local cohomology

In the most general geometric form of the theory, sections $\backslash Gamma\_Y$ 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 $\backslash Gamma\_Y$ form local cohomology groups

:$H\_Y^i(X,F)$

In the theory's algebraic form, the space X is the spectrum Spec(R) of a commutative ring R (assumed to be Noetherian throughout this article) and the sheaf F is the quasicoherent sheaf associated to an R-module M, denoted by $\backslash tilde\; M$. The closed subscheme Y is defined by an ideal I. In this situation, the functor Γ_Y(F) corresponds to the I-torsion functor, a union of annihilators

:$\backslash Gamma\_I(M)\; :=\; \backslash bigcup\_\{n\; \backslash ge\; 0\}\; (0\; :\_M\; I^n),$

i.e., the elements of M which are annihilated by some power of I. As a right derived functor, the i^th local cohomology module with respect to I is the i^th cohomology group $H^i(\backslash Gamma\_I(E^\backslash bullet))$ of the chain complex $\backslash Gamma\_I(E^\backslash bullet)$ obtained from taking the I-torsion part $\backslash Gamma\_I(-)$ of an injective resolution $E^\backslash bullet$ of the module $M$.{{harvtxt|Brodmann|Sharp|1998|loc=1.2.2}} Because $E^\backslash bullet$ consists of R-modules and R-module homomorphisms, the local cohomology groups each have the natural structure of an R-module.

The I-torsion part $\backslash Gamma\_I(M)$ may alternatively be described as

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

and for this reason, the local cohomology of an R-module M agrees{{harvtxt|Brodmann|Sharp|1998|loc=Theorem 1.3.8}} with a direct limit of Ext modules,

:$H\_I^i(M)\; :=\; \backslash varinjlim\_\{n\; \backslash in\; N\}\; \backslash operatorname\; \{Ext\}\_R^i(R/I^n,\; M).$

It follows from either of these definitions that $H^i\_I(M)$ would be unchanged if $I$ were replaced by another ideal having the same radical.{{harvtxt|Brodmann|Sharp|1998|loc=Remark 1.2.3}} It also follows that local cohomology does not depend on any choice of generators for I, a fact which becomes relevant in the following definition involving the Čech complex.

The derived functor definition of local cohomology requires an injective resolution of the module $M$, which can make it inaccessible for use in explicit computations. The Čech complex is seen as more practical in certain contexts. {{harvtxt|Iyengar|Leuschke|Leykin|Miller|Miller|Singh|Walther|2007}}, for example, state that they "essentially ignore" the "problem of actually producing any one of these [injective] kinds of resolutions for a given module"{{harvtxt|Iyengar|Leuschke|Leykin|Miller|Miller|Singh|Walther|2007|pp. 67}} prior to presenting the Čech complex definition of local cohomology, and {{harvtxt|Hartshorne|1977}} describes Čech cohomology as "giv[ing] a practical method for computing cohomology of quasi-coherent sheaves on a scheme."{{harvtxt|Hartshorne|1977|p=218}} and as being "well suited for computations."{{harvtxt|Hartshorne|1977|p=219}}

The Čech complex can be defined as a colimit of Koszul complexes $K^\backslash bullet(f\_1,\backslash ldots,f\_m)$ where $f\_1,\backslash ldots,\; f\_n$ generate $I$. The local cohomology modules can be described{{harvtxt|Brodmann|Sharp|1998|loc=Theorem 5.2.9}} as:

:$H\_I^i(M)\; \backslash cong\; \backslash varinjlim\_m\; H^i\; \backslash left\; (\backslash operatorname\{Hom\}\_R\; \backslash left\; (K^\backslash bullet\; \backslash left\; (f\_1^m,\; \backslash dots,\; f\_n^m\; \backslash right\; ),\; M\; \backslash right\; )\; \backslash right\; )$

Koszul complexes have the property that multiplication by $f\_i$ induces a chain complex morphism $\backslash cdot\; f\_i\; :\; K^\backslash bullet(f\_1,\backslash ldots,\; f\_n)\; \backslash to\; K^\backslash bullet(f\_1,\backslash ldots,\; f\_n)$ that is homotopic to zero,{{cite web|title=Lemma 15.28.6 (0663)—The Stacks project|url=https://stacks.math.columbia.edu/tag/0663|website=stacks.math.columbia.edu|access-date=2020-05-01}} meaning $H^i(K^\backslash bullet(f\_1,\backslash ldots,\; f\_n))$ is annihilated by the $f\_i$. A non-zero map in the colimit of the $\backslash operatorname\{Hom\}$ sets contains maps from the all but finitely many Koszul complexes, and which are not annihilated by some element in the ideal.

This colimit of Koszul complexes is isomorphic to{{cite web|title=Lemma 15.28.13 (0913)—The Stacks project|url=https://stacks.math.columbia.edu/tag/0913|website=stacks.math.columbia.edu|access-date=2020-05-01}} the Čech complex, denoted $\backslash check\{C\}^\backslash bullet(f\_1,\backslash ldots,f\_n;M)$, below.

where the i^th local cohomology module of $M$ with respect to $I=(f\_1,\backslash ldots,f\_n)$ is isomorphic to{{harvtxt|Brodmann|Sharp|1998|loc=Theorem 5.1.19}} the i^th cohomology group of the above chain complex,

:$H^i\_I(M)\backslash cong\; H^i(\backslash check\{C\}^\backslash bullet(f\_1,\backslash ldots,f\_n;M)).$

The broader issue of computing local cohomology modules (in characteristic zero) is discussed in {{harvtxt|Leykin|2002}} and {{harvtxt|Iyengar|Leuschke|Leykin|Miller|Miller|Singh|Walther|2007|loc=Lecture 23}}.

Since local cohomology is defined as derived functor, for any short exact sequence of R-modules $0\backslash to\; M\_1\backslash to\; M\_2\backslash to\; M\_3\backslash to\; 0$, there is, by definition, a natural long exact sequence in local cohomology

:$\backslash cdots\backslash to\; H^i\_I(M\_1)\backslash to\; H^i\_I(M\_2)\backslash to\; H^i\_I(M\_3)\backslash to\; H^\{i+1\}\_I(M\_1)\backslash to\backslash cdots$

There is also 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 modules. For a quasicoherent sheaf F defined on X, this has the form

:$\backslash cdots\backslash to\; H^i\_Y(X,F)\backslash to\; H^i(X,F)\backslash to\; H^i(U,F)\backslash to\; H^\{i+1\}\_Y(X,F)\backslash to\backslash cdots$

In the setting where X is an affine scheme $\backslash text\{Spec\}(R)$ and Y is the vanishing set of an ideal I, the cohomology groups $H^i(X,F)$ vanish for $i>0$.{{harvtxt|Hartshorne|1977|loc=Theorem 3.7}} If $F=\backslash tilde\{M\}$, this leads to an exact sequence

:$0\; \backslash to\; H\_I^0(M)\; \backslash to\; M\; \backslash stackrel\; \{\backslash text\{res\}\}\; \backslash to\; H^0(U,\; \backslash tilde\; M)\; \backslash to\; H^1\_I(M)\; \backslash to\; 0,$

where 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

:$H^\{n\}(U,\; \backslash tilde\; M)\; \backslash stackrel\; \backslash cong\; \backslash to\; H^\{n+1\}\_I(M).$

Because of the above isomorphism with sheaf cohomology, local cohomology can be used to express a number of meaningful topological constructions on the scheme $X=\backslash operatorname\{Spec\}(R)$ in purely algebraic terms. For example, there is a natural analogue in local cohomology of the Mayer–Vietoris sequence with respect to a pair of open sets U and V in X, given by the complements of the closed subschemes corresponding to a pair of ideal I and J, respectively.{{harvtxt|Brodmann|Sharp|1998|loc=Theorem 3.2.3}} This sequence has the form

:$\backslash cdots\; H^i\_\{I+J\}(M)\backslash to\; H^i\_I(M)\backslash oplus\; H^i\_J(M)\backslash to\; H^i\_\{I\backslash cap\; J\}(M)\backslash to\; H^\{i+1\}\_\{I+J\}(M)\backslash to\backslash cdots$

for any $R$-module $M$.

The vanishing of local cohomology can be used to bound the least number of equations (referred to as the arithmetic rank) needed to (set theoretically) define the algebraic set $V(I)$ in $\backslash operatorname\{Spec\}(R)$. If $J$ has the same radical as $I$, and is generated by $n$ elements, then the Čech complex on the generators of $J$ has no terms in degree $i\; >\; n$. The least number of generators among all ideals $J$ such that $\backslash sqrt\{J\}=\backslash sqrt\{I\}$ is the arithmetic rank of $I$, denoted $\backslash operatorname\{ara\}(I)$.{{harvtxt|Brodmann|Sharp|1998|loc=Definition 3.3.2}} Since the local cohomology with respect to $I$ may be computed using any such ideal, it follows that $H^i\_I(M)=0$ for $i>\backslash operatorname\{ara\}(I)$.{{harvtxt|Brodmann|Sharp|1998|loc=Remark 5.1.20}}

When $R$ is graded by $\backslash mathbb\{N\}$, $I$ is generated by homogeneous elements, and $M$ is a graded module, there is a natural grading on the local cohomology module $H^i\_I(M)$ that is compatible with the gradings of $M$ and $R$.{{harvtxt|Brodmann|Sharp|1998|loc=Corollary 12.3.3}} All of the basic properties of local cohomology expressed in this article are compatible with the graded structure.{{harvtxt|Brodmann|Sharp|1998|loc=Chapter 13}} If $M$ is finitely generated and $I=\backslash mathfrak\{m\}$ is the ideal generated by the elements of $R$ having positive degree, then the graded components $H^i\_\{\backslash mathfrak\{m\}\}(M)\_n$ are finitely generated over $R$ and vanish for sufficiently large $n$.{{harvtxt|Brodmann|Sharp|1998|loc=Proposition 15.1.5}}

The case where $I=\backslash mathfrak\; m$ is the ideal generated by all elements of positive degree (sometimes called the irrelevant ideal) is particularly special, due to its relationship with projective geometry.{{harvtxt|Eisenbud|1995|loc=§A.4}} In this case, there is an isomorphism

:$H^\{i+1\}\_\{\backslash mathfrak\; m\}(M)\backslash cong\; \backslash bigoplus\_\{k\; \backslash in\; \backslash mathbf\; Z\}\; H^i(\backslash text\{Proj\}(R),\; \backslash tilde\; M(k))$

where $\backslash text\{Proj\}(R)$ is the projective scheme associated to $R$, and $(k)$ denotes the Serre twist. This isomorphism is graded, giving

:$H^\{i+1\}\_\{\backslash mathfrak\; m\}(M)\_n\; \backslash cong\; H^i(\backslash text\{Proj\}(R),\; \backslash tilde\; M(n))$

in all degrees $n$.{{harvtxt|Brodmann|Sharp|1998|loc=Theorem 20.4.4}}

This isomorphism relates local cohomology with the global cohomology of projective schemes. For example, the Castelnuovo–Mumford regularity can be formulated using local cohomology{{harvtxt|Brodmann|Sharp|1998|loc=Definition 15.2.9}} as

:$\backslash text\{reg\}(M)\; =\; \backslash text\{sup\}\backslash \{\backslash text\{end\}(H^i\_\{\backslash mathfrak\{m\}\}(M))+i\backslash ,|\backslash ,\; 0\backslash leq\; i\backslash leq\; \backslash text\{dim\}(M)\backslash \}$

where $\backslash text\{end\}(N)$ denotes the highest degree $t$ such that $N\_t\backslash neq\; 0$. Local cohomology can be used to prove certain upper bound results concerning the regularity.{{harvtxt|Brodmann|Sharp|1998|loc=Chapter 16}}

Using the Čech complex, if $I=(f\_1,\backslash ldots,f\_n)R$ the local cohomology module $H^n\_I(M)$ is generated over $R$ by the images of the formal fractions

:$\backslash left[\backslash frac\{m\}\{f\_1^\{t\_1\}\backslash cdots\; f\_n^\{t\_n\}\}\backslash right]$

for $m\backslash in\; M$ and $t\_1,\backslash ldots,t\_n\backslash geq\; 1$.{{harvtxt|Iyengar|Leuschke|Leykin|Miller|Miller|Singh|Walther|2007|loc=Corollary 7.14}} This fraction corresponds to a nonzero element of $H^n\_I(M)$ if and only if there is no $k\backslash geq\; 0$ such that $(f\_1\backslash cdots\; f\_t)^k\; m\; \backslash in\; (f\_1^\{t\_1+k\},\backslash ldots,f\_t^\{t\_n+k\})M$.{{harvtxt|Brodmann|Sharp|1998|loc=Exercise 5.1.21}} For example, if $t\_i=1$, then

:$f\_i\backslash cdot\; \backslash left[\backslash frac\{m\}\{f\_1^\{t\_1\}\backslash cdots\; f\_i\backslash cdots\; f\_n^\{t\_n\}\}\backslash right]=0.$

• If $K$ is a field and $R=K[x\_1,\backslash ldots,x\_n]$ is a polynomial ring over $K$ in $n$ variables, then the local cohomology module $H^n\_\{(x\_1,\backslash ldots,x\_n)\}(K[x\_1,\backslash ldots,x\_n])$ may be regarded as a vector space over $K$ with basis given by (the Čech cohomology classes of) the inverse monomials $\backslash left[x\_1^\{-t\_1\}\backslash cdots\; x\_n^\{-t\_n\}\backslash right]$ for $t\_1,\backslash ldots,t\_n\backslash geq\; 1$.{{harvtxt|Iyengar|Leuschke|Leykin|Miller|Miller|Singh|Walther|2007|loc=Exercise 7.16}} As an $R$-module, multiplication by $x\_i$ lowers $t\_i$ by 1, subject to the condition $x\_i\backslash cdot\; \backslash left[x\_1^\{-t\_1\}\backslash cdots\; x\_i^\{-1\}\backslash cdots\; x\_n^\{-t\_n\}\backslash right]=0.$ Because the powers $t\_i$ cannot be increased by multiplying with elements of $R$, the module $H^n\_\{(x\_1,\backslash ldots,x\_n)\}(K[x\_1,\backslash ldots,x\_n])$ is not finitely generated.

If $H^0(U,\backslash tilde\; R)$ is known (where $U=\backslash operatorname\{Spec\}(R)-V(I)$), the module $H^1\_I(R)$ can sometimes be computed explicitly using the sequence

:$0\; \backslash to\; H\_I^0(R)\; \backslash to\; R\; \backslash to\; H^0(U,\; \backslash tilde\; R)\; \backslash to\; H^1\_I(R)\; \backslash to\; 0.$

In the following examples, $K$ is any field.

• If $R=K[X,Y^2,XY,Y^3]$ and $I=(X,Y^2)R$, then $H^0(U,\backslash tilde\; R)=K[X,Y]$ and as a vector space over $K$, the first local cohomology module $H^1\_I(R)$ is $K[X,Y]/K[X,Y^2,XY,Y^3]$, a 1-dimensional $K$ vector space generated by $Y$.{{harvtxt|Brodmann|Sharp|1998|loc=Exercise 2.3.6(v)}}

• If $R=K[X,Y]/(X^2,XY)$ and $\backslash mathfrak\{m\}=(X,Y)R$, then $\backslash Gamma\_\{\backslash mathfrak\{m\}\}(R)=xR$ and $H^0(U,\backslash tilde\; R)=K[Y,Y^\{-1\}]$, so $H^1\_\{\backslash mathfrak\{m\}\}(R)=K[Y,Y^\{-1\}]/K[Y]$ is an infinite-dimensional $K$ vector space with basis $Y^\{-1\},Y^\{-2\},Y^\{-3\},\backslash ldots${{harvtxt|Eisenbud|2005|loc=Example A1.10}}

The dimension dim_R(M) of a module (defined as the Krull dimension of its support) provides an upper bound for local cohomology modules:{{harvtxt|Brodmann|Sharp|1998|loc=Theorem 6.1.2}}

:$H\_I^n(M)\; =\; 0\; \backslash text\{\; for\; all\; \}n>\backslash dim\_R(M).$

If R is local and M finitely generated, then this bound is sharp, i.e., $H^n\_\backslash mathfrak\{m\}(M)\; \backslash ne\; 0$.

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{{harvtxt|Hartshorne|1967|loc=Theorem 3.8}}, {{harvtxt|Brodmann|Sharp|1998|loc=Theorem 6.2.7}}, M is finitely generated, IM ≠ M

:$H^n\_I(M)\; \backslash ne\; 0.$

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

The local duality theorem is a local analogue of Serre duality. For a Cohen-Macaulay local ring $R$ of dimension $d$ that is a homomorphic image of a Gorenstein local ring{{harvtxt|Bruns|Herzog|1998|loc=Theorem 3.3.6}} (for example, if $R$ is complete{{harvtxt|Bruns|Herzog|1998|loc=Corollary 3.3.8}}), it states that the natural pairing

:$H^n\_\backslash mathfrak\; m(M)\; \backslash times\; \backslash operatorname\{Ext\}\_R^\{d-n\}(M,\; \backslash omega\_R)\; \backslash to\; H^d\_\backslash mathfrak\; m(\backslash omega\_R)$

is a perfect pairing, where $\backslash omega\_R$ is a dualizing module for $R$.{{harvtxt|Hartshorne|1967|loc=Theorem 6.7}} In terms of the Matlis duality functor $D(-)$, the local duality theorem may be expressed as the following isomorphism.{{harvtxt|Brodmann|Sharp|1998|loc=Theorem 11.2.8}}

:$H^n\_\backslash mathfrak\; m(M)\; \backslash cong\; D(\backslash operatorname\{Ext\}\_R^\{d-n\}(M,\backslash omega\_R))$

The statement is simpler when $\backslash omega\_R\; \backslash cong\; R$, which is equivalent{{harvtxt|Bruns|Herzog|1998|loc=Theorem 3.3.7}} to the hypothesis that $R$ is Gorenstein. This is the case, for example, if $R$ is regular.

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 {{harvtxt|Fulton|Hansen|1979}} and {{harvtxt|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.{{harvtxt|Brodmann|Sharp|1998|loc=§19.6}}

In polyhedral geometry, a key ingredient of Stanley’s 1975 proof of the simplicial form of McMullen’s Upper bound theorem involves showing that the Stanley-Reisner ring of the corresponding simplicial complex is Cohen-Macaulay, and local cohomology is an important tool in this computation, via Hochster’s formula.{{cite journal |title=How the Upper Bound Conjecture Was Proved|last=Stanley|first=Richard|year=2014|journal=Annals of Combinatorics|pages=533–539|volume=18|issue=3 |doi=10.1007/s00026-014-0238-5|s2cid=253585250 |hdl=1721.1/93189|hdl-access=free}}{{harvtxt|Iyengar|Leuschke|Leykin|Miller|Miller|Singh|Walther|2007|loc=Lecture 16}}

• Local homology - gives topological analogue and computation of local homology of the cone of a space
• Faltings' annihilator theorem

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Category:Sheaf theory

Category:Topological methods of algebraic geometry

Category:Cohomology theories

Category:Commutative algebra

Category:Duality theories