2022/07/05 by Xiaoyan Yang, Yang, Xiaoyan, Jingwen Shen +1
Mathematics · #13D09 #13D45 #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2207.01785
openalex publication_date 2022/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathfraka 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 dimR=d or dimR/\mathfraka=d-1 or ara(\mathfraka)=d-1, show that if d≤2 then an R-complex X\inD_\sqsubset(R) is \mathfraka-cofinite if and only if each homology module Hi(X) is \mathfraka-cofinite; if \mathfraka is a perfect ideal and R is regular local with d≤2 then an R-complex X\inD(R) is \mathfraka-cofinite if and only if Hi(X) is \mathfraka-cofinite for every i∈ℤ; if d≥3 then for an R-complex X of \mathfraka-cofinite R-modules, each Hi(X) is \mathfraka-cofinite if and only if ExtjR(R/\mathfraka,cokerdi) are finitely generated for j≤ d-2. We also study cofiniteness of local cohomology Hi_\mathfraka(X) for an R-complex X\inD_\sqsubset(R) in the above cases. The crucial step to achieve these is to recruit the technique of spectral sequences.