A Smashing Subcategory of the Homotopy Category of Gorenstein Projective Modules

Let A be an artin algebra of finite CM-type. In this paper, we show that if A is virtually Gorenstein, then the homotopy category of Gorenstein projective \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A\mbox{-}$\end{document}modules, denote \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K(A\mbox{-}{\mathcal {GP}})$\end{document}, is always compactly generated. Based on this result, it will be proved that the homotopy category of projective \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A\mbox{-}$\end{document}modules, denote \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K(A\mbox{-}{\mathcal P})$\end{document}, is a smashing subcategory of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K(A\mbox{-}{\mathcal {GP}})$\end{document} and the corresponding Verdier quotient is also compactly generated. Furthermore, it turns out that the inclusion functor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$i: K(A\mbox{-}{\mathcal P})\to K(A\mbox{-}{\mathcal {GP}})$\end{document} induces a recollement of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K(A\mbox{-}{\mathcal {GP}})$\end{document}.


Introduction
Let X be a class of left modules over an associative ring R which is closed under set-indexed coproducts and direct summands. Holm and Jørgensen [13] study the general question of when the homotopy category K(X ) of X is compactly generated. They give a number of sufficient conditions on R and X which ensure that K(X ) is compactly generated.
Let A be an artin algebra and A-Mod the category of A-modules. Denote by A-P the full subcategory of projective A-modules, A-GP the full subcategory of Gorenstein projective A-modules, and A-G proj the full subcategory of all finitelygenerated Gorenstein projective modules. As is well known, the homotopy category K(A-P) is compactly generated [15,Theorem 2.4].
Gorenstein projective modules and algebras of finite Cohen-Macaulay type receive a lot of attention (See e.g. [1, 4-6, 8-10, 12, 14, 16, 17, 19]). Recall from [4,6] that an artin algebra A is of finite Cohen-Macaulay type (simply, CM-type) if there are only finitely many isomorphism classes of finitely-generated indecomposable Gorenstein projective A-modules. We are interested in the compact generatedness of the homotopy category K(A-GP) of an artin algebra A of finite CM-type.
In Section 2, we first show that if A is virtually Gorenstein of finite CM-type, then K(A-GP) is compactly generated. Next, based on this result, we show that K(A-P) is a smashing subcategory of K(A-GP) and the Verdier quotient K(A-GP)/K(A-P) is also compactly generated.
The concept of recollement goes back to the work of Beilinson et al. [2]. In Section 3, we show the existence of recollements of the homotopy category K(A-GP).

Conditions for Compact Generatedness
Our aim in this section is to show that K(A-GP) is compactly generated provided A is virtually Gorenstein of finite CM-type. So based on the result of Bruns and Herzog [6,Proposition 2.11], and the result of Jørgensen [16],

K(A-P) is a smashing subcategory of K(A-GP) and the Verdier quotient K(A-GP)/K(A-P)
is also compactly generated.
Our strategy for the compact generatedness of K(A-GP) is to give sufficient conditions on A. We will use the following lemma.

Lemma 2.1 [4, Theorem 4.10] Let A be an artin algebra. Then A is virtually Gorenstein of f inite CM-type if and only if any Gorenstein projective A-module is a direct sum of f initely-generated modules.
Now we are ready to state and prove our first main theorem in this section.

Theorem 2.2 Let A be a virtually Gorenstein artin algebra of f inite CM-type. Then K(A-GP) is a compactly generated triangulated category.
Proof Since A is virtually Gorenstein of finite CM-type, we get from Lemma 2.1 that A-GP = Add(A-G proj) which means that A-GP is contravariantly finite in A-Mod, and also each Gorenstein projective module is pure projective which means that every pure exact sequence of modules from A-GP is split exact. This implies that K(A-GP) is a compactly generated triangulated category by [13, Theorem 3.1].
Recall from [11] that a complex X • is A-GP-acyclic if the induced complex Hom A (G, X • ) is acyclic for each module in A-GP, and the Gorenstein derived category D gp (A-Mod) of an artin algebra A is defined to be the Verdier quotient of the homotopy category K(A-mod) with respect to the thick subcategory K gpac (A-Mod) which consists of all A-GP-acyclic complexes.

Corollary 2.3 Let A be a Gorenstein artin algebra of f inite CM-type. Then D gp (A-Mod) is compactly generated.
Proof By the assumption on A, we see from [3, Corollary 8.3 and Corollary 8.5] that A satisfies the conditions on Theorem 2.2. Hence we get that K(A-GP) is a compactly generated triangulated category. By [7,Proposition 3.5] there is a triangleequivalence D gp (A-Mod) ∼ = K(A-GP). This implies that D gp (A-Mod) is compactly generated.
For our second main theorem we need a definition and some lemmas. Recall from [18] that a full subcategory B of a compactly generated triangulated category T is smashing if the inclusion B → T has a right adjoint which preserves coproducts.

Lemma 2.5 [5, Proposition 2.11] Let T and T be compactly generated triangulated categories, and let F : T → T be a fully faithful triangle functor which preserves coproducts and compact objects. Then F admits a right adjoint G : T → T which preserves coproducts.
So in view of the above lemmas, we have the following theorem.

Theorem 2.6 Let A be a virtually Gorenstein artin algebra of f inite CM-type. Then K(A-P) is a smashing subcategory of K(A-GP). Moreover, K(A-GP)/K(A-P) is a compactly generated triangulated category.
Proof By the assumpotion on A, we get from Theorem 2.2 that K(A-GP) is compactly generated, and from [15,Theorem 2.4] that K(A-P) is compactly generated and each compact object P • is exactly the upper bounded complex of finitelygenerated projective modules. Let i : K(A-P) → K(A-GP) be the inclusion functor. Note that i naturally preserves coproducts. Let {G • i } i∈I be any family objects in K(A-GP). Then we have This implies that i preserves compact objects. Hence by Lemma 2.5 we get that i admits a right adjoint R :

K(A-GP) → K(A-P) which preserves coproducts. This means K(A-P) is a smashing subcategory of K(A-GP). This implies by Lemma 2.4 that K(A-GP)/K(A-P)
is a compactly generated triangulated category.

Recollements for the Homotopy Category K( A-GP)
In this section, let A be an artin algebra. Based on the compact generatedness of the full subcategory K(A-P) of K(A-GP), we will apply the arguments of Neeman to prove the existence of a recollement of K(A-GP).
So in view of the above theorem, we have the following result. Let us begin by recalling some definitions.
Let T be a triangulated category with the suspension functor . Recall from [5, Section 2] that a torsion pair in T is a pair of strict full subcategories (X .Y) of T satisfying the following conditions:

− → Y T h T
− → (X T ). Then X is called a torsion class and Y is called a torsion-free class. A torsion, torsion-free triple, TTF-triple for short, in T is a triple (X , Y, Z) of full subcategories of T such that the pairs (X , Y) and (Y, Z) are torsion pairs. Now we give a TTF-triple in K(A-GP).