Quasi-hereditary covers of Temperley-Lieb algebras and relative dominant dimension

To make our results precise, we need further notation. In general, assume that $B$ is a finite-dimensional algebra over an algebraically closed field. A pair $(A,P)$ is a quasi-hereditary cover of $B$ if $A$ is a quasi-hereditary algebra, $P$ is a finitely generated projective $A$-module such that $B=\operatorname{End}_{A}(P)^{op}$, and in addition the restriction of the associated Schur functor $F:=\operatorname{Hom}_{A}(P,-)\colon A\!\operatorname{-mod}\rightarrow B\!\operatorname{-mod}$ to the subcategory of finitely generated projective $A$-modules is full and faithful.

Let $\mathcal{F}(\Delta)$ be the category of $A$-modules which have a filtration by standard modules. We would like the functor $F$ to be faithful on $\mathcal{F}(\Delta)$ and to induce isomorphisms

for $X,Y$ modules in $\mathcal{F}(\Delta)$. If this is the case for $0\leq j\leq i$ then $(A,P)$ is called an $i-\mathcal{F}(\Delta)$ cover of $B$. The largest $n$ such that $(A,P)$ is an $n-\mathcal{F}(\Delta)$ cover of $B$, is called the Hemmer-Nakano dimension of $\mathcal{F}(\Delta)$ in [26]. When $B$ is self-injective, Fang and Koenig showed that this dimension is controlled by the dominant dimension of a characteristic tilting module. In addition, they proved that if $B$ is a symmetric algebra and the quasi-hereditary cover admits a certain simple preserving duality, then the dominant dimension of a characteristic tilting module is exactly half of the dominant dimension of $A$.

Recently, in [8], the situation was generalised to include cases where $B$ is not necessarily self-injective. Moreover, it was proved in [8] that the Hemmer-Nakano dimension of $\mathcal{F}(\Delta)$ associated with a $0-\mathcal{F}(\Delta)$ cover can be determined using the relative codominant dimension of a characteristic tilting module with respect to a certain summand of the characteristic tilting module.

The concepts of relative dominant and relative codominant dimension (see the definition below in Subsection 2.3) and the concept of quasi-hereditary cover can be considered in an integral setup, that is, both of these concepts can be studied for Noetherian algebras which are finitely generated and projective as modules over a regular commutative Noetherian ring. In [7], methods were developed to reduce the computations of Hemmer-Nakano dimensions in the integral setup to computations of Hemmer-Nakano dimensions in the setup where the ground ring is an algebraically closed field. So, it will be enough for our purposes to concentrate our attention on the case when the coefficient ring is an algebraically closed field.

The new approach to construct quasi-hereditary covers is [8, Theorem 5.3.1.] and [8, Theorem 8.1.5] when applied to Schur algebras (and $q$-Schur algebras). The novelty is that it uses the Ringel dual of a Schur algebra, rather than a Schur algebra, and works for arbitrary parameters $n,d$.

This can, in particular, be applied to the study of Temperley-Lieb algebras. Indeed, the Temperley-Lieb algebras can be viewed as centraliser algebras of $S(2,d)$ in the endomorphism algebra of the tensor power $(K^{2})^{\otimes d}$ (over a field $K$) and their $q$-analogues. Here $S(n,d)$ can be regarded as the centraliser algebra of $\mathcal{S}_{d}$ in the endomorphism algebra of the tensor power $(K^{n})^{\otimes d}$ over, where $(K^{n})^{\otimes d}$ affords a module structure over $\mathcal{S}_{d}$ by place permutation. Furthermore, $V^{\otimes d}:=(K^{n})^{\otimes d}$ belongs in the additive closure of a characteristic tilting module over $S(n,d)$. Our cases of interest have a simple preserving duality, and in such a case, for this situation, we can without ambiguity interchange the concepts: relative dominant dimension and relative codominant dimension.

Denote by $Q\!\operatorname{-domdim}_{A}X$ the relative dominant dimension of an $A$-module $X$ with respect to $Q$. In this context, the following questions arise:

1. (1)

   What is the value of $V^{\otimes d}\!\operatorname{-domdim}_{S(2,d)}T$, where $T$ is a characteristic tilting module of the quasi-hereditary algebra $S(2,d)$? What happens to this value when we replace a Schur algebra by a $q$-Schur algebra?

2. (2)

   The Ringel duals of Schur algebras as well as Schur algebras have a simple preserving duality. Can we expect, like in the classical case (see [26, Theorem 4.3.]), the equality

   $$V^{\otimes d}\!\operatorname{-domdim}S(n,d)=2\cdot V^{\otimes d}\!\operatorname{-domdim}_{S(n,d)}T$$

   to hold in general?

3. (3)

   Can we expect the quasi-hereditary cover of the Temperley-Lieb algebra constructed in [8, Theorem 8.1.5] to be unique, in some meaningful way?

Our goal in this paper is to give answers to these three questions.

Surprisingly, the answer to (2) is positive without using extra structure on $S(n,d)$ besides the quasi-hereditary structure and the existence of a simple preserving duality.

This result generalises [26, Theorem 4.3.] and our methods give a new proof to their case without using any information on $A$ being gendo-symmetric, that is, an endomorphism algebra of a faithful module over a symmetric algebra. In particular, our result also works for dominant dimension exactly zero. Our approach exploits basic properties of relative injective dimensions, $\Delta$-filtration dimensions, some tools that were used to prove the main result of [40] and general properties connecting relative dominant dimensions with relative codominant dimensions with respect to a fixed module. Observe that the left hand side of the equation in Theorem A is exactly the faithful dimension of $Q$ in sense of [2]. This means that, under these conditions, if the faithful dimension of $Q$ is greater or equal to 4, then the faithful dimension controls the Hemmer-Nakano dimension of $\mathcal{F}(\Delta)$ associated with a quasi-hereditary cover of the endomorphism algebra of $Q$. Theorem A is applied to prove a more general case of Conjecture 6.2.4 of [6], that is, that the faithful dimension of a summand of a characteristic tilting module is an upper bound for the dominant dimension of the algebra provided that the former is greater or equal than two.

Combining techniques of Frobenius twisted tensor products with Theorem A we obtain a complete answer to (1):

The same approach can be used for the $q$-analogue. In this case, the algebra $A$ is the $q$-Schur algebra $S_{K,q}(2,d)$, which can be defined as the centraliser of the Hecke algebra $H_{q}(d)$ acting on $V^{\otimes d}$, again for $\dim V=2$ and in the theorem the characteristic is replaced by the quantum characteristic. When we have $v\in R$ such that $v^{2}=q$ and $\delta=-v-v^{-1}$, the Temperley-Lieb algebra $TL_{K,d}(\delta)$ is a quotient of this action. In both cases, the real difficulty lies in the case in which the characteristic (resp. quantum characteristic) is two.

From Theorem B and its $q$-analogue, it follows that the Temperley-Lieb algebra $TL_{K,q}(\delta)$ is quasi-hereditary, and, in fact, it is the Ringel dual of a $q$-Schur algebra if $\delta\neq 0$ or $d$ is odd. Otherwise, from Theorem B follows the value of Hemmer-Nakano dimension of $\mathcal{F}(\Delta)$ associated with the quasi-hereditary cover of $TL_{K,q}(0)$ formed by the Ringel dual of a $q$-Schur algebra (see [8, Theorem 8.1.5]). So, we have obtained a Hemmer-Nakano type result (see Corollary 6.2.4) now between the Ringel dual of a $q$-Schur algebra and the Temperley-Lieb algebra. In particular, this generalises [10, Theorem C (3), (4)] for Temperley-Lieb algebras. In addition, the full subcategory of costandard modules over a $q$-Schur algebra is equivalent to the full subcategory of cell modules of the Temperley-Lieb algebra whenever $d$ is greater or equal to 6.

If $d=2$, the Temperley-Lieb algebra is exactly an Iwahori-Hecke algebra, so nothing is new for this case. We obtain a positive answer to question (3) when we consider the Laurent polynomial ring over the integers as coefficient ring and $d>2$ (see Section 7 and Corollary 7.2.1). In such a case, the (integral) Schur functor $F$ induces an exact equivalence $\mathcal{F}(\tilde{\Delta})\rightarrow\mathcal{F}(F\tilde{\Delta}),$ where the first category denotes the subcategory of modules admitting a filtration by direct summands of direct sums of standard modules over the Ringel dual of an integral $q$-Schur algebra. The quasi-hereditary cover of the integral Temperley-Lieb algebra formed by the Ringel dual of a $q$-Schur algebra is the unique quasi-hereditary cover which induces this exact equivalence.

We emphasize that the specialisation of Theorem A to projective-injective modules played a keyrole to determine the dominant dimension of Schur algebras of the form $S(n,d)$ with $n\geq d$ in [26] (also their $q$-analogues [27]) and it also gives an easier method to determine the dominant dimension of the blocks of the BGG category $\mathcal{O}$. It is our expectation that its use will be crucial to determine, in particular, $V^{\otimes d}\!\operatorname{-domdim}S(n,d)$ and $\operatorname{domdim}S(n,d)$ also in the cases $2<n<d$ while the latter is also an open problem for $n=2$.

The article is organised as follows: In Section 2, we introduce the notation and the main properties of relative dominant dimension with respect to a module, split quasi-hereditary algebras with a simple preserving duality and cover theory to be used throughout the paper. In Section 3, we discuss elementary results on relative injective dimensions and we give the proof of Theorem A. We then deduce that the dominant dimension is a lower bound for the faithful dimension of a summand of a characteristic tilting module fixed by a simple preserving duality provided the latter is at least two (see Proposition 3.2.3). In Section 4, we collect results on the quasi-hereditary structure of Schur algebras $S(2,d)$, in particular, reduction techniques and how to construct partial tilting and standard modules inductively using the Frobenius twist functor. In Section 5, we compute the relative dominant dimension of $S(2,d)$ with respect to $V^{\otimes d}$ in terms of $V^{\otimes d}\!\operatorname{-domdim}_{S(2,d)}T$, where $T$ is a characteristic tilting module of $S(2,d)$. In particular, we give the proof of Theorem B and its $q$-analogue (see Theorem 5.2.2). In Section 6, we recall that all Temperley-Lieb algebras can be realised as the centraliser algebras of $q$-Schur algebras in the endomorphism algebra of the tensor power $V^{\otimes d}$. As a consequence, we determine the value of Hemmer-Nakano dimension of $\mathcal{F}(\Delta)$ in all cases associated to the cover of the Temperley-Lieb algebra formed by the Ringel dual of a $q$-Schur algebra. This computation is contained in Corollary 6.2.4. In Section 7, we determine the Hemmer-Nakano dimension of the above mentioned quasi-hereditary cover in the integral setup, dividing the study into two cases: the coefficient ring having or not a property of being $2$-partially $q$-divisible (see Subsection 7.1). When the coefficient ring does not have such property, we show that a quasi-hereditary cover with such coefficient ring has better properties. We conclude by addressing the problem of the uniqueness of this cover (see Subsection 7.2).

This follows [8]. 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 $A$-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 functor ${\rm Hom}_{R}(-,R):A\!\operatorname{-mod}\to A^{op}\!\operatorname{-mod}$ 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 $0\to M\stackrel{{\scriptstyle f}}{{\to}}N$.

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 $X_{0},X_{1},\cdots$ are $G$-acyclic. Given $X\in A\!\operatorname{-mod}\cap R\!\operatorname{-proj}$, we denote by $X^{\perp}$ the full subcategory

and by ${}^{\perp}X$ the full subcategory $\{M\in A\!\operatorname{-mod}\cap R\!\operatorname{-proj}\colon\operatorname{Ext}_{A}^{i>0}(M,Z)=0,\forall Z\in\!\operatorname{add}X\}$.

We recall definitions and some general properties relevant to 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

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.

We recall from [8] 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

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 $Q-{\rm codomdim}_{(A,R)}(X)$ 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

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.

The following gives a criterion towards finding $Q\!\operatorname{-domdim}_{(A,R)}M$ for a given module $M$ in $A\!\operatorname{-mod}\cap R\!\operatorname{-proj}$.

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

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 [8, Lemma 3.1.7], for convenience, we recall part of this.

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 [8, Proposition 3.1.11.].

The following application of Lemma 2.3.2 will be useful later.

For the definition and general properties of split quasi-hereditary algebras we refer to [5, 43, 9, 7, 8]. In particular, we follow the notation of [9, 7, 8]. 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 [43], 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 [8, Subsection 2.2.3].

Recall that $\operatorname{Hom}_{A}(T,DA)\simeq DT$ is a characteristic tilting module over $R(A)$.

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 minimal $n\geq 0$ such that there exists an exact sequence

with $M_{0},\ldots,M_{n}\in\mathcal{F}(\nabla)$. Analogously, the $\Delta$-filtration dimension is defined. The $\nabla$-filtration dimensions first appeared in [28] in the study of cohomology of algebraic groups.

$\nabla$ and $\Delta$-filtration dimensions play a crucial role in [23] and [40] 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.

The concept of a cover, and in particular, of a split quasi-hereditary cover was introduced in [43] 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 [7], 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 idea can be traced back to [26] which was later amplified in several directions in [7] and in [8]. This principle is briefly summarized in the following result proved in [8, 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 $(A^{\prime},P^{\prime})$ be an $(-1)-\mathcal{F}(\tilde{\Delta}^{\prime})$ (quasi-hereditary) cover of $B$. We say that $(A,P)$ is equivalent to $(A^{\prime},P^{\prime})$ as quasi-hereditary covers if there exists an equivalence functor $H\colon A\!\operatorname{-mod}\rightarrow A^{\prime}\!\operatorname{-mod}$ which restricts to an equivalence of categories between $\mathcal{F}(\tilde{\Delta})$ and $\mathcal{F}(\tilde{\Delta}^{\prime})$ making the following diagram commutative

for some equivalence of categories $L$. The first application of uniqueness of covers goes back to [43]. Split quasi-hereditary covers with higher values of Hemmer-Nakano dimension associated to them are essentially unique. In fact, this is due to the following result which can be found in [7, Corollary 4.3.6.].

For a more detailed exposition on cover theory and Hemmer-Nakano dimensions we refer to [7, 8].

The following concept of relative injective dimension will be useful as a tool in the proof of Theorem A.

In general, the relative codominant dimension of a characteristic tilting module with respect to a partial tilting module gives a lower bound to the relative dominant dimension of the regular module with respect to a partial tilting module (see [8, Theorem 5.3.1(a)]). In the following, we will see that this lower bound can be sharpened using also the relative dominant dimension of a characteristic tilting module with respect to a partial tilting module.

Surprisingly, the following result generalises [26, Theorem 4.3] without using any techniques on symmetric algebras. In particular, for this proof we do not use the fact that the endomorphism algebra of a faithful projective-injective module over a quasi-hereditary algebra with a simple preserving duality is a symmetric algebra.