Local Cohomology—An Invitation

DOI10.1007/978-3-030-89694-2_26zbMath1498.13051arXiv2106.09796OpenAlexW4213134957MaRDI QIDQ5051900

No author found.

Publication date: 18 November 2022

Published in: Commutative Algebra (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2106.09796

Mathematics Subject Classification ID

Local cohomology and commutative rings (13D45) Research exposition (monographs, survey articles) pertaining to commutative algebra (13-02)

Related Items

On Lyubeznik type invariants, Topological calculation of local cohomological dimension

Uses Software

• Macaulay2
• Kan

Cites Work

• Nilpotence of Frobenius action and the Hodge filtration on local cohomology
• Local cohomology modules supported at determinantal ideals
• Cohomology of sheaves
• Local cohomology of logarithmic forms
• Weights on cohomology, invariants of singularities, and dual complexes
• Local cohomology with support in generic determinantal ideals
• Local cohomology modules of a smooth \(\mathbb{Z}\)-algebra have finitely many associated primes
• The \(F\)-pure threshold of a determinantal ideal
• The Jacobian module, the Milnor fiber, and the \(D\)-module generated by \(f^s\)
• Arithmetical rank of strings and cycles
• \(F\)-injectivity and Frobenius closure of ideals in Noetherian rings of characteristic \(p>0\)
• A characteristic-free proof of a basic result on \(\mathcal D\)-modules
• Lyubeznik numbers of projective schemes
• On the classification of regular holonomic \(\mathcal D\)-modules on skew-symmetric matrices
• Bernstein-Sato polynomials of hyperplane arrangements
• Eulerian graded \(\mathcal{D}\)-modules
• On the vanishing of local cohomology modules
• Dimension projective finie et cohomologie locale. Applications à la demonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck
• On the structure of the endomorphism ring of a certain local cohomology module
• Faisceaux algébriques cohérents
• Algebraic geometry and analytic geometry
• Local cohomology multiplicities in terms of étale cohomology
• Uniform annihilators of local cohomology
• \(D\)-modules, perverse sheaves, and representation theory. Translated from the Japanese by Kiyoshi Takeuchi
• Absolute integral closure in positive characteristic
• On the Matlis duals of local cohomology modules
• On injective resolutions of local cohomology modules
• Test, multiplier and invariant ideals
• Irregularity of hypergeometric systems via slopes along coordinate subspaces
• An introduction to homological algebra
• \(D\)-modules over rings with finite F-representation type
• On cohomologically complete intersections
• Discreteness and rationality of \(F\)-jumping numbers on singular varieties
• Matlis duals of local cohomology modules and their endomorphism rings
• The Frobenius structure of local cohomology
• On connectedness and indecomposibility of local cohomology modules
• \({\mathcal D}\)-modules et faisceaux pervers dont le support singulier est un croisement normal
• The existence of infinitely many supersingular primes for every elliptic curve over \(\mathbb Q\).
• Bemerkungen über Zusammenhangseigenschaften und mengentheoretische Darstellung projektiver algebraischer Mannigfaltigkeiten
• Dualisierende Komplexe in der lokalen Algebra und Buchsbaum-Ringe
• Generalized Euler integrals and \(A\)-hypergeometric functions
• Uniform bounds in noetherian rings
• Infinite integral extensions and big Cohen-Macaulay algebras
• Derived functors of \(I\)-adic completion and local homology
• Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori
• Local cohomological dimension of algebraic varieties
• Local cohomology and the Cousin complex for a commutative Noetherian ring
• B-functions and holonomic systems. Rationality of roots of B-functions
• Local cohomological dimension in characteristic p
• Affine curves in characteristic p are set theoretic complete intersections
• Note on set-theoretic intersections of subvarieties of projective space
• The complexity of Frobenius powers of ideals
• Higher derivations and local cohomology modules
• Weak F-regularity deforms in \(\mathbb{Q}\)-Gorenstein rings
• An algorithm for de Rham cohomology groups of the complement of an affine variety via \(D\)-module computation
• Algorithmic computation of local cohomology modules and the local cohomological dimension of algebraic varieties
• Deformation of F-purity and F-regularity
• Local cohomology of modules of covariants
• Finiteness properties of local cohomology modules (an application of \(D\)- modules to commutative algebra)
• Hypergeometric functions and rings generated by monomials
• Complements of sphere and subspace arrangements
• An algorithm of computing \(b\)-functions
• Stanley-Reisner rings, sheaves, and Poincaré-Verdier duality
• The intersection homology \(D\)-module in finite characteristic
• Cohomology and toric varieties and local cohomology with monomial supports
• Algorithmic computation of de Rham cohomology of complements of complex affine varieties
• Injective dimension of \(D\)-modules: A characteristic-free approach
• Local cohomology of Frobenius images over graded affine algebras
• Characteristic cycles of local cohomology modules of monomial ideals
• Finiteness properties of local cohomology modules: A characteristic-free approach
• \(p\)-torsion elements in local cohomology modules
• Duality and de Rham cohomology for graded \(\mathcal{D}\)-modules
• \(D\)-modules, Bernstein-Sato polynomials and \(F\)-invariants of direct summands
• Support of local cohomology modules over hypersurfaces and rings with FFRT
• \(A\)-hypergeometric modules and Gauss-Manin systems
• \({\mathcal D}\)-modules associated to the determinantal singularities
• Etale cohomological dimension and the topology of algebraic varieties
• Topological computation of local cohomology multiplicities
• The analytic continuation of generalized functions with respect to a parameter
• Every algebraic set in n-space is the intersection of n hypersurfaces
• An example of an infinite set of associated primes of a local cohomology module
• Local cohomology, arrangements of subspaces and monomial ideals.
• Characteristic cycles of local cohomology modules of monomial ideals. II
• Arithmetical ranks of ideals associated to symmetric and alternating matrices
• Annihilating the cohomology of group schemes
• Gröbner deformations of hypergeometric differential equations
• An étale analog of the Goresky-MacPherson formula for subspace arrangements
• On simplicial toric varieties which are set-theoretic complete intersections
• Annihilators of local cohomology modules and simplicity of rings of differential operators
• Computing the monodromy and pole order filtration on Milnor fiber cohomology of plane curves
• Faithfulness of top local cohomology modules in domainss
• Bockstein cohomology of associated graded rings
• On categories of equivariant \(\mathcal{D}\)-modules
• Algebraic aspects of hypergeometric differential equations
• Koszul and local cohomology, and a question of Dutta
• An asymptotic vanishing theorem for the cohomology of thickenings
• On asymptotic socle degrees of local cohomology modules
• F-stable secondary representations and deformation of F-injectivity
• On Lyubeznik type invariants
• Iterated local cohomology groups and Lyubeznik numbers for determinantal rings
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Roots of Bernstein-Sato polynomials of certain homogeneous polynomials with two-dimensional singular loci
• Simple \(\mathcal{D}\)-module components of local cohomology modules
• Dependence of Lyubeznik numbers of cones of projective schemes on projective embeddings
• Quasi-cyclic modules and coregular sequences
• Intersection homology \(\mathcal{D}\)-module and Bernstein polynomials associated with a complete intersection
• Dualizing, projecting, and restricting GKZ systems
• Equivariant \(\mathcal{D}\)-modules on \(2 \times 2 \times 2\) hypermatrices
• Line and rational curve arrangements, and Walther's inequality
• \(F\)-injectivity and Buchsbaum singularities
• On some local cohomology modules
• On invariants and endomorphism rings of certain local cohomology modules
• Local cohomology and pure morphisms
• Nilpotence of Frobenius actions on local cohomology and Frobenius closure of ideals
• Torus equivariant \(D\)-modules and hypergeometric systems
• On a class of holonomic \(\mathcal D\)-modules on \(M_n(\mathbb C)\) related to the action of \(\mathrm{GL}_n(\mathbb C)\times \mathrm{GL}_n (\mathbb C)\)
• Local cohomology with support in ideals of maximal minors and sub-maximal Pfaffians
• Uniform annihilators of local cohomology of excellent rings
• Operations with regular holonomic \(\mathcal D\)-modules with support a normal crossing
• Differential operators on orbifolds
• Computing the support of local cohomology modules
• A linear bound for Frobenius powers and an inclusion bound for tight closure
• On toric varieties which are almost set-theoretic complete intersections
• On some local cohomology invariants of local rings
• Local cohomology modules of polynomial or power series rings over rings of small dimension
• Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas (Quatrième partie). Rédigé avec la colloboration de J. Dieudonné
• Cohomological dimension of algebraic varieties
• The Cousin complex for a module over a commutative Noetherian ring
• Affine duality and cofiniteness
• Residues and duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Havard 1963/64. Appendix: Cohomology with supports and the construction of the \(f^!\) functor by P. Deligne
• Théorie de Hodge. II. (Hodge theory. II)
• Théorie de Hodge. III
• Generators of \(D\)-modules in positive characteristic
• Hypergeometric functions and toric varieties
• Local cohomology and F-stability
• On the support of local cohomology
• On the discreteness and rationality of \(F\)-jumping coefficients
• Hypergeometric D-modules and twisted Gauß-Manin systems
• On the Lyubeznik numbers of a local ring
• Resonant Hypergeometric Systems and Mirror Symmetry
• The saturation of frobenius powers of ideals
• Finiteness properties of local cohomology modules for regular local rings of mixed characteristic: the unramified case
• On the commutation of the test ideal with localization and completion
• Set-theoretic complete intersections on binomials
• An extension of a theorem of Hartshorne
• On the arithmetical rank of certain segre embeddings
• Cohomological and projective dimensions
• Rings of Frobenius operators
• Laurent polynomials, GKZ-hypergeometric systems and mixed Hodge modules
• Annihilators of Top Local Cohomology
• Graded F -modules and local cohomology
• Finiteness Properties of Local Cohomology for F-Pure Local Rings
• Deformation of F-injectivity and local cohomology
• Algorithms for Bernstein--Sato polynomials and multiplier ideals
• Local cohomology with support in ideals of symmetric minors and Pfaffians
• Survey on the $D$-module $f^s$
• On the Endomorphism Rings of Local Cohomology Modules
• Test ideals in non-$\mathbb{Q}$-Gorenstein rings
• Cartier modules: Finiteness results
• LYUBEZNIK NUMBERS OF LOCAL RINGS AND LINEAR STRANDS OF GRADED IDEALS
• FROBENIUS ACTIONS ON LOCAL COHOMOLOGY MODULES AND DEFORMATION
• F-rational rings have rational singularities
• Bass Numbers of Local Cohomology Modules
• Differential Operators on Hyperplane Arrangements
• A survey on the Lyubeznik numbers
• Complete Intersections in Characteristic p > 0
• On the vanishing of local cohomology in characteristic p > 0
• On the Hartshorne–Speiser–Lyubeznik theorem about Artinian modules with a Frobenius action
• On the highest Lyubeznik number of a local ring
• Graded annihilators of modules over the Frobenius skew polynomial ring, and tight closure
• Matlis Duals of Top Local Cohomology Modules and the Arithmetic Rank of an Ideal
• Lyubeznik Table of Sequentially Cohen–Macaulay Rings
• Tight Closure, Invariant Theory, and the Briancon-Skoda Theorem
• A connectedness result in positive characteristic
• On endomorphism rings of local cohomology modules
• On simplicial toric varieties of codimension 2
• A Simple Characterization of Du Bois Singularities
• On endomorphism rings and dimensions of local cohomology modules
• F-injective singularities are Du Bois
• On b-function, spectrum and multiplier ideals
• F-Purity and Rational Singularity
• Set-Theoretic Complete Intersections
• Some Algebraic Sets of High Local Cohomological Dimension in Projective Space
• Rings of Differential Operators on Invariant Rings of Tori
• Studies in Duality on Noetherian Formal Schemes and Non-Noetherian Ordinary Schemes
• Connectedness and Lyubeznik Numbers
• Cohomological dimension of ideals defining Veronese subrings
• Mixed Hodge modules
• On the De Rham cohomology of algebraic varieties
• Minimal resultant systems
• A characterization of rational singularities
• On the De Rham cohomology of algebraic varieties
• Algorithms for \(D\)-modules -- restriction, tensor product, localization, and local cohomology groups
• The associated primes of local cohomology modules over rings of small dimension
• \({\mathcal D}\)-modules on smooth toric varieties
• Complexe de de Rham filtré d'une variété singulière
• Mengentheoretisch vollständige Durchschnitte verschiedener rationaler Raumkurven im P³ über Körpern von Primzahlcharakteristik
• The Integrability of the Characteristic Variety
• Cohomological annihilators
• Cofiniteness and vanishing of local cohomology modules
• The Number of Defining Equations of Affine Algebraic Sets
• Varieties of small codimension in projective space
• Two applications of dualizing complexes over local rings
• Projective Varieties of Low Codimension in Characteristic p > 0
• Über lokale Kohomologiegruppen hoher Ordnung.
• A characterization of rational singularities in terms of injectivity of Frobenius maps
• Normal ideals in regular rings
• F-regularity does not deform
• Rings of differential operators on toric varieties
• F-Regularity, Test Elements, and Smooth Base Change
• Differential operators on Stanley-Reisner rings
• F-modules: applications to local cohomology and D-modules in characteristic p>0.
• Singularities of Pairs
• On the equations defining projective monomial curves
• 𝐹-injective rings and 𝐹-stable primes
• On the highest Lyubeznik number
• $F$-singularities: applications of characteristic $p$ methods to singularity theory
• On the Matlis duals of local cohomology modules
• Completion, Čech and Local Homology and Cohomology
• Content of local cohomology, parameter ideals, and robust algebras
• On the de Rham homology and cohomology of a complete local ring in equicharacteristic zero
• The weak ordinarity conjecture and $F$-singularities
• Bernstein–Sato polynomial versus cohomology of the Milnor fiber for generic hyperplane arrangements
• $D$-module and $F$-module length of local cohomology modules
• Algebraic study of systems of partial differential equations
• On Some Local Cohomology Spectral Sequences
• Length of Local Cohomology in Positive Characteristic and Ordinarity
• On a conjecture of Lynch
• Hypergeometric Hodge modules
• Lyubeznik numbers for nonsingular projective varieties
• Bockstein homomorphisms in local cohomology
• The Number of Equations Defining a Determinantal Variety
• Lyubeznik numbers of irreducible projective varieties depend on the embedding
• The Support of Local Cohomology Modules
• On derived functors of graded local cohomology modules
• F-singularities via alterations
• Local Cohomology Modules Supported on Monomial Ideals
• A remark on bigness of the tangent bundle of a smooth projective variety and D-simplicity of its section rings
• de Rham cohomology of local cohomology modules: The graded case
• A note on Lynch’s conjecture
• Homological methods for hypergeometric families
• On injective dimension of F‐finite F‐modules and holonomic D‐modules
• On the relationship between depth and cohomological dimension
• Characters of equivariant -modules on spaces of matrices
• Gorenstein-duality for one-dimensional almost complete intersections-with an application to non-isolated real singularities
• Stabilization of the cohomology of thickenings
• On determinantal ideals and algebraic dependence
• Equivariant D-modules on binary cubic forms
• A dichotomy for the injective dimension ofF-finiteF-modules and holonomicD-modules
• Connectivity of hyperplane sections of domains
• Lyubeznik numbers and injective dimension in mixed characteristic
• $D$-module structure of local cohomology modules of toric algebras
• Lyubeznik numbers of monomial ideals
• Matlis duals of top Local cohomology modules
• Finiteness Properties of Duals of Local Cohomology Modules
• Set-theoretic complete intersections in characteristic 𝑝
• On the Associated Primes of Matlis Duals of Top Local Cohomology Modules
• Singular Points of Complex Hypersurfaces. (AM-61)
• Characterizations of Regular Local Rings of Characteristic p
• The integrability of characteristics
• Transplanting Cohomology Classes in Complex-Projective Space
• Rings of Invariants of Tori, Cohen-Macaulay Rings Generated by Monomials, and Polytopes