11.1 Local cohomology

Local cohomology connects algebra and geometry by studying modules with support 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. Matlis duality links finitely generated and artinian modules, while local duality relates local cohomology to Ext modules. 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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• Local cohomology functor $H^i_I(M)$ associates to an $R$-module $M$ and an ideal $I \subset R$ the $i$-th local cohomology module of $M$ with support in $I$
• Constructed using the Čech complex, a cochain complex associated to a cover of a topological space
  • For an ideal $I = (f_1, \ldots, f_n)$, the Čech complex is $\check{C}^\bullet(f_1, \ldots, f_n; M)$
  • The $i$-th local cohomology module is the $i$-th cohomology of this complex $H^i_I(M) = H^i(\check{C}^\bullet(f_1, \ldots, f_n; M))$
• Local cohomology measures the depth of a module, the length of a maximal $M$-sequence in $I$
  • $\operatorname{depth}_I(M) = \inf\{i \mid H^i_I(M) \neq 0\}$

Properties and Vanishing Theorems

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

Duality in Local Cohomology

Support and Matlis Duality

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

Local Duality Theorem

• Local duality relates the local cohomology of a finitely generated module $M$ over a local ring $(R,\mathfrak{m})$ with the Matlis dual of the local cohomology of the Matlis dual $M^\vee$
  • For a Cohen-Macaulay local ring of dimension $d$, there are isomorphisms $H^i_\mathfrak{m}(M) \cong \operatorname{Ext}^{d-i}_R(M,\omega_R)^\vee$ where $\omega_R$ is the canonical module
• Consequences and applications of local duality
  • Relates the depth of a module to the smallest non-vanishing Ext module (Auslander-Buchsbaum formula)
  • Gives a duality between the local cohomology of a ring and its canonical module (Grothendieck's Local Duality Theorem)
• Examples illustrating local duality for specific local rings and modules (regular local rings, Gorenstein local rings)