Local cohomology

In algebraic geometry, local cohomology is an analog of relative cohomology. Alexander Grothendieck introduced it in seminars in Harvard in 1961 written up by Hartshorne (1967), and in 1961-2 at IHES written up as SGA2 - Grothendieck (1968), republished as Grothendieck (2005).

Definition

In the geometric form of the theory, sections Γ_Y are considered of a sheaf F of abelian groups, on a topological space X, with support in a closed subset Y. The derived functors of Γ_Y form local cohomology groups

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 ${\tilde M}$ . The closed subscheme Y is defined by an ideal I. In this situation, the functor Γ_Y(F) corresponds to the annihilator

\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,

\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

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

Basic properties

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.

If Y is defined by an ideal I=(f₁, ..., f_n), then local cohomology can be computed by means of Koszul complexes:

H_{I}^{i}(M)=\varinjlim _{m}H^{i}\left(\operatorname {Hom} _{R}\left(K\left(f_{1}^{m},\dots ,f_{n}^{m}\right),M\right)\right),

where K denotes the Koszul complex, obtained as the tensor product of the Koszul complex for the individual $f_{i}^{m}$ , defined as

R{\stackrel {f_{i}^{m}}{\longrightarrow }}R.

In particular, this leads to an exact sequence

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

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.^[1] In this case, the cohomology of U above can be identified with the cohomology groups

\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.^[2]

Relation to invariants of modules

The dimension dim_R(M) of a module (defined as the Krull dimension of its support) provides an upper bound for local cohomology groups:^[3]

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., $H_{\mathfrak {m}}^{n}(M)\neq 0$ .

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^[4]

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 $H_{\mathfrak {m}}^{n}(M)$ vanishes for all but one n.

Local duality

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

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

is a perfect pairing, where ω denotes a dualising module.^[5]

Applications

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.^[6]

Notes

↑ Eisenbud (1995, §A.4)
↑ Brodman & Sharp (1998, §16)
↑ Brodman & Sharp (1998, Theorem 6.1.2)
↑ Hartshorne (1967, Theorem 3.8), Brodman & Sharp (1998, Theorem 6.2.7), M is finitely generated, IM ≠ M
↑ Hartshorne (1967, Theorem 6.3), see also Hartshorne (1967, Theorem 6.7) for a converse statement.
↑ Brodman & Sharp (1998, §19.6)

Introductory Reference

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

References

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

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[1] Eisenbud (1995, §A.4)

[2] Brodman & Sharp (1998, §16)

[3] Brodman & Sharp (1998, Theorem 6.1.2)

[4] Hartshorne (1967, Theorem 3.8), Brodman & Sharp (1998, Theorem 6.2.7), M is finitely generated, IM ≠ M

[5] Hartshorne (1967, Theorem 6.3), see also Hartshorne (1967, Theorem 6.7) for a converse statement.

[6] Brodman & Sharp (1998, §19.6)