Local Cohomology
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In
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, local cohomology is an algebraic analogue of relative cohomology.
Alexander Grothendieck Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 â€“ 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...
introduced it in seminars in Harvard in 1961 written up by , and in 1961-2 at IHES written up as SGA2 - , republished as . Given a function (more generally, a section of a
quasicoherent sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
) defined on an
open subset In mathematics, an open set is a generalization of an open interval in the real line. In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point in it, contains all points of the met ...
of an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...
(or scheme), local cohomology measures the obstruction to extending that function to a larger domain. The
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
1/x, for example, is defined only on the complement of 0 on the
affine line In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
\mathbb^1_K over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
K, and cannot be extended to a function on the entire space. The local cohomology module H^1_(K (where K /math> is the
coordinate ring In algebraic geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space. More formally, an affine algebraic set is the set of the common zeros over an algeb ...
of \mathbb^1_K) detects this in the nonvanishing of a
cohomology class In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
/x/math>. In a similar manner, 1/xy is defined away from the x and y
axes Axes, plural of ''axe'' and of ''axis'', may refer to * ''Axes'' (album), a 2005 rock album by the British band Electrelane * a possibly still empty plot (graphics) See also * Axis (disambiguation) An axis (: axes) may refer to: Mathematics ...
in the
affine plane In geometry, an affine plane is a two-dimensional affine space. Definitions There are two ways to formally define affine planes, which are equivalent for affine planes over a field. The first way consists in defining an affine plane as a set on ...
, but cannot be extended to either the complement of the x-axis or the complement of the y-axis alone (nor can it be expressed as a sum of such functions); this obstruction corresponds precisely to a nonzero class /xy/math> in the local cohomology module H^2_(K ,y. Outside of algebraic geometry, local cohomology has found applications in
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...
,
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
, and certain kinds of
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how ...
.


Definition

In the most general geometric form of the theory, sections \Gamma_Y are considered of a
sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics) In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open s ...
F of
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
s, on a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
X, with
support Support may refer to: Arts, entertainment, and media * Supporting character * Support (art), a solid surface upon which a painting is executed Business and finance * Support (technical analysis) * Child support * Customer support * Income Su ...
in a
closed subset In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
Y, The
derived functor In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in vari ...
s of \Gamma_Y form local cohomology groups :H_Y^i(X,F) In the theory's algebraic form, the space ''X'' is the
spectrum A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...
Spec(''R'') of a commutative ring ''R'' (assumed to be
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...
throughout this article) and the sheaf ''F'' is the
quasicoherent sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
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 ''I''-torsion functor, a union of annihilators :\Gamma_I(M) := \bigcup_ (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 In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
H^i(\Gamma_I(E^\bullet)) of the
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel o ...
\Gamma_I(E^\bullet) obtained from taking the ''I''-torsion part \Gamma_I(-) of an injective resolution E^\bullet of the module M. Because E^\bullet consists of ''R''-modules and ''R''-module
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
s, the local cohomology groups each have the natural structure of an ''R''-module. The ''I''-torsion part \Gamma_I(M) may alternatively be described as :\Gamma_I(M) := \varinjlim_ \operatorname _R(R/I^n, M), and for this reason, the local cohomology of an ''R''-module ''M'' agrees with a
direct limit In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cate ...
of ''Ext'' modules, :H_I^i(M) := \varinjlim_ \operatorname _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 Radical (from Latin: ', root) may refer to: Politics and ideology Politics *Classical radicalism, the Radical Movement that began in late 18th century Britain and spread to continental Europe and Latin America in the 19th century *Radical politics ...
. 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 ÄŒech complex.


Using Koszul and ÄŒech complexes

The
derived functor In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in vari ...
definition of local cohomology requires an injective resolution of the module M, which can make it inaccessible for use in explicit computations. The
ÄŒech complex In algebraic topology and topological data analysis, the ÄŒech complex is an abstract simplicial complex constructed from a point cloud in any metric space which is meant to capture topological information about the point cloud or the distributi ...
is seen as more practical in certain contexts. , for example, state that they "essentially ignore" the "problem of actually producing any one of these njectivekinds of resolutions for a given module" prior to presenting the ÄŒech complex definition of local cohomology, and describes ÄŒech cohomology as "giv nga practical method for computing cohomology of quasi-coherent sheaves on a scheme." and as being "well suited for computations." The ÄŒech complex can be defined as a colimit of
Koszul complex In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology). It turned out to be a useful general construction in homological algebra. As a tool, its ho ...
es K^\bullet(f_1,\ldots,f_m) where f_1,\ldots, f_n generate I. The local cohomology modules can be described as: :H_I^i(M) \cong \varinjlim_m H^i \left (\operatorname_R \left (K^\bullet \left (f_1^m, \dots, f_n^m \right ), M \right ) \right ) Koszul complexes have the property that multiplication by f_i induces a chain complex morphism \cdot f_i : K^\bullet(f_1,\ldots, f_n) \to K^\bullet(f_1,\ldots, f_n) that is homotopic to zero, meaning H^i(K^\bullet(f_1,\ldots, f_n)) is annihilated by the f_i. A non-zero map in the colimit of the \operatorname 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 the
ÄŒech complex In algebraic topology and topological data analysis, the ÄŒech complex is an abstract simplicial complex constructed from a point cloud in any metric space which is meant to capture topological information about the point cloud or the distributi ...
, denoted \check^\bullet(f_1,\ldots,f_n;M), below.
0\to M \to \bigoplus_ M_ \to \bigoplus_ M_ \to \cdots \to M_\to 0
where the ''i''th local cohomology module of M with respect to I=(f_1,\ldots,f_n) is isomorphic to the ''i''th
cohomology group In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
of the above
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel o ...
, :H^i_I(M)\cong H^i(\check^\bullet(f_1,\ldots,f_n;M)). The broader issue of computing local cohomology modules (in characteristic zero) is discussed in and .


Basic properties

Since local cohomology is defined as
derived functor In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in vari ...
, for any short exact sequence of ''R''-modules 0\to M_1\to M_2\to M_3\to 0, there is, by definition, a natural
long exact sequence In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ...
in local cohomology :\cdots\to H^i_I(M_1)\to H^i_I(M_2)\to H^i_I(M_3)\to H^_I(M_1)\to\cdots There is also a
long exact sequence In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ...
of
sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions (holes) to solving a geometric problem glob ...
linking the ordinary sheaf cohomology of ''X'' and of the
open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
''U'' = ''X'' \''Y'', with the local cohomology modules. For a
quasicoherent sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
''F'' defined on ''X'', this has the form :\cdots\to H^i_Y(X,F)\to H^i(X,F)\to H^i(U,F)\to H^_Y(X,F)\to\cdots In the setting where ''X'' is an
affine scheme In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative ring R is the set of all prime ideals of R, and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with ...
\text(R) and ''Y'' is the vanishing set of an ideal ''I'', the cohomology groups H^i(X,F) vanish for i>0. If F=\tilde, this leads to an exact sequence :0 \to H_I^0(M) \to M \stackrel \to H^0(U, \tilde M) \to H^1_I(M) \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^(U, \tilde M) \stackrel \cong \to H^_I(M). Because of the above isomorphism with
sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions (holes) to solving a geometric problem glob ...
, local cohomology can be used to express a number of meaningful
topological Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...
constructions on the scheme X=\operatorname(R) in purely algebraic terms. For example, there is a natural analogue in local cohomology of the
Mayer–Vietoris sequence In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces. The result is due to two Austrian mathematicians, Walther Mayer an ...
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. This sequence has the form :\cdots H^i_(M)\to H^i_I(M)\oplus H^i_J(M)\to H^i_(M)\to H^_(M)\to\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 \operatorname(R). If J has the same radical as I, and is generated by n elements, then the ÄŒech complex on the generators of J has no terms in degree i > n. The least number of generators among all ideals J such that \sqrt=\sqrt is the arithmetic rank of I, denoted \operatorname(I). 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>\operatorname(I).


Graded local cohomology and projective geometry

When R is graded by \mathbb, 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. All of the basic properties of local cohomology expressed in this article are compatible with the graded structure. If M is finitely generated and I=\mathfrak is the ideal generated by the elements of R having positive degree, then the graded components H^i_(M)_n are finitely generated over R and vanish for sufficiently large n. The case where I=\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. In this case, there is an isomorphism :H^_(M)\cong \bigoplus_ H^i(\text(R), \tilde M(k)) where \text(R) is the
projective scheme In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, the de ...
associated to R, and (k) denotes the Serre twist. This isomorphism is graded, giving :H^_(M)_n \cong H^i(\text(R), \tilde M(n)) in all degrees n. This isomorphism relates local cohomology with the global cohomology of projective schemes. For example, the Castelnuovo–Mumford regularity can be formulated using local cohomology as :\text(M) = \text\ where \text(N) denotes the highest degree t such that N_t\neq 0. Local cohomology can be used to prove certain upper bound results concerning the regularity.


Examples


Top local cohomology

Using the ÄŒech complex, if I=(f_1,\ldots,f_n)R the local cohomology module H^n_I(M) is generated over R by the images of the formal fractions :\left frac\right/math> for m\in M and t_1,\ldots,t_n\geq 1. This fraction corresponds to a nonzero element of H^n_I(M) if and only if there is no k\geq 0 such that (f_1\cdots f_t)^k m \in (f_1^,\ldots,f_t^)M. For example, if t_i=1, then :f_i\cdot \left frac\right0. * If K is a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
and R=K _1,\ldots,x_n/math> is a
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
over K in n variables, then the local cohomology module H^n_(K _1,\ldots,x_n may be regarded as a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
over K with basis given by (the
ÄŒech cohomology In mathematics, specifically algebraic topology, ÄŒech cohomology is a cohomology theory based on the intersection properties of open set, open cover (topology), covers of a topological space. It is named for the mathematician Eduard ÄŒech. Moti ...
classes of) the inverse monomials \left _1^\cdots x_n^\right/math> for t_1,\ldots,t_n\geq 1. As an R-module, multiplication by x_i lowers t_i by 1, subject to the condition x_i\cdot \left _1^\cdots x_i^\cdots x_n^\right0. Because the powers t_i cannot be increased by multiplying with elements of R, the module H^n_(K _1,\ldots,x_n is not finitely generated.


Examples of H1

If H^0(U,\tilde R) is known (where U=\operatorname(R)-V(I)), the module H^1_I(R) can sometimes be computed explicitly using the sequence :0 \to H_I^0(R) \to R \to H^0(U, \tilde R) \to H^1_I(R) \to 0. In the following examples, K is any
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
. * If R=K ,Y^2,XY,Y^3/math> and I=(X,Y^2)R, then H^0(U,\tilde R)=K ,Y/math> and as a vector space over K, the first local cohomology module H^1_I(R) is K ,YK ,Y^2,XY,Y^3/math>, a 1-dimensional K vector space generated by Y. * If R=K ,Y(X^2,XY) and \mathfrak=(X,Y)R, then \Gamma_(R)=xR and H^0(U,\tilde R)=K ,Y^/math>, so H^1_(R)=K ,Y^K /math> is an infinite-dimensional K vector space with basis Y^,Y^,Y^,\ldots


Relation to invariants of modules

The dimension dim''R''(M) of a module (defined as the
Krull dimension In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally ...
of its support) provides an upper bound for local cohomology modules: :H_I^n(M) = 0 \textn>\dim_R(M). If ''R'' is
local Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States Arts, entertainment, and media * ''Local'' (comics), a limited series comic book by Bria ...
and ''M'' finitely generated, then this bound is sharp, i.e., H^n_\mathfrak(M) \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 :H^n_I(M) \ne 0. These two bounds together yield a characterisation of Cohen–Macaulay modules over local rings: they are precisely those modules where H^n_\mathfrak(M) vanishes for all but one ''n''.


Local duality

The local duality theorem is a local analogue of
Serre duality In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Ale ...
. For a Cohen-Macaulay
local ring In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of ...
R of dimension d that is a homomorphic image of a
Gorenstein local ring In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring ''R'' with finite injective dimension as an ''R''-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring i ...
(for example, if R is
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
), it states that the natural pairing :H^n_\mathfrak m(M) \times \operatorname_R^(M, \omega_R) \to H^d_\mathfrak m(\omega_R) is a
perfect pairing In mathematics, a pairing is an ''R''-bilinear map from the Cartesian product of two ''R''- modules, where the underlying ring ''R'' is commutative. Definition Let ''R'' be a commutative ring with unit, and let ''M'', ''N'' and ''L'' be ''R' ...
, where \omega_R is a
dualizing module In abstract algebra, a dualizing module, also called a canonical module, is a module over a commutative ring that is analogous to the canonical bundle of a smooth variety. It is used in Grothendieck local duality. Definition A dualizing module f ...
for R. In terms of the Matlis duality functor D(-), the local duality theorem may be expressed as the following isomorphism. :H^n_\mathfrak m(M) \cong D(\operatorname_R^(M,\omega_R)) The statement is simpler when \omega_R \cong R, which is equivalent to the hypothesis that R is Gorenstein. This is the case, for example, if R is regular.


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 Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...
, except for some 'loss' that can be controlled. These results applied to the algebraic fundamental group and to the
Picard group In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ver ...
. 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 In mathematics, the Fulton–Hansen connectedness theorem is a result from intersection theory in algebraic geometry, for the case of subvarieties of projective space with codimension large enough to make the intersection have components of dimensi ...
due to and . The latter asserts that for two
projective varieties In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, the ...
''V'' and ''W'' in P''r'' over an
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...
, 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 Complement may refer to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class collections into complementary sets * Complementary color, in the visu ...
''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''. In polyhedral geometry, a key ingredient of Stanley’s 1975 proof of the simplicial form of McMullen’s
Upper bound theorem In mathematics, the upper bound theorem states that cyclic polytopes have the largest possible number of faces among all convex polytopes with a given dimension and number of vertices. It is one of the central results of polyhedral combinatorics. ...
involves showing that the Stanley-Reisner ring of the corresponding
simplicial complex In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...
is Cohen-Macaulay, and local cohomology is an important tool in this computation, via Hochster’s formula.


See also

*
Local homology In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intui ...
- gives topological analogue and computation of local homology of the cone of a space * Faltings' annihilator theorem


Notes


Introductory Reference

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


References


Book review by Hartshorne
* * * * * * * * * * * *{{cite book , last=Leykin , first=Anton , editor1-last=Lyubeznik , editor1-first=Gennady , chapter=Computing Local Cohomology in Macaulay 2 , title=Local Cohomology and its applications , year=2002 , publisher=Marcel Dekker, isbn=0-8247-0741-9, pages=195–206 Sheaf theory Topological methods of algebraic geometry Cohomology theories Commutative algebra Duality theories