Glossary

11.1 Local cohomology

3 min read•august 7, 2024

connects algebra and geometry by studying modules with in ideals. It uses Čech complexes to measure how deeply an ideal "cuts into" a module, revealing important structural information about rings and modules.

Local cohomology has deep connections to duality theories. links finitely generated and , while relates local cohomology to . These tools provide powerful insights into module structure and ring properties.

Local Cohomology and Čech Complex

Definition and Construction of Local Cohomology

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  • HIi(M)H^i_I(M) associates to an RR-module MM and an ideal I⊂RI \subset R the ii-th local cohomology module of MM with support in II
  • Constructed using the , a cochain complex associated to a cover of a topological space
    • For an ideal I=(f1,…,fn)I = (f_1, \ldots, f_n), the Čech complex is Cˇ∙(f1,…,fn;M)\check{C}^\bullet(f_1, \ldots, f_n; M)
    • The ii-th local cohomology module is the ii-th cohomology of this complex HIi(M)=Hi(Cˇ∙(f1,…,fn;M))H^i_I(M) = H^i(\check{C}^\bullet(f_1, \ldots, f_n; M))
  • Local cohomology measures the of a module, the length of a maximal MM-sequence in II
    • depth⁡I(M)=inf⁡{i∣HIi(M)≠0}\operatorname{depth}_I(M) = \inf\{i \mid H^i_I(M) \neq 0\}

Properties and Vanishing Theorems

  • Local cohomology modules HIi(M)H^i_I(M) are II- annihilated by a power of II
  • Vanishing theorems give conditions for the local cohomology modules to be zero
    • For a noetherian ring RR, HIi(M)=0H^i_I(M) = 0 for i>dim⁡R(M)i > \dim_R(M) ()
    • If II is generated by nn elements, then HIi(M)=0H^i_I(M) = 0 for i>ni > n regardless of the dimension of MM
  • Computing examples of local cohomology using the Čech complex for specific ideals and modules (polynomial ring k[x,y]k[x,y] and the ideal (x,y)(x,y))

Duality in Local Cohomology

Support and Matlis Duality

  • The support of an RR-module MM is the set of primes Supp⁡(M)={P∈Spec⁡(R)∣MP≠0}\operatorname{Supp}(M) = \{P \in \operatorname{Spec}(R) \mid M_P \neq 0\}
    • Equivalently, the support is the variety of the annihilator ideal Ann⁡(M)\operatorname{Ann}(M)
  • Matlis duality establishes a duality between finitely generated modules over a complete local ring and artinian modules
    • For a complete local ring (R,m)(R,\mathfrak{m}) and a finitely generated RR-module MM, the Matlis dual is M∨=Hom⁡R(M,E(R/m))M^\vee = \operatorname{Hom}_R(M,E(R/\mathfrak{m})) where E(R/m)E(R/\mathfrak{m}) is the of the residue field
    • Matlis duality gives an equivalence of categories between finitely generated RR-modules and artinian RR-modules

Local Duality Theorem

  • Local duality relates the local cohomology of a MM over a local ring (R,m)(R,\mathfrak{m}) with the Matlis dual of the local cohomology of the Matlis dual M∨M^\vee
    • For a of dimension dd, there are isomorphisms Hmi(M)≅Ext⁡Rd−i(M,ωR)∨H^i_\mathfrak{m}(M) \cong \operatorname{Ext}^{d-i}_R(M,\omega_R)^\vee where ωR\omega_R is the
  • Consequences and applications of local duality
    • Relates the depth of a module to the smallest non-vanishing Ext module ()
    • Gives a duality between the local cohomology of a ring and its canonical module ()
  • Examples illustrating local duality for specific local rings and modules (regular local rings, Gorenstein local rings)

Key Terms to Review (17)

Artinian modules: Artinian modules are modules that satisfy the descending chain condition on submodules, meaning any descending chain of submodules eventually stabilizes. This property leads to several important features, such as the ability to decompose modules into simple components and their relevance in various homological contexts, particularly in local cohomology where understanding module structure is crucial for computations and the study of depth and dimension.
Auslander-Buchsbaum Formula: The Auslander-Buchsbaum formula is a fundamental result in homological algebra that relates the depth of a module over a Noetherian ring to its projective dimension and the dimensions of the local cohomology modules. This formula captures the interaction between these important invariants, providing insight into the structure of modules and their associated cohomological properties.
Canonical module: The canonical module of a ring is a crucial object that serves as a dualizing complex, providing insights into the homological properties of the ring. It plays an important role in understanding local cohomology and has deep connections to Cohen-Macaulay and Gorenstein rings, particularly in how it describes their singularities and duality properties.
Čech Complex: The Čech complex is a construction in algebraic topology that associates a simplicial complex to a topological space, allowing for the computation of its homology and cohomology. It is particularly useful in the study of local cohomology, as it captures the properties of sheaves over a space by creating a resolution that can be analyzed algebraically. This approach helps to establish connections between topological properties and algebraic structures, providing tools for understanding how local features affect global behavior.
Cohen-Macaulay Local Ring: A Cohen-Macaulay local ring is a type of commutative ring that satisfies certain depth and dimension conditions, specifically having a depth equal to its Krull dimension. This property indicates that the ring has nice homological characteristics, making it easier to study the properties of modules over it. In essence, Cohen-Macaulay rings allow for better understanding of their structure and the behavior of local cohomology.
Depth: In algebra, depth refers to the length of the longest regular sequence in a module over a ring, which essentially measures the 'size' of the module in a homological sense. This concept is crucial as it links to both local cohomology and the structure of rings, particularly in determining whether rings are Cohen-Macaulay or Gorenstein. A higher depth indicates a richer structural feature of the module, reflecting its connections to various important properties like dimension and regular sequences.
Ext Modules: Ext modules are an important concept in homological algebra that measure the extent to which certain sequences fail to be exact. Specifically, they provide a way to characterize the derived functor of the Hom functor, capturing information about extensions of modules and the relationships between them. In particular, they help in understanding how modules can be constructed or decomposed through extension problems, linking directly to concepts like local cohomology.
Finitely generated module: A finitely generated module is a module that can be expressed as a finite combination of elements from a generating set. This means there exists a finite subset of the module such that every element in the module can be written as a linear combination of those generators. Finitely generated modules play a significant role in various areas of algebra, particularly in understanding the structure and properties of modules over rings, including how they relate to cohomology theories and local properties.
Grothendieck's Local Duality Theorem: Grothendieck's Local Duality Theorem provides a powerful connection between local cohomology and the derived category of sheaves. It essentially states that for a Noetherian ring and a finitely generated module, the local cohomology groups can be viewed as derived functors of the section functor, allowing us to understand how these groups behave under duality. This theorem has crucial implications for the study of sheaf cohomology and the interplay between local and global properties in algebraic geometry.
Grothendieck's Vanishing Theorem: Grothendieck's Vanishing Theorem states that for a given coherent sheaf on a projective scheme, the higher cohomology groups vanish when considered with respect to sufficiently high twisting by an ample line bundle. This theorem is pivotal as it provides a crucial link between algebraic geometry and homological algebra, particularly in understanding local cohomology and the behavior of sheaves under various conditions.
Injective hull: An injective hull is the smallest injective module that contains a given module as an essential submodule. It can be thought of as a way to 'enlarge' a module so that it becomes injective, which is useful in various algebraic contexts. The concept plays an important role in understanding injective modules, providing essential properties and characterizations that relate to other structures, such as local cohomology and Cohen-Macaulay rings.
Local cohomology: Local cohomology is a powerful tool in algebra that studies the properties of modules over a ring with respect to a specific prime ideal. It captures information about the local behavior of sheaves or modules, especially in relation to support, and plays a crucial role in various areas such as commutative algebra and algebraic geometry.
Local Cohomology Functor: The local cohomology functor is an important tool in algebraic geometry and commutative algebra, which associates a graded module to a given module over a ring and a prime ideal. It captures information about the behavior of a module near the specified prime ideal, allowing for a refined understanding of its structure and properties. This functor is essential for studying local properties of modules and sheaves, particularly in relation to depth and support.
Local duality: Local duality is a concept in algebra that connects local cohomology and the behavior of sheaves at prime ideals. It provides a framework for understanding how local cohomology modules relate to the duality of certain algebraic structures, particularly in the context of commutative algebra and topology. This notion extends to applications such as the study of Koszul complexes, where it plays a crucial role in understanding the relationships between different algebraic invariants.
Matlis Duality: Matlis duality is a concept in commutative algebra that provides a correspondence between certain modules over a Noetherian ring, specifically relating a module to its Matlis dual, which captures information about the module's structure and local properties. This duality is particularly important in the study of local cohomology, as it helps connect various algebraic structures and provides insight into their properties through duality, allowing for a deeper understanding of their cohomological aspects.
Support: In the context of local cohomology, support refers to the set of prime ideals in a ring that correspond to the points where a module or a sheaf is not locally free. This concept helps to identify the behavior of modules at specific locations in a given topological space, particularly with respect to properties like vanishing and non-vanishing of sections.
Torsion Modules: Torsion modules are modules over a ring where every element is annihilated by some non-zero element of the ring. This means that for each element in the module, there exists a non-zero scalar from the ring that when multiplied with the element results in zero. Torsion modules play a crucial role in understanding the structure of modules and their relationships, particularly when examining local cohomology, as they can influence the behavior of sheaves and the associated support in algebraic geometry.
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