本文選題：三角范疇 + 同倫范疇��； 參考：《南京大學(xué)》2016年博士論文

【摘要】：在Abelian范疇中,對(duì)于具有某種投射(或內(nèi)射)性質(zhì)的對(duì)象的研究是相對(duì)同調(diào)代數(shù)的主要課題.現(xiàn)在經(jīng)典同調(diào)代數(shù)的研究已經(jīng)廣泛的應(yīng)用了三角范疇的工具,Bousfield的局部化理論是其基礎(chǔ)之一,而這一理論實(shí)質(zhì)上給出三角范疇的一個(gè)粘合[45].另一方面,在同倫范疇中,由投射和內(nèi)射對(duì)象誘導(dǎo)的粘合給出復(fù)形的同調(diào)分解[54],進(jìn)而可以引入并研究復(fù)形的同調(diào)維數(shù).在本文的第二章,我們研究了一個(gè)平衡對(duì)如何給出同倫范疇的一個(gè)粘合.在第三章,我們考慮了由純性誘導(dǎo)的平衡對(duì)在同倫范疇中的表現(xiàn),定義并研究了復(fù)形的純同調(diào)維數(shù).Auslander-Reiten理論是Artin代數(shù)表示理論研究的有力工具.而這一理論已經(jīng)被廣泛的應(yīng)用到三角范疇的研究中,在第四章,我們證明了Artin代數(shù)上的有限表現(xiàn)模的有界同倫范疇存在Auslander-Reiten三角,并且給出一些應(yīng)用.在Gorenstein環(huán)上,關(guān)于有限生成Gorenstein投射模的奇點(diǎn)等價(jià)被很多學(xué)者所關(guān)心.我們?cè)噲D將這一等價(jià)的條件放到最弱.在第五章,我們將這一等價(jià)在任意環(huán)上對(duì)于任意的Gorenstein投射模給出證明.
[Abstract]:In the Abelian category, the study of objects with certain projective (or injective) properties is the main subject of relative homology algebra. Now the research of classical homology algebra has widely applied the tool of triangular category. The localization theory of Bousfield is one of its foundations, and this theory essentially gives a binding of the triangle category. 45]. on the other hand, in the homotopy category, the homotopy decomposition [54] is given by the adhesion induced by projective and injective objects, and then the homology dimension of the complex can be introduced and studied. In the second chapter of this paper, we study how a balance gives a bonding of the homotopy category. In the third chapter, we consider the pure induced leveling. The.Auslander-Reiten theory is a powerful tool for the study of Artin algebra representation theory, which has been widely applied to the study of trigonometric categories. In the fourth chapter, we prove the bounded homotopy category of finite expression modules on Artin algebras. In the Auslander-Reiten triangle, and give some applications. On the Gorenstein ring, the odd point equivalence of the finite generating Gorenstein projective modules is concerned by many scholars. We try to put this equivalent condition to the weakest. In the fifth chapter, we give a proof of this equivalence to any Gorenstein projective module on any ring.
【學(xué)位授予單位】：南京大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O154.1

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