论文标题
Čech（共同）复合物作为Koszul复合物和应用
Čech (co-) complexes as Koszul complexes and applications
论文作者
论文摘要
令$ \ check {c} _ {\ loessline {x}} $表示čech复合体相对于元素系统$ \ usevenline {x} = x_1，\ ldots，x_r $ r $ r $。我们构建了一个有限的复杂$ \ Mathcal {l} _ {\下划线{x}} $的免费$ r $ -modules和Quasi-Isomorphism $ \ Mathcal {l} _ {\ supperline {x}}} \ check {c} _ {\ usewissline {x}} $ and isomorphisms $ \ mathcal {l} _ {\ usewiseline {x}} \ otimes_r x \ cong k^\ cong k^{\ bulter}（\ buleast}（ x [\下划线{ $ r $ - complex $ x $。这里$ \下划线{x} - \下划线{u} $表示元素的顺序$ x_1-u_1，\ ldots，x_r-u_r $ in polyenmial Ring $ r [\ useverline {u} u_1，\ ldots，u_r $ of $ r $。此外，$ x [[\ usewissline {u}] $表示$ \ usepline {u} $和$ x [\ usepline {u}^{ - 1}] $ in $ x $的正式功率系列复合体表示$ \ \ upessline {u} $的$ x $ $ x $的相反多项式的复杂。此外，$ k _ {\ bullet}（\下划线{x} - \ usepline {u}; x [[\ upsine {u}]]）$ resp。 $ k^{\ bullet}（\下划线{x} - \ usewissline {u}; x [\ usepline {u}^{ - 1}]）$表示相应的koszul复合物。相应的Koszul共络合物。特别是，由某个Koszul Complects有一个有限的$ r $ - free分辨率的$ \ check {c} _ {\ usewissline {x}} $。这有各种后果，例如在$ \ underline {x} $的情况下，是一个弱的pro-try-temular序列。在这一其他假设下，请参见本地的同胞$ h^i _ {\下划线{x} r}（x）$以及完成$λ_i^{\ unesevenlline {x} r}（x} r}（x）（x），i \ in \ mathbbb {z}，$ koszul cos的左派生函数。这为扭转的右衍生函数提供了新的方法，并提供了各种应用程序的左派生函数。
Let $\check{C}_{\underline{x}}$ denote the Čech complex with respect to a system of elements $\underline{x} = x_1,\ldots,x_r$ of a commutative ring $R$. We construct a bounded complex $\mathcal{L}_{\underline{x}}$ of free $R$-modules and a quasi-isomorphism $\mathcal{L}_{\underline{x}} \stackrel{\sim}{\longrightarrow} \check{C}_{\underline{x}}$ and isomorphisms $\mathcal{L}_{\underline{x}} \otimes_R X \cong K^{\bullet}(\underline{x}-\underline{U}; X[\underline{U}^{-1}])$ and $\operatorname{Hom}_R(\mathcal{L}_{\underline{x}},X) \cong K_{\bullet}(\underline{x}-\underline{U};X[[\underline{U}]])$ for an $R$-complex $X$. Here $\underline{x} - \underline{U}$ denotes the sequence of elements $x_1-U_1,\ldots,x_r-U_r$ in the polynomial ring $R[\underline{U}] = R[U_1,\ldots,U_r]$ in the variables $\underline{U}= U_1,\ldots,U_r$ over $R$. Moreover $X[[\underline{U}]]$ denotes the formal power series complex of $X$ in $\underline{U}$ and $X[\underline{U}^{-1}]$ denotes the complex of inverse polynomials of $X$ in $\underline{U}$. Furthermore $K_{\bullet}(\underline{x}-\underline{U};X[[\underline{U}]])$ resp. $K^{\bullet}(\underline{x}-\underline{U}; X[\underline{U}^{-1}])$ denotes the corresponding Koszul complex resp. the corresponding Koszul co-complex. In particular, there is a bounded $R$-free resolution of $\check{C}_{\underline{x}}$ by a certain Koszul complex. This has various consequences e.g. in the case when $\underline{x}$ is a weakly pro-regular sequence. Under this additional assumption it follows that the local cohomology $H^i_{\underline{x} R}(X)$ and the left derived functors of the completion $Λ_i^{\underline{x} R}(X), i \in \mathbb{Z},$ is a certain Koszul cohomology and Koszul homology resp. This provides new approaches to the right derived functor of torsion and the left derived functor of completion with various applications.
