Dimension

of a topological space $ X $

An integral invariant $ \mathop{\rm dim} X $ defined as follows. $ \mathop{\rm dim} X = - 1 $ if and only if $ X = \emptyset $. A non-empty topological space $ X $ is said to be at most $ n $- dimensional, written as $ \mathop{\rm dim} X \leq n $, if in any finite open covering of $ X $ one can inscribe a finite open covering of $ X $ of multiplicity $ \leq n + 1 $, $ n = 0, 1 , . . . $. If $ \mathop{\rm dim} X \leq n $ for some $ n = - 1, 0, 1 \dots $ then $ X $ is said to be finite-dimensional, written as $ \mathop{\rm dim} X < \infty $, and one defines

$$ \mathop{\rm dim} X = \min \{ {n } : { \mathop{\rm dim} X \leq n } \} . $$

Here if $ \mathop{\rm dim} X = n $, then the space is called $ n $- dimensional. The concept of the dimension of a topological space generalizes the elementary geometrical concept of the number of coordinates of a Euclidean space (and a polyhedron), since the dimension of an $ n $- dimensional Euclidean space (and any $ n $- dimensional polyhedron) is equal to $ n $( the Lebesgue–Brouwer theorem).

The importance of the concept of the dimension of a topological space is revealed by the Nöbeling–Pontryagin–Hurewicz–Kuratowski theorem: An $ n $- dimensional metrizable space with a countable base can be imbedded in the $ ( 2n + 1) $- dimensional Euclidean space. Thus, the class of spaces that are topologically equivalent to subspaces of all possible $ n $- dimensional Euclidean spaces, $ n = 1, 2 \dots $ coincides with the class of finite-dimensional metrizable spaces with a countable base.

The dimension $ \mathop{\rm dim} X $ is sometimes called the Lebesgue dimension, since its definition arises from Lebesgue's theorem on tilings: An $ n $- dimensional cube has, for any $ \epsilon > 0 $, a finite closed covering of multiplicity $ \leq n + 1 $ such that all elements have diameter $ < \epsilon $; there exists an $ \epsilon _ {0} > 0 $ for which the multiplicity of any finite closed covering of an $ n $- dimensional cube is $ \geq n + 1 $ if the diameters of the elements of this covering are $ < \epsilon _ {0} $.

Another, inductive, approach (see Inductive dimension) to the definition of the dimension of a topological space is possible, based on the separation of the space by subspaces of smaller dimension. This approach to the concept of dimension originates from H. Poincaré, L.E.J. Brouwer, P.S. Urysohn, and K. Menger. In the case of metrizable spaces it is equivalent to Lebesgue's definition.

The foundations of dimension theory were laid in the first half of the twenties of the 20th century in papers of Urysohn and Menger. In the later thirties, the dimension theory of metrizable spaces with a countable base was constructed, and by the start of the sixties the dimension theory of arbitrary metrizable spaces was finished.

Below, all topological spaces under consideration are supposed to be normal and Hausdorff (cf. Hausdorff space; Normal space). In this case, in the definition of dimension one can without harm replace the open coverings to be inscribed by closed ones.

Lebesgue's approach to the definition of dimension (in contrast to the inductive approach) makes it possible to geometrize the concept of dimension for any space by comparing the original topological space with most simple geometrical formations — polyhedra (cf. Polyhedron). Roughly speaking, a space is $ n $- dimensional if and only if it differs arbitrarily little from an $ n $- dimensional polyhedron. More precisely, there is Aleksandrov's theorem on $ \omega $- mappings: $ \mathop{\rm dim} X \leq n $ if and only if for any finite open covering $ \omega $ of $ X $ there is an $ \omega $- mapping from $ X $ onto an at most $ n $- dimensional, $ n = 0, 1 \dots $( compact) polyhedron. This theorem can be particularly visualized for compacta: A compactum $ X $ has $ \mathop{\rm dim} X \leq n $ if and only if for any $ \epsilon > 0 $ there is an $ \epsilon $- mapping from $ X $ onto an at most $ n $- dimensional polyhedron. If $ X $ also lies in a Euclidean or Hilbert space, then the $ \epsilon $- mapping can be replaced by an $ \epsilon $- shift (Aleksandrov's theorem on $ \epsilon $- mappings and $ \epsilon $- shifts).

The following statement makes it possible to determine the dimension of a space by comparing it with all possible $ n $- dimensional cubes: $ \mathop{\rm dim} X \geq n $ if and only if the space has an essential mapping onto an $ n $- dimensional cube, $ n = 0, 1 , . . . $( Aleksandrov's theorem on essential mappings).

This theorem can be given the following form: $ \mathop{\rm dim} X \leq n $ if and only if, for any set $ A $ closed in $ X $ and for any continuous mapping $ f: A \rightarrow S ^ {n} $ into the $ n $- dimensional sphere, there is a continuous extension $ F: X \rightarrow S ^ {n} $, $ n = 0, 1 \dots $ of $ f $.

The following characterization of dimension indicates the role of this concept in problems of the existence of solutions to systems of equations: $ \mathop{\rm dim} X \geq n $, $ n = 1, 2 \dots $ if and only if $ X $ has a system of disjoint pairs of closed sets $ A _ {i} $, $ B _ {i} $, $ i = 1 \dots n $, such that for any functions $ f _ {i} $ continuous on $ X $ and satisfying the conditions $ f _ {i} \mid _ {A _ {i} } > 0 $, $ f _ {i} \mid _ {B _ {i} } < 0 $, $ i = 1 \dots n $, there is a point $ x \in X $ at which $ f _ {i} ( x) = 0 $, $ i = 1 \dots n $( this is the Otto–Eilenberg–Hemmingsen theorem on partitions).

One of the most important properties of dimension is expressed by the Menger–Urysohn–Čech countable closed sum theorem: If the space $ X $ is a finite or countable sum of closed subsets of dimension $ \leq n $, then also $ \mathop{\rm dim} X \leq n $, $ n = 0, 1 , . . . $. In this theorem, the condition that the sum be finite or countable may be replaced by the condition of local finiteness. The statement for the large and small inductive dimensions analogous to this sum theorem already fails in the class of Hausdorff compacta. The following statements are among the fundamental general facts of dimension theory, and make it possible to reduce the consideration of arbitrary spaces to that of Hausdorff compacta. For any normal space

a) $ \mathop{\rm dim} \beta X = \mathop{\rm dim} X $, $ \mathop{\rm Ind} \beta X = \mathop{\rm Ind} X $, where $ \beta X $ is the Stone–Čech compactification of $ X $; at the same time, the inequality $ \mathop{\rm ind} \beta X > \mathop{\rm ind} X $ is possible;

b) there exists a compactification $ bX $ of $ X $ with weight (cf. Weight of a topological space) $ w ( bX) $ equal to the weight $ wX $ and with dimension $ \mathop{\rm dim} bX $ equal to the dimension $ \mathop{\rm dim} X $; the analogous statement also holds for the large inductive dimension. The case of a countable weight of the space is especially interesting, since in this case the extension $ bX $ is metrizable.

Statement b) can be strengthened: For any $ n = 0, 1 \dots $ and any infinite cardinal number $ \tau $ there is a Hausdorff compactum $ \Pi _ \tau ^ {n} $ of weight $ \tau $ and dimension $ \mathop{\rm dim} \Pi _ \tau ^ {n} = n $ containing a homeomorphic image of every normal space $ X $ of weight $ \leq \tau $ and dimension $ \mathop{\rm dim} X \leq n $( the theorem on the universal Hausdorff compactum of given weight and dimension). The analogous statement also holds for the large inductive dimension. Here for $ \Pi _ {\aleph _ {0} } ^ {0} $ one can take the perfect Cantor set, and as $ \Pi _ {\aleph _ {0} } ^ {1} $ the Menger universal curve.

It would seem that dimension should possess the monotonicity property: $ \mathop{\rm dim} A \leq \mathop{\rm dim} X $ if $ A \subset X $. This is so if a) the set $ A $ is closed in $ X $ or is strongly paracompact; or b) the space $ X $ is metrizable (and even perfectly normal). However, already for a subset $ A $ of a hereditarily normal space $ X $ one may have $ \mathop{\rm dim} A > \mathop{\rm dim} X $ and $ \mathop{\rm Ind} A > \mathop{\rm Ind} X $. But always $ \mathop{\rm ind} A \leq \mathop{\rm ind} X $ for $ A \subset X $.

One of the main problems in dimension theory is the behaviour of dimension under continuous mappings. In the case of closed mappings (these also include all continuous mappings of Hausdorff compacta) the answer is given by the formulas of W. Hurewicz, which he originally obtained for the class of spaces with a countable base.

Hurewicz' formula for mappings raising the dimension: If a mapping $ f: X \rightarrow Y $ is continuous and closed, then

$$ \mathop{\rm mult} f + \mathop{\rm dim} X - 1 \geq \mathop{\rm dim} Y, $$

where $ \mathop{\rm mult} f = \sup \{ {| f ^ { - 1 } y | } : {y \in Y } \} $ is the multiplicity of $ f $.

Hurewicz' formula for mappings lowering the dimension: For a continuous closed mapping $ f: X \rightarrow Y $ onto a paracompactum $ Y $, the inequality

$$ \tag{1 } \mathop{\rm dim} X - \mathop{\rm dim} f \leq \mathop{\rm dim} Y $$

holds, where

$$ \mathop{\rm dim} f = \ \sup \{ { \mathop{\rm dim} f ^ { - 1 } y } : {y \in Y } \} . $$

For an arbitrary normal space $ Y $ this formula is, in general, false.

In the case of continuous mappings of finite-dimensional compacta, it has been established that a continuous mapping $ f $ of dimension $ \mathop{\rm dim} f = k $ is a superposition of $ k $ continuous mappings of dimension 1 (this is a precization of formula (1), and an analogue of the fact that a $ k $- dimensional cube is the product of $ k $ intervals).

In the case of open mappings one can show that the image of a zero-dimensional Hausdorff compactum is zero-dimensional and, at the same time, that the Hilbert cube is the image of a one-dimensional compactum, even if the corresponding mapping $ f $ has dimension $ \mathop{\rm dim} f $ equal to zero. However, in the case of an open mapping $ f: X \rightarrow Y $ of Hausdorff compacta $ X $ and $ Y $ with multiplicity $ \leq \aleph _ {0} $, the equality $ \mathop{\rm dim} X = \mathop{\rm dim} Y $ holds.

The behaviour of dimension under topological products is described by the following assertions:

a) there exist finite-dimensional compacta $ X $ and $ Y $ for which $ \mathop{\rm dim} X \times Y < \mathop{\rm dim} X + \mathop{\rm dim} Y $;

b) if one of the factors of the product $ X \times Y $ is a Hausdorff compactum or metrizable, then $ \mathop{\rm dim} X \times Y \leq \mathop{\rm dim} X + \mathop{\rm dim} Y $;

c) there exist normal spaces $ X $ and $ Y $ for which $ \mathop{\rm dim} X \times Y > \mathop{\rm dim} X + \mathop{\rm dim} Y $.

In the case of Hausdorff compacta $ X $ and $ Y $ one always has $ \mathop{\rm Ind} X \times Y < \infty $ if $ \mathop{\rm Ind} X < \infty $ and $ \mathop{\rm Ind} Y < \infty $, but one may have $ \mathop{\rm Ind} X \times Y > \mathop{\rm Ind} X + \mathop{\rm Ind} Y $. If, however, the Hausdorff compacta $ X $ and $ Y $ are perfectly normal or one-dimensional, then $ \mathop{\rm Ind} X \times Y \leq \mathop{\rm Ind} X + \mathop{\rm Ind} Y $.

Dimension theory is most meaningful, first, for the class of metric spaces with a countable base, and, secondly, for the class of all metric spaces. In the class of metric spaces with a countable base one has the Urysohn equalities

$$ \tag{2 } \mathop{\rm dim} X = \ \mathop{\rm ind} X = \ \mathop{\rm Ind} X. $$

In the class of arbitrary metric spaces one has the Katětov equality

$$ \tag{3 } \mathop{\rm dim} X = \mathop{\rm Ind} X , $$

and $ \mathop{\rm ind} X = 0 < \mathop{\rm Ind} X = 1 $ is possible.

In the case of metric spaces the concept of an $ n $- dimensional space can be reduced to the concept of a zero-dimensional space by the following two methods. For a metric space $ X $, $ \mathop{\rm dim} X \leq n $, $ n = 0, 1 \dots $ if and only if

a) $ X $ can be represented by at most $ n + 1 $ zero-dimensional summands; or

b) there exists a continuous closed mapping of multiplicity $ \leq n + 1 $ from a zero-dimensional metric space onto $ X $.

For any subset $ A $ of a metric space $ X $ there is a subset $ B \supset A $ of type $ G _ \delta $ in $ X $ for which $ \mathop{\rm dim} B = \mathop{\rm dim} A $.

In the class of metric spaces of weight $ \leq \tau $ and dimension $ \leq n $ there exists a universal space (in the sense of imbedding). Dowker's theorem has played an important role in the dimension theory of metric (and more general) spaces: $ \mathop{\rm dim} X \leq n $ if and only if in any locally finite open covering of $ X $ one can inscribe an open covering of multiplicity $ \leq n + 1 $.

One of the most important problems in dimension theory is the problem of the relations between the Lebesgue dimension and the inductive dimensions. Although for an arbitrary space $ X $ the values of the dimensions $ \mathop{\rm dim} X $, $ \mathop{\rm ind} X $, $ \mathop{\rm Ind} X $ are, in general, pairwise distinct, for some classes of spaces that are in some sense close to metric spaces one has, e.g., the following:

a) if the space $ X $ admits a continuous closed mapping $ f $ of dimension $ \mathop{\rm dim} f = 0 $ onto a metric space, then (3) holds, whence follow the equalities (2) for locally compact Hausdorff groups and their quotient spaces;

b) if there exists a continuous closed mapping from a metric space onto $ X $, then (2) holds.

One more general condition for equality (3) to hold for a paracompactum $ X $ is as follows: $ \mathop{\rm dim} X = n $ and $ X $ is the image of a zero-dimensional space under a closed mapping of multiplicity $ \leq n + 1 $, $ n = 0, 1 , . . . $.

In the case of an arbitrary space $ X $ one always has the inequalities $ \mathop{\rm dim} X \leq \mathop{\rm Ind} X $ and $ \mathop{\rm ind} X \leq \mathop{\rm Ind} X $, while the equalities $ \mathop{\rm dim} X = 0 $ and $ \mathop{\rm Ind} X = 0 $ are equivalent. For a strongly paracompact (in particular, for a Hausdorff compact or Lindelöf compact) space $ X $ one has the inequality $ \mathop{\rm dim} X \leq \mathop{\rm ind} X $. For Hausdorff compacta the equalities $ \mathop{\rm ind} X = 1 $ and $ \mathop{\rm Ind} X = 1 $ are equivalent. There exist Hausdorff compacta satisfying the first axiom of countability (and even perfectly-normal Hausdorff compacta, if one assumes the continuum hypothesis), for which $ \mathop{\rm dim} X = 1 $ and $ \mathop{\rm ind} X = n $, $ n = 2, 3 , . . . $. An example of a topologically homogeneous Hausdorff compactum with $ \mathop{\rm dim} X < \mathop{\rm ind} X $ has been constructed. For perfectly-normal Hausdorff compacta one always has $ \mathop{\rm ind} X = \mathop{\rm Ind} X $. There exist Hausdorff compacta, satisfying even the first axiom of countability, for which $ \mathop{\rm ind} X < \mathop{\rm Ind} X $. It is not known (1983) whether there exists an $ m $ such that for every $ n > m $ there is a Hausdorff compactum (a metric space) $ X $ with $ \mathop{\rm ind} X = m $, $ \mathop{\rm Ind} X = n $.

In the case of non-metrizable spaces, the dimension may not only fail to be monotone, but it also has other pathological properties. For any $ n = 2, 3 \dots $ an example of a Hausdorff compactum $ \theta _ {n} $ in which any closed set has dimension either 0 or $ n = \mathop{\rm dim} \theta _ {n} $ has been constructed. An analogous example for the inductive dimension is impossible. Also, for each $ n = 1, 2 \dots $ an example of a Hausdorff compactum $ \Phi _ {n} $ for which any closed set separating $ \Phi _ {n} $ has dimension $ n = \mathop{\rm dim} \Phi _ {n} $ has been constructed. Thus, the approach to the definition of dimension in the case of a non-metrizable space differs in principle from the inductive approach of Poincaré based on the separation of the space by spaces of a smaller number of coordinates. The Hausdorff compacta $ \Phi _ {n} $ are directly related to the following statement: Any $ n $- dimensional Hausdorff compactum contains an $ n $- dimensional Cantor manifold.

A subset of an $ n $- dimensional Euclidean space $ E ^ {n} $ is $ n $- dimensional if and only if it contains interior points with respect to $ E ^ {n} $. A compactum has dimension $ \leq n $ if and only if it has a mapping of dimension zero into $ E ^ {n} $, hence, up to zero-dimensional mappings, $ n $- dimensional compacta are indistinguishable from the bounded closed subsets of $ E ^ {n} $ containing interior points (with respect to $ E ^ {n} $).

See also Dimension theory.

References

[1]	P.S. Aleksandrov, B.A. Pasynkov, "Introduction to dimension theory" , Moscow (1973) (In Russian) MR0365524 Zbl 0441.55002
[2]	W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948)
[3]	P.S. Urysohn, "Works on topology and other areas of mathematics" , 1–2 , Moscow-Leningrad (1951) (In Russian)

Comments

Let $ X , Y $ be topological spaces and let $ \omega $ be a covering of $ X $. A continuous mapping $ f : X \rightarrow Y $ is an $ \omega $- mapping if each point $ y \in Y $ has a neighbourhood $ U _ {y} $ such that $ f ^ { - 1 } ( U _ {y} ) $ is included in some element of $ \omega $. Let $ X $ be metric; then $ f $ is called an $ \epsilon $- mapping for an $ \epsilon \in \mathbf R $, $ \epsilon > 0 $, if the diameter of each $ f ^ { - 1 } ( y) $ is $ < \epsilon $. Finally, a continuous mapping of a subset $ X $ of a Euclidean or Hilbert space $ E $ into $ E $ is called an $ \epsilon $- shift if each point of $ X $ gets displaced at most by $ \epsilon $.

Lebesgue's theorem on tilings is also called the Lebesgue–Brouwer theorem on tilings or the Pflastersatz.

The Katětov equality (3) is also called the Katetov–Morito equality (of dimensions of metrizable spaces).

The Lebesgue dimension is also called the covering dimension. A metrizable space $ X $ with $ \mathop{\rm ind} X < \mathop{\rm Ind} X $ was constructed by P. Roy [a2].

For still other notions of dimension cf. also Fractal dimension and Hausdorff dimension.

References

[a1]	R. Engelking, "Dimension theory" , North-Holland & PWN (1978) MR0482696 MR0482697 Zbl 0401.54029
[a2]	P. Roy, "Nonequality of dimensions for metric spaces" Trans. Amer. Math. Soc. , 134 (1968) pp. 117–132 MR0227960 Zbl 0181.26002
[a3]	J.-I. Nagata, "Modern dimension theory" , Heldermann (1983) MR0715431 Zbl 0518.54002

Dimension of an associative ring.

Comments

A number associated to a ring or module in such a way that its behaviour under some classical operations, e.g. subobjects, quotient objects, direct sums or products, extensions $ \dots $ may be studied. It is possible to introduce many different notions of dimension; the success of the theory one can develop depends, however, on the properties with respect to the kind of operations mentioned above, so that good techniques for giving proofs by induction on the dimension become available. Several of the dimensions most used in algebra and ring theory may be defined on the lattice of submodules of some module, globalizing the definition by considering the supremum (or a similar invariant) of the dimension of all modules (perhaps restricting to a certain class of modules). In this way one may define the Goldie dimension, the dual Goldie dimension, the Krull dimension, and the Gabriel dimension, as well as relative versions of these defined by restricting to suitable subcategories of modules. Certain dimensions are defined starting from the principle of resolutions in the category of modules, these dimensions include the homological dimensions, e.g. the projective dimension of a module (or ring), the injective dimension of a module (or ring) and the weak or flat dimensions of a module (cf. also Homological dimension). Many concrete problems concerning rings or modules may be solved by the introduction of the appropriate notion of dimension. As another example one can mention the so-called $ G K $- dimension (Gel'fand–Kirillov) that is related to a notion of a non-commutative transcendence degree and that can be used to obtain information about the imbedding of free algebras in the ring considered, e.g. in certain rings of differential operators (the simplest cases of which are the Weyl algebras. The Weyl algebra $ A _ {n} ( k) $ over a field of characteristic zero $ k $ is the algebra $ k < x _ {1} \dots x _ {n} $; $ \partial / {\partial x _ {1} } \dots \partial / {\partial x _ {n} } > $, i.e. the associative algebra generated by $ 2 n $ symbols $ X _ {1} \dots X _ {n} $, $ Y _ {1} \dots Y _ {n} $ subject to the relations $ Y _ {i} X _ {j} - X _ {j} Y _ {i} = \delta _ {ij} $ where $ \delta _ {ij} $ is the Kronecker symbol).

Below the definitions of those dimensions most commonly used in algebra are give.

Krull dimension. For a partially ordered set $ ( L , \leq ) $, let $ \Gamma ( L) $ be the set $ \{ {( a , b ) } : {a \leq b, a , b \in L } \} $. By transfinite recursion one may define on $ \Gamma ( L) $ a filtration:

$$ \Gamma _ {-} 1 ( L) = \{ ( a , b ) \in \Gamma ( L) , a = b \} . $$

$$ \Gamma _ {0} ( L) = \{ ( a , b ) \in \Gamma ( L) : [ a , b ] \textrm{ is Artinian } \} $$

$$ \Gamma _ \alpha = \{ ( a , b ) \in ( L) : \forall b \geq b _ {1} \geq \dots \geq b _ {n} \geq \dots \geq a, $$

$$ \left . \exists n \in \mathbf N , \forall i \geq n : ( b _ {i+} 1 , b _ {i} ) \in \cup _ {\beta < \alpha } \Gamma _ \beta ( L) \right \} . $$

In this way one obtains an ascending chain:

$$ \Gamma _ {-} 1 ( L) \subset \Gamma _ {0} ( L) \subset \dots $$

$$ \dots \subset \Gamma _ \alpha ( L) \subset \dots . $$

Since $ L $ is a set it follows that $ \Gamma _ \xi ( L) = \Gamma _ {\xi + 1 } ( L) = \dots $. If $ \Gamma ( L) = \Gamma _ \alpha ( L) $, one says that the Krull dimension $ \mathop{\rm Kdim} $ of $ L $ is defined. If $ R $ is a ring and $ M $ is a left $ R $- module, one says that $ M $ has Krull dimension if the lattice $ L ( M) $ of left submodules of $ M $ has Krull dimension. If the left $ R $- module $ {} _ {R} R $ has Krull dimension, one says that the ring $ R $ has Krull dimension.

Gabriel dimension. For a modular upper-continuous lattice (cf. Continuous lattice; Modular lattice) $ L $ having 0 and 1 one defines $ \mathop{\rm Gdim} L $ by transfinite recursion. $ \mathop{\rm Gdim} L = 0 $ if and only if $ L = \{ 0 \} $. Let $ \alpha $ be a non-limit ordinal number and assume that the Gabriel dimension $ \mathop{\rm Gdim} L ^ \prime = \beta $ has already been defined for lattices with $ \beta < \alpha $. One says that $ L $ is $ \alpha $- simple if for each $ a \neq 0 $ in $ L $ one has: $ \mathop{\rm Gdim} [ 0 , a ] $ is not smaller that $ \alpha $ but $ \mathop{\rm Gdim} [ a , 1 ] < \alpha $. One says that $ \mathop{\rm Gdim} L = \alpha $ if $ \mathop{\rm Gdim} L $ is not smaller than $ \alpha $ but for every $ a \neq 1 $ in $ L $ there is some $ b > 0 $ such that $ [ a , b ] $ is $ \beta $- simple for some $ \beta \leq \alpha $. If $ a \in L $ and $ \mathop{\rm Gdim} [ 0 , a ] = \alpha $, one says that $ a $ has Gabriel dimension $ \alpha $. If $ L $ has Krull dimension, then $ L $ also has Gabriel dimension, and $ \mathop{\rm Kdim} L \leq \mathop{\rm Gdim} L \leq 1 + \mathop{\rm Kdim} L $.

If $ L $ is a Noetherian lattice, then $ \mathop{\rm Gdim} L = 1 + \mathop{\rm Kdim} L $.

If $ R $ is a ring and $ M $ is a left $ R $- module, then $ \mathop{\rm Gdim} M $ is defined to be $ \mathop{\rm Gdim} L ( M) $.

It is somewhat remarkable that affine PI-rings (cf. PI-algebra) need not have Krull dimension, while on the other hand these rings have finite Gabriel dimensions.

Projective dimension. A projective resolution of a left $ R $- module $ M $ is an exact sequence

$$ \dots \rightarrow P _ {n} \rightarrow \dots \rightarrow P _ {0} \rightarrow M \rightarrow 0 , $$

where each $ P _ {i} $ is a projective left $ R $- module (cf. also Resolution). If $ P _ {k} \neq 0 $ but $ P _ {n} = 0 $ for all $ n > k $, then one says that the resolution has infinite length. It is easy to prove that each module $ M $ has a projective resolution and so one may define $ \mathop{\rm pd} _ {R} ( M) $, the projective dimension of $ M $, to be the least $ n $ for which $ M $ has a projective resolution of length $ n $. If such an $ n $ does not exist, one puts $ \mathop{\rm pd} _ {R} ( M) = \infty $; clearly, $ \mathop{\rm pd} _ {R} ( M) = 0 $ if and only if $ M $ is projective (cf. Projective module). The (left) global dimension $ \mathop{\rm gl}\AAh dim R $ of $ R $ is defined to be $ \sup \{ { \mathop{\rm pd} _ {R} ( M) } : {M \textrm{ an } R \textrm{ \AAh module } } \} $; in fact, this global dimension is the same if one uses right modules for its determination.

One may define the injective dimension of a module $ M $ in a completely dual way, using injective resolutions, such that $ fnnme {inj\AAh dim } M $ is the length of a minimal injective resolution for $ M $. The global (left) injective dimension of $ R $ is defined to be the supremum of the injective dimensions of arbitrary (left) $ R $- modules, but one can establish that this dimension of the ring $ R $ is the same as $ \mathop{\rm gl}\AAh dim R $ defined as above using projective resolutions.

Moreover, if $ R $ is a left and right Noetherian ring, then the left and right global dimensions of $ R $ are the same. Note that the semi-simple Artinian rings are characterized by the fact that they have global dimension zero (cf. also Artinian ring).

Instead of considering projective resolutions one may look at resolutions of $ M $ in terms of flat $ R $- modules (cf. Flat module). The dimension defined in this way is the flat dimension, or weak dimension, of $ M $, denoted by $ \mathop{\rm fd} ( M) $. The left weak dimension $ \mathop{\rm l}.wdim ( R) $ of $ R $ is defined to be $ \sup \{ { \mathop{\rm fd} ( M) } : {M \textrm{ a } \textrm \left R \textrm{ \AAh module } } \} \right .$; the right weak dimension $ \mathop{\rm r}.wdim ( R) $ is defined similarly. Cf. also Homological dimension.

For a left Noetherian ring $ R $, $ \mathop{\rm gl}\AAh dim R = \mathop{\rm l}.wdim R $; for a right Noetherian ring $ R $, $ \mathop{\rm gl}\AAh dim R = \mathop{\rm r}.wdim R $. So for left and right Noetherian rings the projective global dimension, the injective global dimension and the weak dimension coincide; this is not true for arbitrary $ R $- modules though.

The global dimension is important in the study of commutative regular local rings that play an important part in basic algebraic geometry. Note that a local commutative ring is regular exactly then when it has finite global dimension, and in this case the global dimension equals the Krull dimension.

Gel'fand–Kirillov dimension. For an algebra $ A $ over a field $ k $ one considers subalgebras $ k [ V] $ in $ A $ generated over $ k $ by a vector space $ V $ over $ k $ contained in $ A $. If $ V $ is finite dimensional over $ k $ and $ 1 \in V $, then $ V $ is called a frame for $ A $ if $ A = k [ V] $ and a subframe otherwise. Let $ V = k v _ {1} + \dots + k v _ {d} $ be a subframe of $ A $ and let $ V ^ {i} $ be the set of monomials of length $ i $ in $ v _ {1} \dots v _ {d} $. Write $ F _ {n} ^ { V } ( A) = k + V + \dots + V ^ {n} $. Then $ \{ {F _ {n} ^ { V } ( A) } : {n \in \mathbf Z } \} $ defines a filtration on $ k [ V] $, where by definition $ F _ {m} ^ { V } ( A) = 0 $ for $ m \leq - 1 $. The associated graded ring of this filtration is isomorphic to $ \oplus _ {i = - 1 } ^ \infty V ^ {i+} 1 / V ^ {i} $.

Define $ d _ {V} ( n) = \mathop{\rm dim} _ {k} F _ {n} ^ { V } ( A) $ and $ \mathop{\rm GKdim} k [ V] = \lim\limits \sup ( \mathop{\rm log} d _ {V} ( n) \mathop{\rm log} ( n) ^ {-} 1 ) $. This number is well-defined and it depends only on $ k ( V) $ but not on the choice of $ V $.

Put $ \mathop{\rm GKdim} A = \mathop{\rm GKdim} k [ V] $ if $ V $ is a frame for $ A $. If $ A $ is algebraic over $ k $, then $ \mathop{\rm GKdim} A = 0 $( cf. also Algebraic algebra). Note that $ \mathop{\rm GKdim} A $ is a real number but not necessarily an integer; W. Borho and H. Kraft have shown that any $ r \in \mathbf R $ with $ r \geq 2 $ can appear as the GKdim of some $ k $- algebra. In the interval $ [ 0 , 1 ] $ only 0 and 1 can appear as GKdim of a $ k $- algebra; G. Bergman proved that numbers in $ ( 1 , 2 ) $ cannot appear as the GKdim of some $ k $- algebra. That this dimension is sometimes related to the Krull dimension is not a big surprise, at least in the commutative case.

If $ M $ is a finitely-generated module over an affine commutative $ k $- algebra, then $ \mathop{\rm GKdim} M = \mathop{\rm Kdim} M $.

If $ \mathfrak g $ is a solvable Lie algebra over $ k $, say $ \mathop{\rm dim} _ {k} \mathfrak g = n $, then the GKdim of the universal enveloping algebra $ U ( \mathfrak g ) $ satisfies $ \mathop{\rm GKdim} U ( \mathfrak g ) = \mathop{\rm Kdim} U ( \mathfrak g ) = n $.

If $ A $ is a prime PI-algebra, then $ \mathop{\rm GKdim} A = \mathop{\rm tdeg} _ {k} A $ and in the affine case this number also equals the classical Krull dimension, $ \mathop{\rm cl}.Kdim A $, defined in terms of the length of maximal chains of prime ideals of $ A $.

The Gel'fand–Kirillov transcendence degree $ fnnem GKtdeg ( A ) $ is defined to be $ \sup _ {V} \inf _ {V} \mathop{\rm GKdim} k [ b V ] $, where $ V $ ranges over the subframes of $ A $ and $ b $ ranges over the regular elements of $ A $. If $ D _ {n} ( k) $ is the $ n $- th Weyl field, the quotient division algebra of $ A _ {n} ( k) $, then $ fnnem GKtdeg ( D _ {n} ( k) ) = 2 n $, whereas $ \mathop{\rm GKdim} D _ {n} ( k) = \infty $, so the transcendence degree is somewhat better behaved when dealing with rings of differential operators.

There are a large number of concepts called "dimension" in many parts of mathematics. The three principal groups appear to be the topological concepts (including the dimension of differentiable and analytic manifolds) and the algebraic ones described above, and the ideas of dimension in algebraic and analytic geometry; cf. also Analytic space; Rational function; Analytic set; Cohomological dimension; Spectrum of a ring. The last group of dimension ideas, i.e. those of algebraic and analytic geometry, are intermediate between the other two and form something of a bridge.

A topological space is irreducible if it can not be written as a union of two proper closed subspaces $ X _ {1} \cup X _ {2} = X $, $ X _ {i} \neq X $. A topological space $ X $ is called Noetherian if it satisfies the descending chain condition for closed subsets: For any sequence of closed subsets $ Y _ {1} \supset Y _ {2} \supset \dots $ there is an $ r $ such that $ Y _ {r} = Y _ {r+} 1 = \dots $. Now define the (algebraic-geometrically inspired) dimension of $ X $ as the supremum of all integers $ n $ such that there exists a chain of irreducible closed subsets

$$ \emptyset \neq Z _ {0} \subset \dots \subset Z _ {n} $$

(proper inclusions everywhere) in $ X $. To avoid confusion, this notion of dimension is written $ \mathop{\rm adim} ( X) $ here. This is not a notion which makes a great deal of sense for Hausdorff spaces (the only irreducible Hausdorff spaces are one-point spaces), but it is just right for algebraic varieties and schemes (with the Zariski topology).

Indeed, if $ X = \mathop{\rm Spec} ( A) $, where $ A $ is a commutative Noetherian ring with unit element, then this is the Krull dimension of $ A $: $ \mathop{\rm Kdim} ( A) = \mathop{\rm adim} ( X) $.

Let $ X $ be an irreducible algebraic variety; then $ \mathop{\rm adim} ( X) $ is also the transcendence degree of the field of rational functions on $ X $, another frequently used concept to define the dimension of an algebraic variety. The local dimension of $ X $ at a point $ x $ is defined as $ \mathop{\rm dim} _ {x} X = \mathop{\rm dim} _ {k(} x) ( \mathfrak m _ {x} / \mathfrak m _ {x} ^ {2} ) $, where $ \mathfrak m _ {x} $ is the maximal ideal of the local ring $ {\mathcal O} _ {X,x} $ at $ x $ and $ k ( x) = {\mathcal O} _ {X,x} / \mathfrak m _ {x} $. One has $ \mathop{\rm adim} ( X) = \min _ {x} \mathop{\rm dim} _ {x} X $ and $ \mathop{\rm adim} ( X) = \mathop{\rm dim} _ {x} X $ if and only if $ x $ is a regular point if and only if $ {\mathcal O} _ {X,x} $ is a regular local ring.

If $ X $ is an algebraic variety over $ \mathbf C $ and $ X ^ {*} $ is its open subvariety of smooth points, then $ X ^ {*} $ is also a complex manifold over $ \mathbf C $, of $ \mathbf C $- dimension $ \mathop{\rm adim} ( X) $( meaning that locally one needs $ n $ complex coordinates to describe it) and hence of dimension $ 2 n $ as a topological manifold.

Finally it is possible to describe the topological dimension of a completely-regular space in terms of the algebra $ C _ {b} ( X) $ of bounded real-valued functions on $ X $. The metric topology on $ C _ {b} ( X) $ is defined by the norm

$$ \| f \| = \sup _ { x } | f ( x) | , $$

which is also determined algebraically by

$$ \| f \| = \sup _ {\mathfrak m } | f ( x) | $$

where $ \mathfrak m $ runs through the maximal ideals of $ C _ {b} ( X) $. A subring $ A $ of $ C _ {b} ( X) $ will be called an analytic subring if

i) all constant functions belong to $ A $;

ii) $ f ^ { 2 } \in A \Rightarrow f \in A $;

iii) $ A $ is closed in the metric topology on $ C _ {b} ( X) $.

A set of functions $ B $ is said to be an analytic base for an analytic subring $ A \subset C _ {b} ( X) $ if $ A $ is the smallest analytic subring containing $ B $.

The following are equivalent for a completely-regular space $ X $: 1) $ \mathop{\rm dim} X \leq n $; 2) every countable set in $ C _ {b} ( X) $ is contained in an analytic subring with analytic base of cardinality $ \leq n $; and 3) every finite subfamily of $ C _ {b} ( X) $ is contained in an analytic subring with an analytic base of cardinality $ \leq n $( Katětov's theorem). If $ X $ is a compact metric space, these three properties are also equivalent to $ C _ {b} ( X) $ itself having an analytic base of cardinality $ \leq n $.

References

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[a2]	R. Gordon, J.C. Robson, "Krull dimension" , Amer. Math. Soc. (1973) MR0352177 MR0321980 MR0316489 Zbl 0269.16017 Zbl 0266.16022 Zbl 0257.16004
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[a4]	C. Năstăsecu, F. van Oystaeyen, "Dimensions of ring theory" , Reidel (1987)
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