论文标题
Hartshorne关于Cofinite综合体的问题
Hartshorne's question on cofinite complexes
论文作者
论文摘要
让$ \ mathfrak {a} $是通勤的Noetherian环$ r $和$ d $ a正整数的适当理想。在$ \ mathrm {dim} r = d $或$ \ mathrm {dim} r/\ mathfrak {a} = d-1 $或$ \ mathrm {ara}（ara}（ara}（ara}（$ nathfrak {a a} $ d $ d $ d）中$ x \ in \ mathrm {d} _ \ sqsubset（r）$ is $ \ mathfrak {a} $ - cofinite且仅当每个同源模块$ \ mathrm {h} _i（x）$ is $ \ mathrm {如果$ \ mathfrak {a} $是一个完美的理想，而$ r $是普通本地的，则$ d \ leq2 $，则a $ r $ -complex $ x \ in \ mathrm {d}（r）$ as $ \ m athfrak {a} $ \ mathfrak {a} $ - cofinite in \ mathbb {z} $ in \ mathbb in \ mathbb;如果$ d \ geq3 $，则对于$ r $ - complex $ x $ $ \ MATHFRAK {a} $ - cofinite $ r $ -Modules，每个$ \ Mathrm {h} _i（x）$是$ \ Mathfrak {a} $ - cofInite，并且仅当$ \ mathrm {extrm {ext} $ J \ leq D-2 $。我们还研究了本地共同体学$ \ mathrm {h}^i_ \ mathfrak {a}（x）$的cofinitientens $ r $ -complex $ x \ in \ mathrm {d} _ \ sqsubset（r）$。实现这些目标的关键步骤是募集光谱序列的技术。
Let $\mathfrak{a}$ be a proper ideal of a commutative noetherian ring $R$ and $d$ a positive integer. We answer Hartshorne's question on cofinite complexes completely in the cases $\mathrm{dim}R=d$ or $\mathrm{dim}R/\mathfrak{a}=d-1$ or $\mathrm{ara}(\mathfrak{a})=d-1$, show that if $d\leq2$ then an $R$-complex $X\in\mathrm{D}_\sqsubset(R)$ is $\mathfrak{a}$-cofinite if and only if each homology module $\mathrm{H}_i(X)$ is $\mathfrak{a}$-cofinite; if $\mathfrak{a}$ is a perfect ideal and $R$ is regular local with $d\leq2$ then an $R$-complex $X\in\mathrm{D}(R)$ is $\mathfrak{a}$-cofinite if and only if $\mathrm{H}_i(X)$ is $\mathfrak{a}$-cofinite for every $i\in\mathbb{Z}$; if $d\geq3$ then for an $R$-complex $X$ of $\mathfrak{a}$-cofinite $R$-modules, each $\mathrm{H}_i(X)$ is $\mathfrak{a}$-cofinite if and only if $\mathrm{Ext}^j_R(R/\mathfrak{a},\mathrm{coker}d_i)$ are finitely generated for $j\leq d-2$. We also study cofiniteness of local cohomology $\mathrm{H}^i_\mathfrak{a}(X)$ for an $R$-complex $X\in\mathrm{D}_\sqsubset(R)$ in the above cases. The crucial step to achieve these is to recruit the technique of spectral sequences.
