| 郭寿桃,杨晓燕.正合范畴中的复形、 余挠对及粘合[J].数学年刊A辑,2021,42(1):59~74 |
| 正合范畴中的复形、 余挠对及粘合 |
| Complexes, Cotorsion Pairs and Recollements on Exact Categories |
| Received:May 17, 2019 Revised:July 31, 2020 |
| DOI：10.16205/j.cnki.cama.2021.0006 |
| 中文关键词: 正合范畴, 复形, 余挠对, 粘合 |
| 英文关键词:Exact category, Complex, Cotorsion pair, Recollement |
| 基金项目:国家自然科学基金~(No.,11761060) |
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| 中文摘要: |
| 作者在弱幂等完备的正合范畴(mathscr{A},mathscr{E}) 中引入了复形的新的定义, 并且证明了mathscr{E}{-}正合复形的同伦范畴 mathcal{K}_{ex}(mathscr{E}) 是同伦范畴mathcal{K}_mathscr{E}(mathscr{A}) 的厚子范畴.给定(mathscr{A},mathscr{E}) 中的余挠对(mathscr{X},mathscr{Y}),定义了正合范畴(mathcal{C}_mathscr{E}(mathscr{A}),mathcal{C}(mathscr{E}))中的两个余挠对(mathscr{wt{X}}_mathscr{E},dgmathscr{wt{Y}}_mathscr{E}) 和(dgmathscr{wt{X}}_mathscr{E},mathscr{wt{Y}}_mathscr{E}), 并且证明了当mathscr{A} 是可数完备时,mathcal{C}_mathscr{E}(mathscr{A}) 中任意无界复形的dgmathscr{wt{X}}_mathscr{E}{-}分解存在.作为应用, 建立了相对于范畴mathcal{K}_{ex}(mathscr{E}) 和mathcal{D}_mathscr{E}(mathscr{A})的范畴mathcal{K}_mathscr{E}(mathscr{A}) 的左粘合, 给出了R{-}模范畴的粘合的例子. |
| 英文摘要: |
| In this paper, a new definition of complexes on a weakly idempotent complete exact category (A , E ) is introduced. The authors prove that the homotopy category Kex(E ) of E -exact complexes is a thick subcategory of the homotopy category KE (A ). Given a cotorsion pair (X , Y ) in (A , E ), they define two cotorsion pairs ( fXE , dg fYE ) and (dgXfE , YfE ) on the exact category (CE (A ), C(E )), and prove the existence of dg fXE -resolution for any unbounded complex in CE (A ) whenever A is countably complete. As an application of these results, the authors establish a left recollement of KE (A ) relative to Kex(E ) and DE (A ).Some examples of recollements on the category of R-modules are given. |
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