resolution (algebra)
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In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
of
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...
s (or, more generally, of
object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...
s of an
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
), which is used to define invariants characterizing the structure of a specific module or object of this category. When, as usually, arrows are oriented to the right, the sequence is supposed to be infinite to the left for (left) resolutions, and to the right for right resolutions. However, a finite resolution is one where only finitely many of the objects in the sequence are non-zero; it is usually represented by a finite exact sequence in which the leftmost object (for resolutions) or the rightmost object (for coresolutions) is the zero-object. Generally, the objects in the sequence are restricted to have some property ''P'' (for example to be free). Thus one speaks of a ''P resolution''. In particular, every module has free resolutions, projective resolutions and flat resolutions, which are left resolutions consisting, respectively of free modules, projective modules or
flat module In algebra, a flat module over a ring ''R'' is an ''R''-module ''M'' such that taking the tensor product over ''R'' with ''M'' preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact se ...
s. Similarly every module has injective resolutions, which are right resolutions consisting of
injective module In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module ''Q'' that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if ''Q'' is a submodule o ...
s.


Resolutions of modules


Definitions

Given a module ''M'' over a ring ''R'', a left resolution (or simply resolution) of ''M'' is an
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
(possibly infinite) of ''R''-modules :\cdots\oversetE_n\overset\cdots\oversetE_2\oversetE_1\oversetE_0\oversetM\longrightarrow0. The homomorphisms ''di'' are called boundary maps. The map ε is called an augmentation map. For succinctness, the resolution above can be written as :E_\bullet\oversetM\longrightarrow0. The dual notion is that of a right resolution (or coresolution, or simply resolution). Specifically, given a module ''M'' over a ring ''R'', a right resolution is a possibly infinite exact sequence of ''R''-modules :0\longrightarrow M\oversetC^0\oversetC^1\oversetC^2\overset\cdots\oversetC^n\overset\cdots, where each ''Ci'' is an ''R''-module (it is common to use superscripts on the objects in the resolution and the maps between them to indicate the dual nature of such a resolution). For succinctness, the resolution above can be written as :0\longrightarrow M\oversetC^\bullet. A (co)resolution is said to be finite if only finitely many of the modules involved are non-zero. The length of a finite resolution is the maximum index ''n'' labeling a nonzero module in the finite resolution.


Free, projective, injective, and flat resolutions

In many circumstances conditions are imposed on the modules ''E''''i'' resolving the given module ''M''. For example, a ''free resolution'' of a module ''M'' is a left resolution in which all the modules ''E''''i'' are free ''R''-modules. Likewise, ''projective'' and ''flat'' resolutions are left resolutions such that all the ''E''''i'' are projective and
flat Flat or flats may refer to: Architecture * Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries Arts and entertainment * Flat (music), a symbol () which denotes a lower pitch * Flat (soldier), ...
''R''-modules, respectively. Injective resolutions are ''right'' resolutions whose ''C''''i'' are all
injective module In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module ''Q'' that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if ''Q'' is a submodule o ...
s. Every ''R''-module possesses a free left resolution.
A fortiori ''Argumentum a fortiori'' (literally "argument from the stronger eason) (, ) is a form of argumentation that draws upon existing confidence in a proposition to argue in favor of a second proposition that is held to be implicit in, and even more cer ...
, every module also admits projective and flat resolutions. The proof idea is to define ''E''0 to be the free ''R''-module generated by the elements of ''M'', and then ''E''1 to be the free ''R''-module generated by the elements of the kernel of the natural map ''E''0 → ''M'' etc. Dually, every ''R''-module possesses an injective resolution. Projective resolutions (and, more generally, flat resolutions) can be used to compute
Tor functor In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to con ...
s. Projective resolution of a module ''M'' is unique up to a
chain homotopy In homological algebra in mathematics, the homotopy category ''K(A)'' of chain complexes in an additive category ''A'' is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain c ...
, i.e., given two projective resolutions ''P''0 → ''M'' and ''P''1 → ''M'' of ''M'' there exists a chain homotopy between them. Resolutions are used to define homological dimensions. The minimal length of a finite projective resolution of a module ''M'' is called its ''
projective dimension In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizat ...
'' and denoted pd(''M''). For example, a module has projective dimension zero if and only if it is a projective module. If ''M'' does not admit a finite projective resolution then the projective dimension is infinite. For example, for a commutative
local ring In abstract algebra, more specifically 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 varieties or manifolds, or of algebraic n ...
''R'', the projective dimension is finite if and only if ''R'' is regular and in this case it coincides with 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 th ...
of ''R''. Analogously, the
injective dimension In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module ''Q'' that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if ''Q'' is a submodule of ...
id(''M'') and
flat dimension In abstract algebra, the weak dimension of a nonzero right module ''M'' over a ring ''R'' is the largest number ''n'' such that the Tor group \operatorname_n^R(M,N) is nonzero for some left ''R''-module ''N'' (or infinity if no largest such ''n ...
fd(''M'') are defined for modules also. The injective and projective dimensions are used on the category of right ''R'' modules to define a homological dimension for ''R'' called the right
global dimension In ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring ''A'' denoted gl dim ''A'', is a non-negative integer or infinity which is a homological invariant ...
of ''R''. Similarly, flat dimension is used to define
weak global dimension In abstract algebra, the weak dimension of a nonzero right module ''M'' over a ring ''R'' is the largest number ''n'' such that the Tor group \operatorname_n^R(M,N) is nonzero for some left ''R''-module ''N'' (or infinity if no largest such ''n ...
. The behavior of these dimensions reflects characteristics of the ring. For example, a ring has right global dimension 0 if and only if it is a
semisimple ring In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itsel ...
, and a ring has weak global dimension 0 if and only if it is a
von Neumann regular ring In mathematics, a von Neumann regular ring is a ring ''R'' (associative, with 1, not necessarily commutative) such that for every element ''a'' in ''R'' there exists an ''x'' in ''R'' with . One may think of ''x'' as a "weak inverse" of the elemen ...
.


Graded modules and algebras

Let ''M'' be a
graded module In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
over a graded algebra, which is generated over a field by its elements of positive degree. Then ''M'' has a free resolution in which the free modules ''E''''i'' may be graded in such a way that the ''d''''i'' and ε are graded linear maps. Among these graded free resolutions, the minimal free resolutions are those for which the number of basis elements of each ''E''''i'' is minimal. The number of basis elements of each ''E''''i'' and their degrees are the same for all the minimal free resolutions of a graded module. If ''I'' is a
homogeneous ideal In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
in a
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
over a field, the Castelnuovo-Mumford regularity of the projective algebraic set defined by ''I'' is the minimal integer ''r'' such that the degrees of the basis elements of the ''E''''i'' in a minimal free resolution of ''I'' are all lower than ''r-i''.


Examples

A classic example of a free resolution is given by the
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 ...
of a
regular sequence In commutative algebra, a regular sequence is a sequence of elements of a commutative ring which are as independent as possible, in a precise sense. This is the algebraic analogue of the geometric notion of a complete intersection. Definitions Fo ...
in a
local ring In abstract algebra, more specifically 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 varieties or manifolds, or of algebraic n ...
or of a homogeneous regular sequence in a graded algebra finitely generated over a field. Let ''X'' be an aspherical space, i.e., its
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
''E'' is
contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...
. Then every singular (or
simplicial In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
) chain complex of ''E'' is a free resolution of the module Z not only over the ring Z but also over the
group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the giv ...
Z 'π''1(''X'')


Resolutions in abelian categories

The definition of resolutions of an object ''M'' in an
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
''A'' is the same as above, but the ''Ei'' and ''Ci'' are objects in ''A'', and all maps involved are morphisms in ''A''. The analogous notion of projective and injective modules are projective and
injective object In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categori ...
s, and, accordingly, projective and injective resolutions. However, such resolutions need not exist in a general abelian category ''A''. If every object of ''A'' has a projective (resp. injective) resolution, then ''A'' is said to have enough projectives (resp. enough injectives). Even if they do exist, such resolutions are often difficult to work with. For example, as pointed out above, every ''R''-module has an injective resolution, but this resolution is not
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
ial, i.e., given a homomorphism ''M'' → ''M' '', together with injective resolutions :0 \rightarrow M \rightarrow I_*, \ \ 0 \rightarrow M' \rightarrow I'_*, there is in general no functorial way of obtaining a map between I_* and I'_*.


Abelian categories without projective resolutions in general

One class of examples of Abelian categories without projective resolutions are the categories \text(X) of
coherent sheaves 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 ...
on a scheme X. For example, if X = \mathbb^n_S is projective space, any coherent sheaf \mathcal on X has a presentation given by an exact sequence :\bigoplus_ \mathcal_X(s_) \to \bigoplus_ \mathcal_X(s_i) \to \mathcal \to 0. The first two terms are not in general projective since H^n(\mathbb^n_S,\mathcal_X(s)) \neq 0 for s > 0. But, both terms are locally free, and locally flat. Both classes of sheaves can be used in place for certain computations, replacing projective resolutions for computing some derived functors.


Acyclic resolution

In many cases one is not really interested in the objects appearing in a resolution, but in the behavior of the resolution with respect to a given
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
. Therefore, in many situations, the notion of acyclic resolutions is used: given a left exact functor ''F'': ''A'' → ''B'' between two abelian categories, a resolution :0 \rightarrow M \rightarrow E_0 \rightarrow E_1 \rightarrow E_2 \rightarrow \cdots of an object ''M'' of ''A'' is called ''F''-acyclic, if 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 ''R''''i''''F''(''E''''n'') vanish for all ''i'' > 0 and ''n'' ≥ 0. Dually, a left resolution is acyclic with respect to a right exact functor if its derived functors vanish on the objects of the resolution. For example, given a ''R'' module ''M'', the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
  \otimes_R M is a right exact functor Mod(''R'') → Mod(''R''). Every flat resolution is acyclic with respect to this functor. A ''flat resolution'' is acyclic for the tensor product by every ''M''. Similarly, resolutions that are acyclic for all the functors Hom( ⋅ , ''M'') are the projective resolutions and those that are acyclic for the functors Hom(''M'',  ⋅ ) are the injective resolutions. Any injective (projective) resolution is ''F''-acyclic for any left exact (right exact, respectively) functor. The importance of acyclic resolutions lies in the fact that the derived functors ''R''''i''''F'' (of a left exact functor, and likewise ''L''''i''''F'' of a right exact functor) can be obtained from as the homology of ''F''-acyclic resolutions: given an acyclic resolution E_* of an object ''M'', we have :R_i F(M) = H_i F(E_*), where right hand side is the ''i''-th homology object of the complex F(E_*). This situation applies in many situations. For example, for the constant sheaf ''R'' on a differentiable manifold ''M'' can be resolved by the sheaves \mathcal C^*(M) of smooth differential forms: : 0 \rightarrow R \subset \mathcal C^0(M) \stackrel d \rightarrow \mathcal C^1(M) \stackrel d \rightarrow \cdots \mathcal C^(M) \rightarrow 0. The sheaves \mathcal C^*(M) are fine sheaves, which are known to be acyclic with respect to the
global section In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
functor \Gamma: \mathcal F \mapsto \mathcal F(M). Therefore, the sheaf cohomology, which is the derived functor of the global section functor Γ is computed as \mathrm H^i(M, \mathbf R) = \mathrm H^i( \mathcal C^*(M)). Similarly Godement resolutions are acyclic with respect to the global sections functor.


See also

* Standard resolution * Hilbert–Burch theorem *
Hilbert's syzygy theorem In mathematics, Hilbert's syzygy theorem is one of the three fundamental theorems about polynomial rings over fields, first proved by David Hilbert in 1890, which were introduced for solving important open questions in invariant theory, and are at ...
* Free presentation * Matrix factorizations (algebra)


Notes


References

* * * * * {{Weibel IHA Homological algebra Module theory