top of $Se$.

We work mostly with the restrictions of simple modules, (co)standard modules and tilting modules to $SL(2, K)$. Recall that $L(\lambda )$ and $L(\mu )$ are isomorphic as $SL(2, K)$-modules if and only if they can be regarded both as $S(2, d)$-modules for some large $d$ and they are isomorphic as $S(2, d)$-modules. This fact can be seen using the canonical surjective map of $KSL(2, K)$ onto $S(2, d)$. Since every partition $(\lambda _1, \lambda _2)$ of $d$ in at most two parts is completely determined by the value $\lambda _1-\lambda _2$, it follows that $L(\lambda )$ and $L(\mu )$ are isomorphic as $SL(2, K)$-modules if and only if $\lambda _1-\lambda _2 = \mu _1 - \mu _2$. Similarly, we may label the standard modules and tilting modules.

We therefore label these modules by $m =\lambda _1-\lambda _2$ if $\lambda = (\lambda _1, \lambda _2)$ (such labellings can also be found e.g. in [Reference Erdmann and Law24, Subsection 3.2]). This means that we consider Schur algebras $S=S(2,d)$, allowing degrees to vary but keeping the parity. We make the convention that we view tacitly modules for $S(2, d')$ with $d'\leq d$ of the same parity as modules for $S(2, d)$. We say that such a degree $d'$ is admissible for the module defined in degree $d$. With this, the weights labelling the simple modules for $S(2, d)$ are precisely all non-negative integers $m\leq d$ of the same parity. The dominance order when $p=2$ and the degree is even, is the linear order.

The tilting module $T(0)$ is simple, it is the trivial module for $SL(2, K)$. As a building block, the tilting module $T(1)$ appears, which is isomorphic to the natural $SL(2, K)$-module $V$. Furthermore, $T(2) \cong V^{\otimes 2}$. For $d\geq 4$ we have that $V^{\otimes d}$ is the direct sum of $T(k)$ where all $T(k)$ occur for $k$ of the same parity of $d$, except that $T(0)$ does not occur when $d$ is even. (See e.g. [Reference Erdmann27, 4.2], it is implicitly in [Reference Donkin17, 3.4].)

4.2. The category $\mathcal {F}(\Delta )$ and projective modules

Non-split extensions of standard modules satisfy a directedness property, that is:

\[ \operatorname{Ext}_S^1(\Delta(r), \Delta(s))\neq 0\quad \text{implies}\quad r< s. \]

This has the following immediate consequence:

Of main interest for us are the indecomposable projective modules. Let $P_d(m)$ denote the indecomposable projective of $S(2, d)$ with simple quotient $L(m)$. Recall $P_d(m)$ has a $\Delta$-filtration, and that the filtration multiplicities $[P_d(m):\Delta (w)]$ are the same as the decomposition numbers. That is,

\[ [P_d(m): \Delta(w)] = (\nabla(w):L(m)) = (\Delta(w):L(m)). \]

where we write $(M:L(m))$ for the multiplicity of $L(m)$ as a composition factor of the module $M$. Note this also shows that projective modules depend on the degree $d$. In this case, decomposition numbers are always $0$ or $1$, see [Reference Henke34, Prop. 2.2, Theorem 3.2]. We give an example in figure 1.

Figure 1. Decomposition matrix for $S(2, 46)$ for $p=2$. The $(m,n)$-entry denotes $(\Delta (m):L(n))$, the column label is the same as the row label.

It follows that either $P_{d}(m) \cong P_{d-2}(m)$ as a module for $S(2, d)$, or else there is a non-split exact sequence:

(4.2)\begin{equation} 0\to \Delta(d) \to P_{d}(m) \to P_{d-2}(m)\to 0. \end{equation}

Namely, the top of $P_{d-2}(m)$ is $L(m)$, so there is a surjective homomorphism from $P_{d}(m)$ onto $P_{d-2}(m)$. Recall that $\mathcal {F}(\Delta )$ is closed under kernels of epimorphisms. By the filtration property in lemma 4.2 if this is not an isomorphism, then its kernel is a direct sum of copies of $\Delta (d)$ and there is only one since the decomposition numbers are $\leq \! 1$.

4.3. Twisted tensor product methods

Let $(-)^F$ denote the Frobenius twist (see [Reference Donkin20, p. 64]), this is an exact functor. In our setting, that is for even characteristic, we have the following tools, due to [Reference Donkin17]. Odd degrees when $p=2$ are less important. Namely, each block of $S(2, d)$ for $d$ odd is Morita equivalent to some block of some Schur algebra $S(2, x)$ with $x=(d-1)/2$ via the functor $\Delta (1)\otimes (-)^F$, see for example [Reference Erdmann and Henke23, Lemma 1] or [Reference Donkin18, Section 4, Theorem].

1. (1)

   1. (a) Let $m=2t$. There is an exact sequence of $S$-modules:

      \[ 0\to \Delta(t-1)^F\to \Delta(m)\to \Delta(t)^F\to 0 \]
      Taking contravariant duals gives the analogue for costandard modules.

   2. (b) Let $m=2t+1$, then $\Delta (m)\cong L(1)\otimes \Delta (t)^F$.

   (See e.g. [Reference Cox6, Prop. 3.3]).

We note that this determines recursively the decomposition numbers, as input using that $\Delta (t)$ is simple and isomorphic to $L(t)$ for $t=0, 1$, recall $p=2$. This can also be used to show that when $p=2$ and $d$ is even, the algebra $S(2, d)$ is indecomposable. Further, this also implies, by induction, that the decomposition numbers are always $0$ and $1$ when $p=2$ and $d$ is even.

1. (2) We have a complete description of the indecomposable tilting modules in this case. We have already described $T(m)$ for $m\leq 2$. The following is due to Donkin, see [Reference Donkin17, Example 2, p. 47].

Proposition 4.3 Let $m=2s$ and $m\geq 2$, then

\[ T(m)\cong T(2)\otimes T(s-1)^F \]

If $m=2s+1$, then $T(r) \cong T(1) \otimes T(s)^F$.

This describes recursively all indecomposable tilting modules.

The following shows that filtration multiplicities $[T(m):\Delta (w)]$ are $\leq 1$.

Proposition 4.4 The $\Delta$-filtration multiplicities of indecomposable tilting modules in even degree can be computed recursively from:

\[ 0\to \Delta(2t+2) \to T(2)\otimes \Delta(t)^F \to \Delta(2t) \to 0. \]

To prove this, one may specialize [Reference Cox6, Prop. 3.4].

We will see below that modules $T(2)\otimes X^F$ for $X$ in $\mathcal {F}(\Delta )$ have infinite relative dominant dimension with respect to $V^{\otimes d}$. This means that we can use lemma 2.2 (from Lemma 3.1.7 of [Reference Cruz10]) to relate the relative $V^{\otimes d}$-dominant dimension of the end terms, and this suggests a route towards the proof of our second main result.

We define a twisted filtration of a module $M\in \mathcal {F}(\Delta )$ to be a filtration where each quotient is isomorphic to $T(2)\otimes \Delta (t)^F$ for some $t$.

Lemma 4.5 Let $m=2s\geq 1$. Then, the tilting module $T(m)$ has a twisted filtration

\[ 0= M_k \subset M_{k-1} \subset \cdots \subset M_1\subset M_0=T(m) \]

with $M_{i-1}/M_i \cong T(2)\otimes \Delta (s_i)^F$, with quotients $\Delta (2s_i)$ and $\Delta (2s_i+2)$, for $s_1 < s_2<\cdots < s_k$.

Proof. We have $T(m)\cong T(2)\otimes T(s-1)^F$. The module $T(s-1)$ has a $\Delta$-filtration

\[ N_k=0\subset N_{k-1} \subset \ldots \subset N_0=T(s-1) \]

with $N_{i-1}/N_i \cong \Delta (s_i)$ and such that $s_1 < s_2 < \cdots < s_k$, by lemma 4.2. Applying the exact functor $T(2)\otimes (-)^F$ gives the claim.

5.1. The characteristic two case

Lemma 5.1 leaves us to consider $p=2$ and $d$ even $(\neq 0$). In this case, as mentioned above, the components of $V^{\otimes d}$ are the $T(m)$ with $m\neq 0$. The standard sequence (4.1) is an $\!\operatorname {add} T$-approximation, this follows from a special case of proposition 2.4. In particular,

(5.1)\begin{equation} V^{{\otimes} d}\!\operatorname{-domdim}_S \Delta(d) = 1 + V^{{\otimes} d}\!\operatorname{-domdim}_S X(d). \end{equation}

Theorem 5.2 Let $d=2s> 0$. We have

\[V^{\otimes d}\!\operatorname {-domdim}_S (\Delta ({d})) = s + V^{\otimes d}\!\operatorname {-} \operatorname {domdim}_S (T(0)). \]

To prove this, we will use lemma 2.2 on extensions of $\Delta (t)$ by $\Delta (2+t)$.

The cases $d=2$ and $d=4$ are easy.

1. (1) For $d=2$ we have the exact sequence $0\to \Delta (2) \to T(2) \to \Delta (0) = T(0)\to 0$, which proves the statement of the theorem by corollary 2.5.

2. (2) Let $d=4$, we have the exact sequence $0\to \Delta (4) \to T(4) \to \Delta (2)\to 0$. Splicing this with the sequence for $\Delta (2)$ gives the claim.

Degrees $d\geq 6$ need more work. The main ingredient is the observation that subquotients of the form $T(2)\otimes N^F$ with $N\in \mathcal {F}(\Delta )$ are not relevant for a minimal $V^{\otimes d}$-approximation.

Proof. By lemma 2.2 it suffices to prove this when $X=\Delta (s)$. We proceed by induction on $s$. When $s=0$ or $s=1$ we see that $T(2)\otimes \Delta (s)^F$ is a summand of $V^{\otimes d}$. For the inductive step consider the exact sequence:

\[ 0\to T(2)\otimes \Delta(s)^F \to T(2)\otimes T(s)^F \to T(2)\otimes X(s)^F \to 0. \]

The middle term is isomorphic to $T(2 + 2s)$. Since $X(s)$ has a filtration with quotients $\Delta (t)$ for $t< s$ it follows by induction (and lemma 2.2) that the module $T(2)\otimes X(s)^F$ has infinite $V^{\otimes d}$-dominant dimension for any even degree $d\geq 2s+2$. We deduce that the module ${T(2)\otimes \Delta (s)^F}$ has infinite $V^{\otimes d}$-dominant dimension as well.

Proof of theorem 5.2 Assume $d=2s\geq 4$. By proposition 4.4, there exists $S(2, d)$-exact sequences

(5.2)\begin{equation} 0\rightarrow \Delta(2t+2)\rightarrow T(2)\otimes \Delta(t)^F\rightarrow \Delta(2t)\rightarrow 0, \end{equation}

for every $0\leq t\leq s-1$. Moreover, they remain exact over $\operatorname {Hom}_{S(2, d)}(-, V^{\otimes d})$ since $V^{\otimes d}$ is a partial tilting module and so $\operatorname {Ext}_{S(2,d)}^1(\Delta (2t+2), V^{\otimes d})=0$. By lemma 5.3, $T(2)\otimes \Delta (t)^F$ has infinite relative dominant dimension with respect to $V^{\otimes d}$ for $0\leq t\leq s-1$. By lemma 2.2,

(5.3)\begin{align} V^{{\otimes} d}\!\operatorname{-domdim}_{S(2, d)} \Delta(2t+2)=1+V^{{\otimes} d}\!\operatorname{-domdim}_{S(2, d)} \Delta(2t), \quad 0\leq t\leq s-1. \end{align}

Hence, $V^{\otimes d}\!\operatorname {-domdim}_{S(2, d)} \Delta (0)=s+V^{\otimes d}\!\operatorname {-domdim}_{S(2, d)} \Delta (d)$.

We will now determine the relative dominant dimension of $S(2,d)$ with respect to $V^{\otimes d}$. Let $P_d(m)$ be the indecomposable projective $S(2, d)$-module with homomorphic image $L(m)$. Throughout, $d$ and $m$ are even.

Corollary 5.6 With the setting as in lemma 5.4,

if (a) occurs, then $V^{\otimes d}\!\operatorname {-domdim}_S P_d(m) = +\infty$.

If (b) occurs, then for $d=2s$ we have $V^{\otimes d}\!\operatorname {-domdim}_S P_d(m) = V^{\otimes d}\!\operatorname {-domdim}_S \Delta (d).$ In particular,

\[ V^{{\otimes} d}\!\operatorname{-domdim}_S S(2, d) = V^{{\otimes} d}\!\operatorname{-domdim}_S \Delta(d) = (d/2) + V^{{\otimes} d}\!\operatorname{-domdim}_ST(0). \]

This completes the proof of theorem B for algebraically closed fields. By [Reference Cruz10, Lemma 3.2.3], the result also holds over arbitrary fields.

5.2. The quantum case

For the quantum case, it is enough to take $S=S_{K, q}(2, d)$ where $q+1=0$. In this case, everything is exactly the same as over $S(2, d)$ when $\operatorname {char} K=2$. Namely, we may take $S_{K, q}(2, d)$ as $A_q(2, d)^*$ as it is done in [Reference Cox6, Reference Donkin20], and also in [Reference Dipper and Donkin12, Reference Donkin19]. The definition of $A_q(2)$ may be found in [Reference Dipper and Donkin12, p. 16]. This means that one takes the quantum group $G(2)$ as defined in [Reference Dipper and Donkin12] instead of $SL(2, K)$.

As it is explained in [Reference Erdmann and Law24, Sections 3.1 and 3.2], we can use the same labelling for weights; in that paper the parameter $q$ is a primitive $\ell$-th root of $1$ and we only need $\ell =2$. We can regard $S_{K, q}(2, d)$ as a factor algebra of $S_{K, q}(2, d+2)$, using [Reference Donkin20, Section 4.2], and therefore regard modules in degree $d$ again as modules in degree $d'$ for $d'>d$ of the same parity.

There is a Frobenius morphism from the quantum group $G(2)$ to the classical setting, hence if $\Delta (m)$ (resp. $T(m)$) is a standard module (resp. a tilting module) for the classical setting, then $\Delta (m)^F$ (resp. $T(m)^F$) is a module for the quantum group, and so are the tensor products $T(2)\otimes \Delta (m)^F$ and $T(2)\otimes T(m)^F$ modules for the quantum group. The $q$-analogues of the exact sequences in § 4.3 and proposition 4.4 exist by [Reference Cox6, Prop. 3.3 and 3.4]. See also [Reference Erdmann and Law24, Proposition 3.1] (our situation of interest is recovered by fixing $l=2$ in their setup). The $q$-analogue of proposition 4.3 can be found in [Reference Donkin20, Section 3.4, p. 73, (8)].

We note that (1) of § 4.3 and the $q$-analogue imply, by induction, that all decomposition numbers are $0$ or $1$.

In [Reference Dipper and Donkin12], it is shown that this version of the $q$-Schur algebra is the same as our definition, as the endomorphism algebra of the action of the Iwahori–Hecke algebra on the tensor space $V^{\otimes d}$, see § 6. The definition of the Iwahori–Hecke algebra, as we take it is given in 6.2. In particular, we denote by $H$ the Iwahori–Hecke algebra. Their strategy in [Reference Dipper and Donkin12, Section 3] is to show that the action of the Iwahori–Hecke algebra on the tensor space is a comodule homomorphism (see 3.1.6 of [Reference Dipper and Donkin12]). The $H$-action in [Reference Dipper and Donkin12, 3.1.6] is not the same as ours, but it is explained in detail (see 4.4.3 of [Reference Dipper and Donkin12]) that the action we use also can be taken.

Hence, the arguments of § 5 remain valid in the quantum case and therefore, we obtain the following:

Theorem 5.8 Let $K$ be a field and fix $q=u^{-2}$ for some $u\in K$. Let $S$ be the $q$-Schur algebra $S_{K, q}(2, d)$ and $T$ be the characteristic tilting module of $S$. Then,

\[ V^{{\otimes} d}\!\operatorname{-domdim}_S S=2\cdot V^{{\otimes} d}\!\operatorname{-domdim}_S T=\begin{cases} d, & \text{ if } 1+q=0 \text{ and } d \text{ is even}, \\ + \infty, & \text{ otherwise .} \end{cases} \]

5.3. An application of theorem B

In [Reference Buan and Solberg3], Buan and Solberg characterize complements of almost complete (co) tilting modules. Let $Q$ be an $A$-module, then the faithful dimension of $Q$ as they define it, is equal to $Q\!\operatorname {-domdim}_A A$. A partial tilting module $Q$ is almost complete if it is not a tilting module, but there is an indecomposable module $X$ such that $Q\oplus X$ is a tilting module. In this case, call $X$ a complement of $Q$. Recall that a module $M$ is a tilting module over $A$ if and only if $DM$ is a cotilting module over $A^{op}$ and $M\!\operatorname {-domdim} A=DM\!\operatorname {-domdim}_{A^{op}}A$.

Hence, dualizing [Reference Buan and Solberg3, Theorem 3.6] and [Reference Buan and Solberg3, Proposition 3.2] of Buan and Solberg we have the following.

Assume $A=S(2, d)$ for char$(K)=2$ and $d$ even, then $Q=V^{\otimes d}$ is an almost complete tilting module. We have proved that $Q\!\operatorname {-domdim} A=d$, so there should be in general $d+1$ complements.

Example 5.12 Assume $A=S(2, 4)$ when $p=2$, we continue with the notation as in example 5.10. Consider $Q=V^{\otimes 4}$. We know $T(0)$ is a complement of $Q$. By computing minimal right ${\rm add}( Q)$-approximations, we get the following exact sequence:

(5.4)\begin{equation} 0\rightarrow P(0)\xrightarrow{f_2} T(2)\oplus T(4)\xrightarrow{f_1} T(2)\xrightarrow{f_0} T(0)\rightarrow 0.\end{equation}

Namely, the kernel of the surjection $f_0: T(2)\to \Delta (0)\cong T(0)$ is $\Delta (2)$. Take $f_1$ to be the map where the first component has image $L(0)$ and the second component has image $\Delta (2)$, this is a right ${\rm add}\, Q$ approximation, and its kernel is isomorphic to $P(0)$. It is minimal since $P(0)$ is indecomposable. A minimal right ${\rm add}\, Q$ approximation of $P(0)$ cannot be surjective since $P(0)$ is projective but not a direct summand of $Q$.

By (2) of theorem 5.11 on (5.4), we get that ${\rm ker} f_0 = \Delta (2)$ and $P(0)$ are complements of $Q$. Now, applying the simple preserving duality functor to (5.4) we get an exact sequence:

(5.5)\begin{equation} 0\rightarrow T(0)\rightarrow T(2)\rightarrow T(2)\oplus T(4)\rightarrow I(0),\end{equation}

and surjective minimal right approximations are sent to injective minimal left approximations. Thus, the complements of $Q$ are

\[ P(0), \ \Delta(2), \ T(0), \ \nabla(2), \ I(0)\rightarrow 0. \]

6.1. The classical case

We will start by considering the class of Temperley–Lieb algebras of the form $TL_{R, d}(-2)$ which can be viewed as quotients of group algebras of the symmetric group.

The following description of the kernel of $\Phi$ goes back to [Reference Jones42, p. 364].

Theorem 6.3 For each $i=1, 2, \ldots, d-2$ define

\[ x_i:= T_iT_{i+1}T_i - T_iT_{i+1} - T_{i+1}T_i + T_i + T_{i+1} -1 \in R\mathcal{S}_d. \]

Let $I$ be the ideal of $R\mathcal {S}_d$ generated by the $x_i$ for $1\leq i\leq d-2$. Then, there is an exact sequence

\[ 0\to I \to R\mathcal{S}_d \stackrel{\Phi}\to TL_{R,d}({-}2) \to 0 \]

Proof. One checks that $\Phi (x_i)=0$ for each $i=1, \ldots, d-2$. So, we have a commutative diagram:

where $\pi$ maps the image of $T_i$ in $RS_d/I$ to $\Phi (T_i) = U_i+1$, and $\iota$ is the inclusion map. Consider $\pi ': TL_{R, d}(-2) \to R\mathcal {S}_d/I$ defined by taking $U_i$ to the image of $T_i-1$ in $R\mathcal {S}_d/I$. One checks that $\pi '$ preserves the defining relations for $TL_{R, d}(-2)$, so that it is a well-defined map. Finally,

\[ \pi'(\pi(T_i + I)) = T_i+I,\quad \pi(\pi'(U_i)) = U_i. \]

Therefore, ${\rm ker}\, \Phi = I$.

It is nowadays widely known that Temperley–Lieb algebras can be viewed as the centralizer algebras of quantum groups $\mathfrak {sl}_2$ in the endomorphism algebra of a tensor power and this goes back to the work of Martin [Reference Martin44] and Jimbo [Reference Jimbo39]. Recall that over $R$, the Schur algebra $S_R(2, d)$ is defined as the endomorphism algebra $\operatorname {End}_{R\mathcal {S}_d}((R^2)^{\otimes d})$, where $(R^2)^{\otimes d}$ affords a right $R\mathcal {S}_d$-module structure via place permutation. In order to relate the Temperley–Lieb algebra to the Schur algebra, we need a suitable action of the Temperley–Lieb algebra on the tensor space. It is as follows.

Theorem 6.4 Let $V$ be a free $R$-module of rank $2$. Then, $V^{\otimes d}$ is a module over $\Lambda = TL_{R, d}(-2)$ where $U_i$ acts as ${\rm id}_V^{\otimes (i-1)}\otimes \tau \otimes {\rm id}_V^{\otimes (d-i-1)}$. Here, $\tau$ is the endomorphism of $V^{\otimes 2}$ defined by

\[ \tau(v_1\otimes v_2) = v_2\otimes v_1 - v_1\otimes v_2 \]

(for $v_1, v_2\in V$). Moreover, there is an algebra isomorphism

\[ \Lambda \to {\rm End}_{S_R(2, d)}(V^{{\otimes} d})^{op}. \]

Proof. We know that $R\mathcal {S}_d$ acts by place permutations on $V^{\otimes d}$ and we can view this as a right action. With this, $T_i- 1$ acts as $U_i$ on the space $V^{\otimes d}$. This shows that it factors through $\Lambda$. In particular, to show that $\Lambda \rightarrow {\rm End}_{S(2, d)}(V^{\otimes d})^{op}$ is surjective it is enough to check that the canonical map $RS_d\rightarrow {\rm End}_{S(2, d)}(V^{\otimes d})^{op}$ is surjective. But this follows from classical Schur–Weyl duality (see e.g. [Reference König, Slungård and Xi43]). In fact, this can be seen in the following way: let $K$ be a field, then the canonical map $K\mathcal {S}_d\rightarrow \operatorname {End}_{S(2, d)}(V^{\otimes d})^{op}$ fits in the following commutative diagram:

Here, $\psi$ is surjective because $\operatorname {domdim} S_K(d, d)\geq 2$ and $\phi$ is surjective by [Reference Erdmann27, 1.7] and [Reference Donkin20, 4.7] because $(K^d)^{\otimes d}$ is a projective–injective module over $S(d, d)$. Observe that $\operatorname {End}_{S_R(2, d)}(V^{\otimes d})^{op}\in R\!\operatorname {-proj}$ (see e.g. [Reference Cruz9, Proposition A.4.3, Corollary A.4.4]) and it has a base change property (see e.g. [Reference Cruz9, Corollary A.4.6]). In particular, $R(\mathfrak {m})\otimes _R \operatorname {End}_{S_R(2, d)}(V^{\otimes d})^{op}\simeq \operatorname {End}_{S_{R(\mathfrak {m})}(2, d)}(V^{\otimes d})^{op}$ for every maximal ideal $\mathfrak {m}$ of $R$. By the above discussion, the maps $R(\mathfrak {m})S_d\rightarrow \operatorname {End}_{S_{R(\mathfrak {m})}(2, d)}(V^{\otimes d})^{op}$ are surjective for every maximal ideal $\mathfrak {m}$ of $R$. Now, by Nakayama's lemma the map $RS_d\rightarrow \operatorname {End}_{S_R(2, d)}(V^{\otimes d})^{op}$ is surjective.

It remains to show that the action of $\Lambda$ is injective. Let $\sum _i a_iU_i \in \Lambda$ acting as zero on $V^{\otimes d}$. The action of $\sum _i a_iU_i$ in $y_k$, defined as the basis element $e_1\otimes \cdots \otimes e_1\otimes e_2\otimes e_1\otimes \cdots \otimes e_1$ where $e_2$ appears in position $k+1$, yields that $a_k=a_{k+1}=0$. This concludes the proof.