arXiv math/02030372002-03math.RA

重拾和倾斜复合体

Recollement and Tilting Complexes

Jun-ichi Miyachi

备注：: 24 pages

摘要：

首先，我们研究了由部分倾斜复形诱导的无界模复形的导出范畴的复元。其次，我们给出了P^{centerdot}是双元倾斜复形的等价条件，即诱导双元$\{\cat{D}_{A/AeA}(A), \cat{D}(A), \cat{D}(eAe)}$与$\{\cat{D}_{B/BfB}(B), \cat{D}(B), \cat{D}(fBf)}$等价的倾斜复形，其中e，f分别是A，B的幂等元.在这种情况下，存在一个无界双模复形$\varDelta^{\centerdot}_{T}$，它导致$\cat{D}_{A/AeA}(A)$和$\cat{D}_{B/BfB}(B)$之间的等价性。第三，我们将上述应用于对称代数A。我们证明了长度为2的部分倾斜复形$P^{\centerdot}$扩张为倾斜复形，并且$P^{\centerdot}$是倾斜复形当且仅当$P^{\centerdot}$的不可分解类型数是A之一.最后，我们表明，对于一个幂等e的A，倾斜复杂的eAe扩展到一个倾斜复杂的A，它的标准等价诱导之间的等价$\cat{Mod}A/AeA$和$\cat{Mod}B/BfB$。

英文摘要：

First, we study recollement of a derived category of unbounded complexes of modules induced by a partial tilting complex. Second, we give equivalent conditions for P^{centerdot} to be a recollement tilting complex, that is, a tilting complex which induces an equivalence between recollements $\{\cat{D}_{A/AeA}(A), \cat{D}(A), \cat{D}(eAe)}$ and $\{\cat{D}_{B/BfB}(B), \cat{D}(B), \cat{D}(fBf)}$, where e, f are idempotents of A, B, respectively. In this case, there is an unbounded bimodule complex $\varDelta^{\centerdot}_{T}$ which induces an equivalence between $\cat{D}_{A/AeA}(A)$ and $\cat{D}_{B/BfB}(B)$. Third, we apply the above to a symmetric algebra A. We show that a partial tilting complex $P^{\centerdot}$ for A of length 2 extends to a tilting complex, and that $P^{\centerdot}$ is a tilting complex if and only if the number of indecomposable types of $P^{\centerdot}$ is one of A. Finally, we show that for an idempotent e of A, a tilting complex for eAe extends to a recollement tilting complex for A, and that its standard equivalence induces an equivalence between $\cat{Mod}A/AeA$ and $\cat{Mod}B/BfB$.

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摘要：

英文摘要：

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