scholarly journals The Frobenius action on local cohomology modules in mixed characteristic

2007 ◽  
Vol 143 (6) ◽  
pp. 1478-1492 ◽  
Author(s):  
Kazuma Shimomoto
Keyword(s):  
Local Cohomology ◽  
Ring Theory ◽  
Local Rings ◽  
New Approach ◽  
Mixed Characteristic ◽  
Local Cohomology Modules ◽  
Frobenius Map ◽  
The Subject ◽  
Torsion Modules ◽  
Cohomology Modules

AbstractHeitmann’s proof of the direct summand conjecture has opened a new approach to the study of homological conjectures in mixed characteristic. Inspired by his work and by the methods of almost ring theory, we discuss a normalized length for certain torsion modules, which was introduced by Faltings. Using the normalized length and the Frobenius map, we prove some results of local cohomology for local rings in mixed characteristic, which has an immediate implication for the subject of splinters studied by Singh.

scholarly journals Parameter-test-ideals of Cohen–Macaulay rings

2008 ◽  
Vol 144 (4) ◽  
pp. 933-948 ◽  
Author(s):  
Mordechai Katzman
Keyword(s):  
Local Cohomology ◽  
Residue Field ◽  
Injective Hull ◽  
Local Cohomology Modules ◽  
D Modules ◽  
Frobenius Map ◽  
Parameter Test ◽  
Computing Parameter ◽  
Injective Hulls ◽  
Cohomology Modules

AbstractWe describe an algorithm for computing parameter-test-ideals in certain local Cohen–Macaulay rings. The algorithm is based on the study of a Frobenius map on the injective hull of the residue field of the ring and on the application of Sharp’s notion of ‘special ideals’. Our techniques also provide an algorithm for computing indices of nilpotency of Frobenius actions on top local cohomology modules of the ring and on the injective hull of its residue field. The study of nilpotent elements on injective hulls of residue fields also yields a great simplification of the proof of the celebrated result in the article Generators of D-modules in positive characteristic (J. Alvarez-Montaner, M. Blickle and G. Lyubeznik, Math. Res. Lett. 12 (2005), 459–473).

Cohomological annihilators

1982 ◽  
Vol 91 (3) ◽  
pp. 345-350 ◽  
Author(s):  
Peter Schenzel
Keyword(s):  
Maximal Ideal ◽  
Commutative Algebra ◽  
Local Cohomology ◽  
Local Rings ◽  
Intersection Theorem ◽  
Prime Characteristic ◽  
Local Cohomology Modules ◽  
System Of Parameters ◽  
Cohomology Modules ◽  
Unique Maximal Ideal

The local cohomology theory introduced by Grothendieck(1) is a useful tool for attacking problems in commutative algebra and algebraic geometry. Let A denote a local ring with its unique maximal ideal m. For an ideal I ⊂ A and a finitely generated A-module M we consider the local cohomology modules HiI (M), i є ℤ, of M with respect to I, see Grothendieck(1) for the definition. In particular, the vanishing resp. non-vanishing of the local cohomology modules is of a special interest. For more subtle considerations it is necessary to study the cohomological annihilators, i.e. aiI(M): = AnnΔHiI(M), iєℤ. In the case of the maximal ideal I = m these ideals were used by Roberts (6) to prove the ‘New Intersection Theorem’ for local rings of prime characteristic. Furthermore, we used this notion (7) in order to show the amiability of local rings possessing a dualizing complex. Note that the amiability of a system of parameters is the key step for Hochster's construction of big Cohen-Macaulay modules for local rings of prime characteristic, see Hochster(3) and (4).

LOCAL COHOMOLOGY OF MULTI-REES ALGEBRAS, JOINT REDUCTION NUMBERS AND PRODUCT OF COMPLETE IDEALS

2016 ◽  
Vol 228 ◽  
pp. 1-20 ◽  
Author(s):  
PARANGAMA SARKAR ◽  
J. K. VERMA
Keyword(s):  
Local Cohomology ◽  
Local Rings ◽  
Rees Algebras ◽  
Local Cohomology Modules ◽  
Joint Reduction ◽  
Cohomology Modules

We find conditions on the local cohomology modules of multi-Rees algebras of admissible filtrations which enable us to predict joint reduction numbers. As a consequence, we are able to prove a generalization of a result of Reid, Roberts and Vitulli in the setting of analytically unramified local rings for completeness of power products of complete ideals.

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