The Happel functor and homologically well-graded Iwanaga-Gorenstein algebras

Abstract

Happel constructed a fully faithful functor $\mathcal{H}:{\mathsf{D}}^{\mathrm{b}}(\mathrm{mod}\mathrm{\Lambda })\to {\underset{\_}{\mathrm{mod}}}^{\mathbb{Z}}\mathrm{\text{T}}(\mathrm{\Lambda })$ for a finite dimensional algebra Λ. He also showed that this functor $\mathcal{H}$ gives an equivalence precisely when $\mathrm{gldim}\mathrm{\Lambda }<\infty$. Thus if $\mathcal{H}$ gives an equivalence, then it provides a canonical tilting object $\mathcal{H}(\mathrm{\Lambda })$ of ${\mathrm{mod}}^{\mathbb{Z}}\mathrm{\text{T}}(\mathrm{\Lambda })$.

In this paper we generalize the Happel functor $\mathcal{H}$ in the case where $\mathrm{\text{T}}(\mathrm{\Lambda })$ is replaced with a finitely graded IG-algebra A. We study when this functor is fully faithful or is an equivalence. For this purpose we introduce the notion of homologically well-graded (hwg) IG-algebra, which can be characterized as an algebra posses a homological symmetry which, a posteriori, guarantee that the algebra is IG. We prove that hwg IG-algebras is precisely the class of finitely graded IG-algebras that the Happel functor is fully faithful. We also identify the class that the Happel functor gives an equivalence. As a consequence of our result, we see that if $\mathcal{H}$ gives an equivalence, then it provides a canonical tilting object $\mathcal{H}(T)$ of ${\underset{\_}{\mathsf{CM}}}^{\mathbb{Z}}A$. For some special classes of finitely graded IG-algebras, our tilting objects $\mathcal{H}(T)$ coincide with tilting object constructed in previous works.

Introduction

A central theme in the representation theory of Iwanaga-Gorenstein (IG) algebra is the study of the stable category of Cohen-Macaulay (CM) modules. It was initiated by Auslander-Reiten [2], Happel [16] and Buchweitz [6], and has been studied by many researchers. The stable category of CM-modules $\underset{\_}{\mathsf{CM}}A$ has a canonical structure of triangulated category. It is equivalent to the singular derived category Sing A and is a triangulated category that is also important in algebraic geometry and mathematical physics. The situation is the same with graded IG-algebras and the stable category ${\underset{\_}{\mathsf{CM}}}^{\mathbb{Z}}A$ of graded CM-modules. Recently, tilting theory and cluster tilting theory of the stable categories $\underset{\_}{\mathsf{CM}}A$ and ${\underset{\_}{\mathsf{CM}}}^{\mathbb{Z}}A$ are extensively studied and has many interactions with other areas (see the excellent survey [18]).

As will be soon recalled, the original Happel functor $\mathcal{H}$ connects two important triangulated categories: the derived category ${\mathsf{D}}^{\mathrm{b}}(\mathrm{mod}\mathrm{\Lambda })$ and the stable category ${\underset{\_}{\mathrm{mod}}}^{\mathbb{Z}}\mathrm{\text{T}}(\mathrm{\Lambda })$, so served as a powerful tool to study these categories. From tilting theoretic point of view, the functor $\mathcal{H}$ provides the existence of a canonical tilting object $\mathcal{H}(\mathrm{\Lambda })$ in the stable category ${\underset{\_}{\mathrm{mod}}}^{\mathbb{Z}}\mathrm{\text{T}}(\mathrm{\Lambda })$.

In this paper and [28], we generalize the Happel functor $\mathcal{H}$ by replacing the self-injective algebra $\mathrm{\text{T}}(\mathrm{\Lambda })$ with a finitely graded IG-algebra $A={⨁}_{i=0}^{\ell }{A}_i$. In this paper, one of our main concern is the question when the canonical object $\mathcal{H}(T)$ (which is $\mathcal{H}(\mathrm{\Lambda })$ in the simplest case) is a tilting object of ${\underset{\_}{\mathsf{CM}}}^{\mathbb{Z}}A$. For this purpose we introduce the notion of homologically well-graded (hwg) algebras. Our main concern in the paper is a finitely graded hwg IG-algebra $A={⨁}_{i=0}^{\ell }{A}_i$, that is a finitely graded algebra which is IG as well as hwg. We provide several characterizations of a finitely graded hwg IG-algebra. One of our main results characterizes a hwg IG-algebra A as a finitely graded IG-algebra A such that $\mathcal{H}(T)$ is a tilting object, whose endomorphism algebra is the Beilinson algebra ∇A – an algebra canonically constructed from A (see (1.3)). Another result characterizes a finitely graded hwg IG-algebra as an algebra posses a homological symmetry which, a posteriori, guarantee that the algebra is IG. This phenomenon is looked as a generalization of the fact that a Frobenius algebra is an algebra posses a symmetry which, guarantee that the algebra is self-injective. Since a reason why Frobenius algebras are of importance in several areas is its symmetry, we can expect that hwg IG-algebra also play a basic role of other areas.

In [28] we make use of the generalized Happel functor to study general aspect of finitely graded IG-algebras and their stable categories. For example, we show that the Grothendieck group $K_0({\underset{\_}{\mathsf{CM}}}^{\mathbb{Z}}A)$ is free of finite rank. We expect that the generalized Happel functor can become an indispensable tool to study finitely graded IG-algebras.

Now we explain the main results and the notations used throughout the paper.

First we recall, the original Happel functor $\mathcal{H}$. Let Λ be a finite dimensional algebra over some field k and $\mathrm{\text{T}}(\mathrm{\Lambda }):=\mathrm{\Lambda }\oplus \mathrm{\text{D}}(\mathrm{\Lambda })$ the trivial extension algebra of Λ by the bimodule $\mathrm{\text{D}}(\mathrm{\Lambda })={\mathrm{Hom}}_{\mathbf{k}}(\mathrm{\Lambda },\mathbf{k})$, equipped with the grading $\mathrm{deg}\mathrm{\Lambda }=0,\mathrm{deg}\mathrm{\text{D}}(\mathrm{\Lambda })=1$. In his pioneering work, Happel [14], [15] constructed a fully faithful triangulated functor

$$\mathcal{H}:{\mathsf{D}}^{\mathrm{b}}(\mathrm{mod}\mathrm{\Lambda })\hookrightarrow {\underset{\_}{\mathrm{mod}}}^{\mathbb{Z}}\mathrm{\text{T}}(\mathrm{\Lambda })$$

and showed that it gives an equivalence if and only if $\mathrm{gldim}\mathrm{\Lambda }<\infty$. Thus if $\mathcal{H}$ gives an equivalence, then it provides a canonical tilting object $\mathcal{H}(\mathrm{\Lambda })$ of ${\mathrm{mod}}^{\mathbb{Z}}\mathrm{\text{T}}(\mathrm{\Lambda })$.

Although it looks like that the Happel functor $\mathcal{H}$ is determined from Λ, there is a way to construct $\mathcal{H}$ starting from $\mathrm{\text{T}}(\mathrm{\Lambda })$. In Section 3, we generalize the Happel functor $\mathcal{H}$ to the case where $\mathrm{\text{T}}(\mathrm{\Lambda })$ is replaced by a finitely graded IG-algebra $A={⨁}_{i=0}^{\ell }{A}_i$. The generalized Happel functor $\mathcal{H}$ has ${\underset{\_}{\mathsf{CM}}}^{\mathbb{Z}}A$ as its codomain. The domain is the derived category ${\mathsf{D}}^{\mathrm{b}}({\mathrm{mod}}^{[0,\ell -1]}A)$ of the abelian category ${\mathrm{mod}}^{[0,\ell -1]}A\subset {\mathrm{mod}}^{\mathbb{Z}}A$ which is the full subcategory consisting of $M={⨁}_{i\in \mathbb{Z}}{M}_i$ such that $M_i=0$ for $i\notin [0,\ell -1]$.

$$\mathcal{H}:{\mathsf{D}}^{\mathrm{b}}({\mathrm{mod}}^{[0,\ell -1]}A)\to {\underset{\_}{\mathsf{CM}}}^{\mathbb{Z}}A.$$

The first fundamental question about $\mathcal{H}$ is the following.

Question 1.1

When is it fully faithful or an equivalence?

We focus on the special case where A is a graded self-injective algebra. Recall that graded Frobenius algebra is a special class of graded self-injective algebras (for the definition, see Example 5.5). We can deduce an answer to the question from previous works by Chen [7], Mori with the first author [26] and the second author [38]. Namely, the functor $\mathcal{H}$ is fully faithful if and only if A is graded Frobenius. Moreover, if this is the case, $\mathcal{H}$ is an equivalence if and only if $\mathrm{gldim}{A}_0<\infty$.

To state the second question, we need to introduce a graded A-module T, which has been observed to play an important role in the study of the Happel functor.

$$T:=\underset{i=0}{\overset{\ell -1}{⨁}}A{(-i)}_{\le \ell -1}\in {\mathrm{mod}}^{[0,\ell -1]}A.$$

The endomorphism algebra $\mathrm{\nabla }A:=\mathrm{End}(T)$ is called the Beilinson algebra. We may identify it with the upper triangular matrix algebra below via canonical isomorphism.

$$\mathrm{\nabla }A:={\mathrm{End}}_{{\mathrm{mod}}^{\mathbb{Z}}A}(T)\cong \left(\begin{array}{cccc}\hfill A_0\hfill & \hfill A_1\hfill & \hfill \cdots \hfill & \hfill A_{\ell -1}\hfill \\ \hfill 0\hfill & \hfill A_0\hfill & \hfill \cdots \hfill & \hfill A_{\ell -2}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \hfill & \hfill ⋮\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill A_0\hfill \end{array}\right).$$

We denote by γ the algebra homomorphism induced by the Happel functor $\mathcal{H}$.

$$\gamma :\mathrm{\nabla }A={\mathrm{End}}_{{\mathrm{mod}}^{\mathbb{Z}}A}(T)\stackrel{{\mathcal{H}}_{T,T}}{\to }{\mathrm{End}}_{{\underset{\_}{\mathsf{CM}}}^{\mathbb{Z}}A}\mathcal{H}(T).$$

We note that T is a progenerator of ${\mathrm{mod}}^{[0,\ell -1]}A$. Moreover, by Morita theory, the functor $\mathsf{q}:={\mathrm{Hom}}_{{\mathrm{mod}}^{[0,\ell -1]}A}(T,-)$ gives an equivalence

$$\mathsf{q}:{\mathrm{mod}}^{[0,\ell -1]}A\cong \mathrm{mod}\mathrm{\nabla }A\mathrm{\text{such that}}\mathsf{q}(T)=\mathrm{\nabla }A.$$

Thus, we may regard the Happel functor $\mathcal{H}$ as an exact functor from ${\mathsf{D}}^{\mathrm{b}}(\mathrm{mod}\mathrm{\nabla }A)$ to ${\underset{\_}{\mathsf{CM}}}^{\mathbb{Z}}A$.

$$\mathcal{H}:{\mathsf{D}}^{\mathrm{b}}(\mathrm{mod}\mathrm{\nabla }A)\stackrel{{\mathsf{q}}^{-1}\cong }{\to }{\mathsf{D}}^{\mathrm{b}}({\mathrm{mod}}^{[0,\ell -1]}A)\to {\underset{\_}{\mathsf{CM}}}^{\mathbb{Z}}A.$$

The image $\mathcal{H}(T)$ has been studied by many researchers. In the case where A is a graded self-injective algebra, it is shown in [7], [26], [38] that $\mathcal{H}(T)$ is a tilting object of ${\underset{\_}{\mathrm{mod}}}^{\mathbb{Z}}A$ if and only if $\mathrm{gldim}{A}_0<\infty$. Moreover, the morphism γ is an isomorphism if and only if A is graded Frobenius.

As for graded IG-algebras A, it has been shown that the object $\mathcal{H}(T)\in {\underset{\_}{\mathsf{CM}}}^{\mathbb{Z}}A$ is a tilting object or relates to a construction of a tilting object of ${\underset{\_}{\mathsf{CM}}}^{\mathbb{Z}}A$ in many other cases [20], [23], [24], [31]. However we would like to mention that for a graded IG-algebra A, the graded module $\mathcal{H}(T)$ does not give a tilting object of ${\underset{\_}{\mathsf{CM}}}^{\mathbb{Z}}A$ in general (see for example [24, Example 3.7]). Thus our second question naturally arises.

Question 1.2

When is $\mathcal{H}(T)$ a tilting object of ${\underset{\_}{\mathsf{CM}}}^{\mathbb{Z}}A$ which satisfies the condition that the map γ is an isomorphism?

The answers of above two questions are given by the notion of homologically well-graded (hwg) algebras. The prototypical example of hwg algebras is the trivial extension algebra $\mathrm{\text{T}}(\mathrm{\Lambda })=\mathrm{\Lambda }\oplus \mathrm{\text{D}}(\mathrm{\Lambda })$ of a finite dimensional algebra Λ. In the paper [7] mentioned above, Chen introduced the notion of well-gradedness for finitely graded algebra and showed that a well-graded self-injective algebra A is graded Morita equivalent to $\mathrm{\text{T}}(\mathrm{\nabla }A)$. Thus graded representation theory of A is equivalent to that of $\mathrm{\text{T}}(\mathrm{\nabla }A)$ and in particular Happel's results can be applied. However for a finitely graded IG-algebra A which is not self-injective, well-gradedness is not enough to control CM-representation theory. We observed that a key to establish the Happel embedding is the following equation

$${\mathrm{Hom}}_{{\mathrm{mod}}^{\mathbb{Z}}\mathrm{\text{T}}(\mathrm{\Lambda })}(\mathrm{\Lambda },\mathrm{\text{T}}(\mathrm{\Lambda })(i))=0\mathrm{\text{for}}i\ne 1.$$

The relationship between the equation and well-gradedness is explained in the beginning of Section 4.2. The point is that the equation admits a natural homological generalization, which yields the definition of a hwg algebra.

Our main results show that a hwg IG-algebra gives complete answers to the above two questions.

Theorem 1.3 Theorem 6.3, Theorem 6.16

Assume that k is a commutative Noetherian ring and $A={⨁}_{i=0}^{\ell }{A}_i$ is an IG-algebra that is finitely generated as a k-module. Then the following conditions are equivalent.

• A is hwg (resp. A is hwg and $A_0$ satisfies the condition (F)).

• The Happel functor $\mathcal{H}$ is fully faithful (resp. equivalence).

• The morphism γ is an isomorphism and ${\mathrm{Hom}}_{{\underset{\_}{\mathsf{CM}}}^{\mathbb{Z}}A}(\mathcal{H}(T),\mathcal{H}(T)[n])=0$ for $n\ne 0$ (resp. γ is an isomorphism and $\mathcal{H}(T)$ is a tilting object of ${\underset{\_}{\mathsf{CM}}}^{\mathbb{Z}}A$).

The condition (F) is defined in Definition 6.5. It is a condition on finiteness of homological dimensions on $A_0$ which is weaker than the condition $\mathrm{gldim}{A}_0<\infty$. But this condition is equivalent to $\mathrm{gldim}{A}_0<\infty$ in the case where k is a complete local ring and hence in particular is a field.

In the case where k is a field, as we mentioned above a typical example of hwg algebra is $\mathrm{\text{T}}(\mathrm{\Lambda })$ for some finite dimensional algebra Λ. We can apply the equivalence(s) (1) ⇔ (2) to it and recover Happel's original result. However Happel's proof of the implication $(1)\Rightarrow (2)$ of respective cases made use of the fact that the stable category ${\underset{\_}{\mathrm{mod}}}^{\mathbb{Z}}\mathrm{\text{T}}(\mathrm{\Lambda })$ has Auslander-Reiten triangles. Since we do not know that Auslander-Reiten triangles may not make sense in the case where k is not a field, we can not use the Happel method and need to develop our method.

Our method relies on the decompositions of complexes of graded injective or projective modules established in [27]. As a by-product we are able to deal with the case where A is not necessary IG. In Lemma 6.14 and Proposition 6.15, we study the relation between existence of a generator in the singular derived category of A and the finiteness of homological dimensions of $A_0$. To the best of our knowledge, all previous results about such a relation only in the case of graded IG-algebras A. Thus, although it is beyond the main theme of the paper, these results are of their own interest.

To finish the introduction, we explain other results of the paper. A graded algebra which is both hwg and IG has a nice structure. We show that if a finitely graded algebra $A={⨁}_{i=0}^{\ell }{A}_i$ is hwg IG, then the subalgebra $A_0$ of degree 0 elements is Noetherian and the highest degree submodule $A_{\ell }$ is a cotilting bimodule over $A_0$; see Definition 5.1.

Recall that the key property of a cotilting bimodule C over a Noetherian algebra is that the duality ${\mathbb{R}\mathrm{Hom}}_{\mathrm{\Lambda }}(-,C)$ gives a contravariant equivalence between the derived categories ${\mathsf{D}}^{\mathrm{b}}(\mathrm{mod}\mathrm{\Lambda })$ and ${\mathsf{D}}^{\mathrm{b}}(\mathrm{mod}{\mathrm{\Lambda }}^{\mathrm{op}})$.

In Theorem 5.3 we give characterizations of hwg IG-algebras which are neither stated in terms of the stable category nor the Happel functor. Among other things, we verify that a hwg IG-algebra is precisely a finitely graded Noetherian algebra $A={⨁}_{i=0}^{\ell }{A}_i$ which has the following properties: (i) the $A_0$-$A_0$-bimodule $A_{\ell }$ is a cotilting bimodule. (ii) A has a homological symmetry given by the duality induced from the cotilting bimodule $A_{\ell }$.

In the case k is a field and Λ is a finite dimensional k-algebra, the k-dual bimodule $\mathrm{\text{D}}(\mathrm{\Lambda })$ is an example of cotilting bimodule and the duality induced by $\mathrm{\text{D}}(\mathrm{\Lambda })$ is nothing but the k-duality, i.e., ${\mathbb{R}\mathrm{Hom}}_{\mathrm{\Lambda }}(-,\mathrm{\text{D}}(\mathrm{\Lambda }))\cong \mathrm{\text{D}}(-)$. Using this fact, we observe in Example 5.5 that a finite dimensional graded self-injective algebra A is hwg if and only if it is a graded Frobenius. In this sense, a hwg IG-algebra can be looked as a generalization of a graded Frobenius algebra obtained by replacing the bimodule $\mathrm{\text{D}}(\mathrm{\Lambda })$ with a general cotilting bimodule C.

It follows from a classical result by Fossum-Griffith-Reiten [11, Theorem 4.32] that when Λ is a Noetherian algebra and C is a cotilting bimodule, the trivial extension algebra $A=\mathrm{\Lambda }\oplus C$ is IG. We prove that if we equip A with the grading $\mathrm{deg}\mathrm{\Lambda }=0,\mathrm{deg}C=1$, then it becomes hwg IG. Moreover, in Corollary 5.6, we show that a graded algebra $A=A_0\oplus A_1$ concentrated in degree $0,1$ is hwg IG if and only if it is obtained in such a way.

Now it is natural to recall the following result of commutative Gorenstein algebras due to Foxby [9] and Reiten [34]. Namely, the trivial extension algebra $A=\mathrm{\Lambda }\oplus C$ of a commutative Noetherian local algebra Λ by a (bi)module C is IG if and only if C is a cotilting (bi)module. Thus with our terminology this theorem says that, in commutative local setting, every graded IG-algebra $A=A_0\oplus A_1$ concentrated in degree $0,1$ is hwg IG. We prove the same result is true for a commutative finitely graded IG-algebra.

Theorem 1.4 Theorem 8.1

A commutative local finitely graded IG-algebra $A={⨁}_{i=0}^{\ell }{A}_i$ is hwg.

The paper is organized as follows. In Section 2, first we fix notations for graded modules and their derived categories. Then we recall a decomposition of a complex $I\in \mathsf{C}({\mathrm{Inj}}^{\mathbb{Z}}A)$ of graded injective modules introduced in [27]. In Section 3 we give the construction of the Happel functor and recall related results. In Section 4 we introduce a notion of homologically well-graded (hwg) algebras. In Section 5 we give characterizations of hwg algebras and show that it can be looked as a generalization of graded Frobenius algebras. In Section 6, we give characterizations of fully faithfulness of $\mathcal{H}$ (Theorem 6.3) and characterizations of when $\mathcal{H}$ gives an equivalence (Theorem 6.16). In Section 7, we give several examples and constructions of hwg IG-algebras. We observe that being hwg IG is more robust than being IG. For example, even though taking Veronese algebras and Segre products do not preserve IG-algebras, these operations preserve hwg IG-algebras. In Section 8, we focus on the commutative case and generalize a result of Fossum-Griffith-Reiten, Foxby and Reiten [11], [9], [34]. In Section 9 we discuss the definition of hwg IG-algebras.

Acknowledgment

The authors thank anonymous referee for his/her careful reading and numerous comments about mathematical contents and readability. The first author was partially supported by JSPS KAKENHI Grant Number 26610009. The second author was partially supported by JSPS KAKENHI Grant Number 26800007.

Throughout the paper k denotes a commutative ring. An algebra Λ is always a k-algebra. Unless otherwise stated, the word “Λ-modules” means right Λ-modules. We denote by Mod Λ the category of Λ-modules. We denote by Proj Λ (resp. Inj Λ) the full subcategory of projective (resp. injective) Λ-modules. We denote by proj Λ the full subcategory of finitely generated projective Λ-modules.

We set ${\mathrm{Hom}}_{\mathrm{\Lambda }}:={\mathrm{Hom}}_{\mathrm{Mod}\mathrm{\Lambda }}$. Note that ${\mathrm{Hom}}_{\mathrm{\Lambda }}$ also denotes the Hom-space of the derived category $\mathsf{D}(\mathrm{Mod}\mathrm{\Lambda })$.

We denote the opposite algebra by ${\mathrm{\Lambda }}^{\mathrm{op}}$. We identify left Λ-modules with (right) ${\mathrm{\Lambda }}^{\mathrm{op}}$-modules. A Λ-Λ-bimodule D is always assumed to be k-central, i.e., $ad=da$ for $d\in D,a\in \mathbf{k}$. For a Λ-Λ-bimodule D, we denote by $D_{\mathrm{\Lambda }}$ and ${}_{\mathrm{\Lambda }}D$ the underlying right and left Λ-modules respectively.

For an additive category $\mathcal{A}$, we denote by $\mathsf{C}(\mathcal{A})$ and $\mathsf{K}(\mathcal{A})$ the category of cochain complexes and cochain morphisms and its homotopy category respectively. For complexes $X,Y\in \mathsf{C}(\mathcal{A})$, we denote by ${\mathrm{Hom}}_{\mathcal{A}}^{•}(X,Y)$ the Hom-complex. For an abelian category $\mathcal{A}$, we denote by $\mathsf{D}(\mathcal{A})$ the derived category of $\mathcal{A}$.

We denote the derived functor of ${\mathrm{Hom}}_{\mathcal{A}}$ by ${\mathbb{R}\mathrm{Hom}}_{\mathcal{A}}$.

For an algebra Λ, we set ${\mathrm{Hom}}_{\mathrm{\Lambda }}:={\mathrm{Hom}}_{\mathsf{D}(\mathrm{Mod}\mathrm{\Lambda })}$ and ${\mathbb{R}\mathrm{Hom}}_{\mathrm{\Lambda }}:={\mathbb{R}\mathrm{Hom}}_{\mathrm{Mod}\mathrm{\Lambda }}$.

A triangulated category $\mathsf{T}$ is always assumed to be linear over the base commutative ring k. Let $\mathsf{U},\mathsf{V}\subset \mathsf{T}$ be full triangulated subcategories. We denote by $\mathsf{U}⁎\mathsf{V}\subset \mathsf{T}$ to be the full subcategory consisting of those objects X which fit into an exact triangle $U\to X\to V\to$ with $U\in \mathsf{U},V\in \mathsf{V}$. If ${\mathrm{Hom}}_{\mathsf{T}}(U,V)=0$ for all $U\in \mathsf{U}$ and all $V\in \mathsf{V}$, we write $U\perp V$.

Let $X\in \mathsf{T}$ be an object. We denote by $\mathsf{thick}X$ the thick closure of X, that is, the smallest triangulated subcategory of $\mathsf{T}$ containing X that is closed under direct summands. In other words, it is a triangulated subcategory of $\mathsf{T}$ consisting of objects which are constructed from X by taking shifts, cones and direct summands. An object $X\in \mathsf{T}$ is said to be a tilting object of $\mathsf{T}$ if $\mathsf{thick}X=\mathsf{T}$ and ${\mathrm{Hom}}_{\mathsf{T}}(X,X[n])=0$ for $n\ne 0$.