scholarly journals The higher derived functors of the primitive element functor of quasitoric manifolds

2011 ◽  
Vol 158 (16) ◽  
pp. 2103-2110 ◽  
Author(s):  
David Allen ◽  
Jose La Luz
Keyword(s):  
Primitive Element ◽  
Derived Functors ◽  
Quasitoric Manifolds

scholarly journals On the Derived Functors of the Third Symmetric-Power Functor

2009 ◽  
Vol 2 (0) ◽  
pp. 19-39
Author(s):  
Bernhard Köck ◽  
Ramesh Satkurunath
Keyword(s):  
Symmetric Power ◽  
The Third ◽  
Derived Functors ◽  
Power Functor

scholarly journals Proregular sequences, local cohomology, and completion

2003 ◽  
Vol 92 (2) ◽  
pp. 161 ◽  
Author(s):  
Peter Schenzel
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Čech Complex ◽  
Regular Sequences ◽  
Local Cohomology Modules ◽  
Derived Functors ◽  
Adic Completion ◽  
Cech Complex ◽  
Cohomology Modules ◽  
The Ideal

As a certain generalization of regular sequences there is an investigation of weakly proregular sequences. Let $M$ denote an arbitrary $R$-module. As the main result it is shown that a system of elements $\underline x$ with bounded torsion is a weakly proregular sequence if and only if the cohomology of the Čech complex $\check C_{\underline x} \otimes M$ is naturally isomorphic to the local cohomology modules $H_{\mathfrak a}^i(M)$ and if and only if the homology of the co-Čech complex $\mathrm{RHom} (\check C_{\underline x}, M)$ is naturally isomorphic to $\mathrm{L}_i \Lambda^{\mathfrak a}(M),$ the left derived functors of the $\mathfrak a$-adic completion, where $\mathfrak a$ denotes the ideal generated by the elements $\underline x$. This extends results known in the case of $R$ a Noetherian ring, where any system of elements forms a weakly proregular sequence of bounded torsion. Moreover, these statements correct results previously known in the literature for proregular sequences.

scholarly journals Notes on endomorphisms, local cohomology and completion

2021 ◽  
pp. 169-180
Author(s):  
Peter Schenzel
Keyword(s):  
Noetherian Ring ◽  
Local Cohomology ◽  
Local Cohomology Module ◽  
Finitely Generated Module ◽  
Content Type ◽  
Local Cohomology Modules ◽  
Derived Functors ◽  
Adic Completion ◽  
Cohomology Modules

Let M M denote a finitely generated module over a Noetherian ring R R . For an ideal I ⊂ R I \subset R there is a study of the endomorphisms of the local cohomology module H I g ( M ) , g = g r a d e ( I , M ) , H^g_I(M), g = grade(I,M), and related results. Another subject is the study of left derived functors of the I I -adic completion Λ i I ( H I g ( M ) ) \Lambda ^I_i(H^g_I(M)) , motivated by a characterization of Gorenstein rings given in [25]. This provides another Cohen-Macaulay criterion. The results are illustrated by several examples. There is also an extension to the case of homomorphisms of two different local cohomology modules.

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