Local cohomology

In the geometric form of the theory, sections ${\displaystyle \Gamma _{Y}}$ are considered of a sheaf ${\displaystyle F}$ of abelian groups, on a topological space ${\displaystyle X}$, with support in a closed subset ${\displaystyle Y}$, The derived functors of ${\displaystyle \Gamma _{Y}}$ form local cohomology groups

${\displaystyle H_{Y}^{i}(X,F)}$

For applications in commutative algebra, the space X is the spectrum Spec(R) of a commutative ring R (supposed to be Noetherian throughout this article) and the sheaf F is the quasicoherent sheaf associated to an R-module M, denoted by ${\displaystyle {\tilde {M}}}$. The closed subscheme Y is defined by an ideal I. In this situation, the functor Γ_Y(F) corresponds to the annihilator

${\displaystyle \Gamma _{I}(M):=\bigcup _{n\geq 0}(0:_{M}I^{n}),}$

i.e., the elements of M which are annihilated by some power of I. Equivalently,

${\displaystyle \Gamma _{I}(M):=\varinjlim _{n\in N}\operatorname {Hom} _{R}(R/I^{n},M),}$

which also shows that local cohomology of quasi-coherent sheaves agrees with

${\displaystyle H_{I}^{i}(M):=\varinjlim _{n\in N}\operatorname {Ext} _{R}^{i}(R/I^{n},M).}$

There is a long exact sequence of sheaf cohomology linking the ordinary sheaf cohomology of X and of the open set U = X \Y, with the local cohomology groups.

In particular, this leads to an exact sequence

${\displaystyle 0\to H_{I}^{0}(M)\to M{\stackrel {\text{res}}{\to }}H^{0}(U,{\tilde {M}})\to H_{I}^{1}(M)\to 0,}$

where U is the open complement of Y and the middle map is the restriction of sections. The target of this restriction map is also referred to as the ideal transform. For n ≥ 1, there are isomorphisms

${\displaystyle H^{n}(U,{\tilde {M}}){\stackrel {\cong }{\to }}H_{I}^{n+1}(M).}$

An important special case is the one when R is graded, I consists of the elements of degree ≥ 1, and M is a graded module.[3] In this case, the cohomology of U above can be identified with the cohomology groups

${\displaystyle \bigoplus _{k\in \mathbf {Z} }H^{n}({\text{Proj}}(R),{\tilde {M}}(k))}$

of the projective scheme associated to R and (k) denotes the Serre twist. This relates local cohomology with global cohomology on projective schemes. For example, Castelnuovo–Mumford regularity can be formulated using local cohomology.[4]

The dimension dim_R(M) of a module (defined as the Krull dimension of its support) provides an upper bound for local cohomology groups:[5]

${\displaystyle H_{I}^{n}(M)=0{\text{ for all }}n>\dim _{R}(M).}$

If R is local and M finitely generated, then this bound is sharp, i.e., ${\displaystyle H_{\mathfrak {m}}^{n}(M)\neq 0}$.

The depth (defined as the maximal length of a regular M-sequence; also referred to as the grade of M) provides a sharp lower bound, i.e., it is the smallest integer n such that[6]

${\displaystyle H_{I}^{n}(M)\neq 0.}$

These two bounds together yield a characterisation of Cohen–Macaulay modules over local rings: they are precisely those modules where ${\displaystyle H_{\mathfrak {m}}^{n}(M)}$ vanishes for all but one n.

The local duality theorem is a local analogue of Serre duality. For a complete Cohen-Macaulay local ring R, it states that the natural pairing

${\displaystyle H_{\mathfrak {m}}^{n}(M)\times \operatorname {Ext} _{R}^{d-n}(M,\omega )\to H_{\mathfrak {m}}^{d}(R)}$

is a perfect pairing, where ω is a dualizing module for R.[7]

The initial applications were to analogues of the Lefschetz hyperplane theorems. In general such theorems state that homology or cohomology is supported on a hyperplane section of an algebraic variety, except for some 'loss' that can be controlled. These results applied to the algebraic fundamental group and to the Picard group.

Another type of application are connectedness theorems such as Grothendieck's connectedness theorem (a local analogue of the Bertini theorem) or the Fulton–Hansen connectedness theorem due to Fulton & Hansen (1979) and Faltings (1979). The latter asserts that for two projective varieties V and W in P^r over an algebraically closed field, the connectedness dimension of Z = V ∩ W (i.e., the minimal dimension of a closed subset T of Z that has to be removed from Z so that the complement Z \ T is disconnected) is bound by

c(Z) ≥ dim V + dim W − r − 1.

For example, Z is connected if dim V + dim W > r.[8]

• Local homology - gives topological analogue and computation of local homology of the cone of a space

• Huneke, Craig; Taylor, Amelia, Lectures on Local Cohomology

• Brodman, M. P.; Sharp, R. Y. (1998), Local Cohomology: An Algebraic Introduction with Geometric Applications (2nd ed.), Cambridge University Press Book review by Hartshorne
• Eisenbud, David (1995). Commutative algebra with a view toward algebraic geometry. Graduate Texts in Mathematics. 150. New York: Springer-Verlag. xvi+785. ISBN 0-387-94268-8. MR 1322960.
• Faltings, Gerd (1979), "Algebraisation of some formal vector bundles", Ann. of Math., 2, 110 (3): 501–514, doi:10.2307/1971235, MR 0554381
• Fulton, W.; Hansen, J. (1979), "A connectedness theorem for projective varieties with applications to intersections and singularities of mappings", Annals of Mathematics, Annals of Mathematics, 110 (1): 159–166, doi:10.2307/1971249, JSTOR 1971249
• Grothendieck, Alexander (2005) [1968], Séminaire de Géométrie Algébrique du Bois Marie - 1962 - Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux - (SGA 2), Documents Mathématiques (Paris), 4, Paris: Société Mathématique de France, arXiv:math/0511279, Bibcode:2005math.....11279G, ISBN 978-2-85629-169-6, MR 2171939
• Grothendieck, Alexandre (1968) [1962]. Séminaire de Géométrie Algébrique du Bois Marie - 1962 - Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux - (SGA 2) (Advanced Studies in Pure Mathematics 2) (in French). Amsterdam: North-Holland Publishing Company. vii+287.
• Hartshorne, Robin (1967) [1961], Local cohomology. A seminar given by A. Grothendieck, Harvard University, Fall, 1961, Lecture notes in mathematics, 41, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0073971, MR 0224620
• Iyengar, Srikanth B.; Leuschke, Graham J.; Leykin, Anton; Miller, Claudia; Miller, Ezra; Singh, Anurag K.; Walther, Uli (2007), Twenty-four hours of local cohomology, Graduate Studies in Mathematics, 87, Providence, R.I.: American Mathematical Society, doi:10.1090/gsm/087, ISBN 978-0-8218-4126-6, MR 2355715