A Smashing Subcategory of the Homotopy Category of Gorenstein Projective Modules

Transcription

1 Appl Categor Struct (2015) 23: DOI /s of Gorensten Projectve Modules Nan Gao eceved: 26 October 2012 / Accepted: 8 January 2013 / Publshed onlne: 26 July 2013 The Author(s) Ths artcle s publshed wth open access at SprngerLnk.com Abstract Let A be an artn algebra of fnte CM-type. In ths paper, we show that f A s vrtually Gorensten, then the homotopy category of Gorensten projectve A-modules, denote K( A-GP), s always compactly generated. Based on ths result, t wll be proved that the homotopy category of projectve A-modules, denote K( A-P), s a smashng subcategory of K(A-GP) and the correspondng Verder quotent s also compactly generated. Furthermore, t turns out that the ncluson functor : K( A-P) K( A-GP) nduces a recollement of K( A-GP). Keywords Gorensten projectve modules Compactly generated homotopy categores Smashng subcategory ecollements 1 Introducton Let X be a class of left modules over an assocatve rng whch s closed under set-ndexed coproducts and drect summands. Holm and Jørgensen [13] study the generalqueston of whenthe homotopy category K(X ) ofx s compactly generated. They gve a number of suffcent condtons on and X whch ensure that K(X ) s compactly generated. Let A be an artn algebra and A-Mod the category of A-modules. Denote by A-P the full subcategory of projectve A-modules, A-GP the full subcategory of Supported by the Natonal Natural Scence Foundaton of Chna (Grant No ). N. Gao (B) Department of Mathematcs, Shangha Unversty, Shangha , People s epublc of Chna e-mal: gaonanjane@gmal.com

2 88 N. Gao Gorensten projectve A-modules, and A-G proj the full subcategory of all fntelygenerated Gorensten projectve modules. As s well known, the homotopy category K(A-P) s compactly generated [15, Theorem 2.4]. Gorensten projectve modules and algebras of fnte Cohen Macaulay type receve a lot of attenton (See e.g. [1, 4 6, 8 10, 12, 14, 16, 17, 19]). ecall from [4, 6] that an artn algebra A s of fnte Cohen Macaulay type (smply, CM-type) f there are only fntely many somorphsm classes of fntely-generated ndecomposable Gorensten projectve A-modules. We are nterested n the compact generatedness of the homotopy category K(A-GP) of an artn algebra A of fnte CM-type. In Secton 2,we frst showthat f A s vrtually Gorensten of fnte CM-type, then K(A-GP) s compactly generated. Next, based on ths result, we show that K(A-P) s a smashng subcategory of K(A-GP) and the Verder quotent K(A-GP)/K(A-P) s also compactly generated. The concept of recollement goes back to the work of Belnson et al. [2]. In Secton 3, we show the exstence of recollements of the homotopy category K(A-GP). 2 Condtons for Compact Generatedness Our am n ths secton s to show that K(A-GP) s compactly generated provded A s vrtually Gorensten of fnte CM-type. So based on the result of Bruns and Herzog [6, Proposton 2.11], and the result of Jørgensen [16], K(A-P) s a smashng subcategory of K(A-GP) and the Verder quotent K(A-GP)/K(A-P) s also compactly generated. Our strategy for the compact generatedness of K(A-GP) s to gve suffcent condtons on A. We wll use the followng lemma. Lemma 2.1 [4, Theorem 4.10] Let A be an artn algebra. Then A s vrtually Gorensten of f nte CM-type f and only f any Gorensten projectve A-module s a drect sum of f ntely-generated modules. Now we are ready to state and prove our frst man theorem n ths secton. Theorem 2.2 Let A be a vrtually Gorensten artn algebra of f nte CM-type. Then K(A-GP) s a compactly generated trangulated category. Proof Snce A s vrtually Gorensten of fnte CM-type, we get from Lemma 2.1 that A-GP = Add(A-G proj) whch means that A-GP s contravarantly fnte n A-Mod, and also each Gorensten projectve module s pure projectve whch means that every pure exact sequence of modules from A-GP s splt exact. Ths mples that K(A-GP) s a compactly generated trangulated category by [13, Theorem 3.1]. ecall from [11] thatacomplexx s A-GP-acyclc f the nduced complex Hom A (G, X ) s acyclc for each module n A-GP, and the Gorensten derved category D gp (A-Mod) of an artn algebra A s defned to be the Verder quotent of the homotopy category K(A-mod) wth respect to the thck subcategory K gpac (A-Mod) whch conssts of all A-GP-acyclc complexes.

3 89 Corollary 2.3 Let A be a Gorensten artn algebra of f nte CM-type. Then D gp (A-Mod) s compactly generated. Proof By the assumpton on A, weseefrom[3, Corollary 8.3 and Corollary 8.5] that A satsfes the condtons on Theorem 2.2. Hence we get that K(A-GP) s a compactly generatedtrangulated category.by [7, Proposton 3.5] there s a trangleequvalence D gp (A-Mod) = K(A-GP). Ths mples that D gp (A-Mod) s compactly generated. For our second man theorem we need a defnton and some lemmas. ecall from [18] that a full subcategory B of a compactly generated trangulated category T s smashng f the ncluson B T has a rght adjont whch preserves coproducts. Lemma 2.4 [18, Lemma 4.1] Let B be a smashng subcategory of a compactly generated trangulated category T.ThenT/B s a compactly generated trangulated category. Lemma 2.5 [5, Proposton 2.11] Let T and T be compactly generated trangulated categores, and let F : T T be a fully fathful trangle functor whch preserves coproducts and compact objects. Then F admts a rght adjont G : T T whch preserves coproducts. So n vew of the above lemmas, we have the followng theorem. Theorem 2.6 Let A be a vrtually Gorensten artn algebra of f nte CM-type. Then K(A-P) s a smashng subcategory of K(A-GP). Moreover,K(A-GP)/K(A-P) s a compactly generated trangulated category. Proof By the assumpoton on A, we get from Theorem 2.2 that K(A-GP) s compactly generated, and from [15, Theorem 2.4] that K( A-P) s compactly generated and each compact object P s exactly the upper bounded complex of fntelygenerated projectve modules. Let : K(A-P) K(A-GP) be the ncluson functor. Note that naturally preserves coproducts. Let {G } I be any famly objects n K(A-GP). Then we have Hom K(A-GP)(P, I G ) = Hom K(A-GP)(P, I G ) = I Hom K(A-GP)(P, G ) = I Hom K(A-GP)(P, G ). Ths mples that preserves compact objects. Hence by Lemma 2.5 we get that admts a rght adjont : K(A-GP) K(A-P) whch preserves coproducts. Ths means K(A-P) s a smashng subcategory of K(A-GP). Ths mples by Lemma 2.4 that K(A-GP)/K(A-P) s a compactly generated trangulated category. 3 ecollements for the Homotopy Category K(A-GP) In ths secton, let A be an artn algebra. Based on the compact generatedness of the full subcategory K(A-P) of K(A-GP), we wll apply the arguments of Neeman to prove the exstence of a recollement of K(A-GP).

4 90 N. Gao Lemma 3.1 [21, Theorem 4.1], [22, Theorem 8.6.1] Let F : T T be a trangle functor between trangulated categores T and T,whereT s compactly generated. (1) F admts a rght adjont f and only f t preserves all coproducts. (2) F admts a left adjont f and only f t preserves all products. Theorem 3.2 Let A be an artn algebra. Then the ncluson functor : K(A-P) K(A-GP) nduces a recollement of the form K(A-P) Ker such that Ker = K(A-GP)/K(A-P) as trangulated categores. Proof Snce A s an artn algebra, t follows from [15, Theorem 2.4] that K(A-P) s compactly generated. Note that the ncluson functor naturally preserves all coproducts and products. Then admts a rght adjont, also a left adjont. Hence by [20, Theorem 2.2] we have a recollement of the form K( A-P) Ker such that Ker = K(A-GP)/K(A-P) as trangulated categores. So n vew of the above theorem, we have the followng result. Let us begn by recallng some defntons. Let T be a trangulated category wth the suspenson functor. ecall from [5, Secton 2] that a torson par n T s a par of strct full subcategores (X.Y) of T satsfyng the followng condtons: (1) T (X, Y) = 0 ; (2) (X ) X and 1 (Y) f T g Y; (3) For any T T there exsts a trangle X T T T Y T ht (X T ).ThenX s called a torson class and Y s called a torson-free class. A torson, torson-free trple, TTF-trple for short, n T s a trple (X, Y, Z) of full subcategores of T such that the pars (X, Y) and (Y, Z) are torson pars. Now we gve a TTF-trple n K(A-GP). Corollary 3.3 Let A be an artn algebra. Then there exsts a TT F-trple (K(A-P), Ker, (Ker) ) n K(A-GP). Proof By Theorem 3.2 we have the recollement of the form K( A-P) Ker. Hence by [20, Theorem 2.2] we get that (K(A-P), Ker) and (Ker, (Ker) ) are two torson pars n K(A-GP). Ths means K(A-GP) has a TTF-trple (K(A-P), Ker, (Ker) ). Acknowledgement on the work. The author would lke to thank Professor Pu Zhang for hs valuable comments

5 91 Open Access Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton Lcense whch permts any use, dstrbuton, and reproducton n any medum, provded the orgnal author(s) and the source are credted. eferences 1. Avramov, L.L., Martsnkovsky, A.: Absolute, relatve, and Tate cohomology of modules of fnte Gorensten dmenson. Proc. Lond. Math. Soc. 85(3), (2002) 2. Belnson, A.A., Bernsten, J., Delgne, P.: Fasceaux pervers. In: Proceedngs of the Conference Analyss and Topology on Sngular Spaces, vol Lumny, Astérsque (1982) 3. Belganns, A.: Cohen Macaulay modules, (co)torson pars and vrtually Gorensten algebras. J. Algebra 288, (2005) 4. Belganns, A.: On algebras of fnte Cohen Macaulay type. Adv. Math. 226, (2011) 5. Belganns, A., eten, I.: Homologcal and homotopcal aspects of toson theores. Mem. Amer. Math. Soc. 188, (2007) 6. Bruns, W., Herzog, J.: Cohen Macaulay ngs, revsed edton. Cambrdge Studes n Adv. Math., vol. 39. Cambrdge Unv. Press (1998) 7. Chen, X.W.: Homotopy equvalences nduced by balanced pars. J. Algebra 324(10), (2010) 8. Chrstensen, L.W., Frankld, A., Holm, H.: On Gorensten projectve, njectve and flat dmensons-a functoral descrpton wth applcatons. J. Algebra 302(1), (2006) 9. Enochs, E.E., Jenda, O.M.G.: Gorensten njectve and projectve modules. Math. Z. 220(4), (1995) 10. Enochs, E.E., Jenda, O.M.G.: elatve homologcal algebra. De Gruyter Exp. Math., vol. 30. Walter De Gruyter Co. (2000) 11. Gao, N., Zhang, P.: Gorensten derved categores. J. Algebra 323, (2010) 12. Happel, D.: On Gorensten algebras. epresentaton theory of fnte groups and fntedmensonal algebras. Prog. Math. 95, (1991) 13. Holm, H., Jørgensen, P.: Compactly generated homotopy categores. Homology Homotopy Appl. 9(1), (2007) 14. Holm, H.: Gorensten homologcal dmensons. J. Pure Appl. Algebra 189(1 3), (2004) 15. Jørgensen, P.: The homotopy category of complexes of projectve modules. Adv. Math. 193, (2005) 16. Jørgensen, P.: Exstence of Gorensten projectve resolutons and tate cohomology. J. Eur. Math. Soc. 9(1), (2007) 17. Körrer, H.: Cohen Macaulay modules on hpersurface sngulartes. Invent. Math. 88(1), (1987) 18. Krause, H.: Smashng subcategores and the telescope conjecture-an algebrac approach. Invent. Math. 139, (2000) 19. L, Z.W., Zhang, P.: Gorensten algebras of fnte Cohen Macaulay type. Adv. Math. 223(2), (2010) 20. Myach, J.: Localzaton of trangulated categores and derved categores. J. Algebra 141, (1991) 21. Neeman, A.: The Grothendeck dualty theorem va Bousfeld s technques and Brown representablty. J. Am. Math. Soc. 8(1), (1996) 22. Neeman, A.: Trangulated categores. Annals of Mathematcs Syudes, vol. 148, Prnceton Unv. Press (2001)