由n项大倾斜复形导出的导出范畴的重排
Recollements of derived categories from $n$-term big tilting complexes
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中文总结 AI 辅助
针对环A上的n项大倾斜复形T，构造B-Mod的扩张闭正合子范畴ℰ，证明导出范畴DB可由Dℰ和DA重排，还得到ℰ的d-对称性、阿贝尔性条件，将两项经典情形推广到任意有限项，构造了n≥2时的非紧大倾斜复形。
中文摘要 AI 辅助
设A为环，T为A上的n项大倾斜复形，由投射A-模的有界复形$\boldsymbol{\text{cpx}P}$表示。令$B=\text{End}_{\boldsymbol{\text{D}}A}(\boldsymbol{\text{cpx}P})$，$\boldsymbol{\text{Λ}}:=\text{dotEnd}_A(\boldsymbol{\text{cpx}P})$，$\boldsymbol{\text{Δ}}:=\tau_{\text{≤}0}\boldsymbol{\text{Λ}}$。给出A-B-双复形的关联复形$\boldsymbol{\text{cpx}T}=\boldsymbol{\text{cpx}P}\boldsymbol{\text{⊗L}}_{\boldsymbol{\text{Δ}}}B$。我们构造B-Mod范畴的一个扩张闭正合子范畴$\boldsymbol{\text{mathscr E}}$，并证明导出范畴$\boldsymbol{\text{D}}B$可由$\boldsymbol{\text{D}}\boldsymbol{\text{mathscr E}}$和$\boldsymbol{\text{D}}A$得到重排。证明通过$\boldsymbol{\text{cpx}T}$的投射模型的dg双中心化子描述，以及将$\boldsymbol{\text{D}}\boldsymbol{\text{mathscr E}}$与导出张量函子的核等同的正合实现定理完成。我们进一步证明，对于不小于$\boldsymbol{\text{cpx}T}$的完美右B-模型振幅的任意d，$\boldsymbol{\text{mathscr E}}$是d-对称的。此外，$\boldsymbol{\text{mathscr E}}$是阿贝尔范畴当且仅当核在标准t-结构下稳定，等价于该重排由同调环满射诱导。在右B-振幅至多为1时，核因此是阿贝尔的，我们的构造在两项情形下特殊化为泛局部化产生的重排。因此，经典的两项情形推广到任意有限项大倾斜复形，其中正合范畴替代了高振幅下的阿贝尔核。最后，对每个n≥2，我们构造真正的n项非紧大倾斜复形，包括一个明确的三项例子，其左项由代数Calkin型商决定。
英文摘要
Let $A$ be a ring and let $\bf T$ be an $n$-term big tilting complex over $A$, represented by a bounded complex $\cpx{P}$ of projective $A$-modules. Set $B=\End_{\D{A}}(\cpx{P}), Λ:=\dotEnd_A(\cpx{P})$ and $Δ:=τ_{\leq 0}Λ$. The associated complex of $A$-$B$-bimodule $\cpx{T}=\cpx{P}\otimesL_ΔB$ is given. We construct an extension-closed exact subcategory $\mathscr E$ of $B\Modcat$ and prove that the derived category $\D{B}$ admits a recollement by $\D{\mathscr E}$ and $\D{A}$. The proof passes through a dg double-centralizer description of a projective model of $\cpx{T}$ and an exact realisation theorem identifying $\D{\mathscr E}$ with the kernel of the derived tensor functor. We further show that $\mathscr E$ is $d$-symmetric for every $d$ not smaller than the amplitude of a perfect right $B$-model of $\cpx{T}$. Moreover, $\mathscr E$ is abelian if and only if the kernel is stable under the standard $t$-structure, equivalently, the recollement is induced by a homological ring epimorphism. In right $B$-amplitude at most one, the kernel is therefore abelian, and our construction specializes to the recollement arising from universal localisation in the two-term case. Thus the classical two-term picture extends to arbitrary finite-term big tilting complexes, with exact categories replacing abelian kernels in higher amplitude. Finally, we construct genuinely $n$-term non-compact big tilting complexes for every $n\geq2$, including an explicit three-term example whose left-hand term is determined by an algebraic Calkin-type quotient.
