scholarly journals Codes over \frak m -adic completion rings

2019 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Saadoun Mahmoudi ◽  
◽  
Karim Samei
Keyword(s):  
Adic Completion

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.

scholarly journals Formal fibers and birational extensions

1993 ◽  
Vol 131 ◽  
pp. 1-38 ◽  
Author(s):  
William Heinzer ◽  
Christel Rotthaus ◽  
Judith D. Sally
Keyword(s):  
Noetherian Ring ◽  
Important Condition ◽  
Quotient Field ◽  
Noetherian Domain ◽  
Definition Of ◽  
Adic Completion

Suppose (R, m) is a local Noetherian domain with quotient field K and m-adic completion Ȓ. It is well known that the fibers of the morphism Spec(Ȓ) ₒ Spec(R), i.e., the formal fibers of R, encode important information about the structure of R. Perhaps the most important condition in Grothendieck’s definition of R being excellent is that the formal fibers of R be geometrically regular. Indeed, a local Noetherian ring is excellent provided it is universally catenary and has geometrically regular formal fibers [G, (7.8.3), page 214]. But the structure of the formal fibers of R is often difficult to determine. We are interested here in bringing out the interrelatedness of properties of the generic formal fiber of R with the existence of certain local Noetherian domains C birationally dominating R and having C/mC is a finite R-module.

The spectrum (P ⋀ bo) −∞

1984 ◽  
Vol 96 (1) ◽  
pp. 85-93 ◽  
Author(s):  
Donald M. Davis ◽  
Mark Mahowald
Keyword(s):  
Inverse Limit ◽  
Projective Spaces ◽  
Inverse System ◽  
Cohomology Groups ◽  
Image Position ◽  
Adic Completion

There are spectra P−k constructed from stunted real projective spaces as in [1] such that H*(P−k) is the span in ℤ/2[x, x−1] of those xi with i ≥ −k. (All cohomology groups have ℤ/2-coefficients unless specified otherwise.) Using collapsing maps, these form an inverse systemwhich is similar to those of Lin ([15], p. 451). It is a corollary of Lin's work that there is an equivalence of spectrawhere holim is the homotopy inverse limit ([3], ch. 5) and Ŝ–1 the 2-adic completion of a sphere spectrum. One may denote by this holim (P–κ), although one must constantly keep in mind that , but rather

Building Noetherian Domains Inside an Ideal-adic Completion

2019 ◽  
pp. 279-288
Author(s):  
William Heinzer ◽  
Christel Rotthaus ◽  
Sylvia Wiegand
Keyword(s):  
Adic Completion
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