Cohomology of face rings, and torus actions
Abstract.
In this survey article we present several new developments of ‘toric topology’ concerning the cohomology of face rings (also known as Stanley–Reisner algebras). We prove that the integral cohomology algebra of the momentangle complex (equivalently, of the complement of the coordinate subspace arrangement) determined by a simplicial complex is isomorphic to the algebra of the face ring of . Then we analyse Massey products and formality of this algebra by using a generalisation of Hochster’s theorem. We also review several related combinatorial results and problems.
1. Introduction
This article centres on the cohomological aspects of ‘toric topology’, a new and actively developing field on the borders of equivariant topology, combinatorial geometry and commutative algebra. The algebrogeometric counterpart of toric topology, known as ‘toric geometry’ or algebraic geometry of toric varieties, is now a well established field in algebraic geometry, which is characterised by its strong links with combinatorial and convex geometry (see the classical survey paper [10] or more modern exposition [13]). Since the appearance of Davis and Januszkiewicz’s work [11], where the concept of a (quasi)toric manifold was introduced as a topological generalisation of smooth compact toric variety, there has grown an understanding that most phenomena of smooth toric geometry may be modelled in the purely topological situation of smooth manifolds with a nicely behaved torus action.
One of the main results of [11] is that the equivariant cohomology of a toric manifold can be identified with the face ring of the quotient simple polytope, or, for more general classes of torus actions, with the face ring of a certain simplicial complex . The ordinary cohomology of a quasitoric manifold can also be effectively identified as the quotient of the face ring by a regular sequence of degreetwo elements, which provides a generalisation to the wellknown Danilov–Jurkiewicz theorem of toric geometry. The notion of the face ring of a simplicial complex sits in the heart of Stanley’s ‘Combinatorial commutative algebra’ [24], linking geometrical and combinatorial problems concerning simplicial complexes with commutative and homological algebra. Our concept of toric topology aims at extending these links and developing new applications by applying the full strength of the apparatus of equivariant topology of torus actions.
The article surveys certain new developments of toric topology related to the cohomology of face rings. Introductory remarks can be found at the beginning of each section and most subsections. A more detailed description of the history of the subject, together with an extensive bibliography, can be found in [8] and its extended Russian version [9].
The current article represents the work of the algebraic topology and combinatorics group at the Department of Geometry and Topology, Moscow State University, and the author thanks all its members for the collaboration and insight gained from numerous discussions, particularly mentioning Victor Buchstaber, Ilia Baskakov, and Arseny Gadzhikurbanov. The author is also grateful to Nigel Ray for several valuable comments and suggestions that greatly improved this text and his hospitality during the visit to Manchester sponsored by an LMS grant.
2. Simplicial complexes and face rings
The notion of the face ring of a simplicial complex is central to the algebraic study of triangulations. In this section we review its main properties, emphasising functoriality with respect to simplicial maps. Then we introduce the bigraded algebra through a finite free resolution of as a module over the polynomial ring. The corresponding bigraded Betti numbers are important combinatorial invariants of .
2.1. Definition and main properties
Let be an arbitrary dimensional simplicial complex on an element vertex set , which we usually identify with the set of ordinals . Those subsets belonging to are referred to as simplices; we also use the notation . We count the empty set as a simplex of . When it is necessary to distinguish between combinatorial and geometrical objects, we denote by a geometrical realisation of , which is a triangulated topological space.
Choose a ground commutative ring with unit (we are mostly interested in the cases or finite field). Let be the graded polynomial algebra over with . For an arbitrary subset , denote by the squarefree monomial .
The face ring (or Stanley–Reisner algebra) of is the quotient ring
where is the homogeneous ideal generated by all monomials such that is not a simplex of . The ideal is called the Stanley–Reisner ideal of .
Example 2.1.
Let be a 2dimensional simplicial complex shown on Figure 1. Then
Despite its simple construction, the face ring appears to be a very powerful tool allowing us to translate the combinatorial properties of different particular classes of simplicial complexes into the language of commutative algebra. The resulting field of ‘Combinatorial commutative algebra’, whose foundations were laid by Stanley in his monograph [24], has attracted a lot of interest from both combinatorialists and commutative algebraists.
Let and be two simplicial complexes on the vertex sets and respectively. A set map is called a simplicial map between and if for any ; we often identify such with its restriction to (regarded as a collection of subsets of ), and use the notation .
Proposition 2.2.
Let be a simplicial map. Define a map by
Then induces a homomorphism , which we will also denote by .
Proof.
We have to check that . Suppose is not a simplex of . Then
(2.1) 
We claim that is not a simplex of for any monomial in the right hand side of the above identity. Indeed, if , then by the definition of simplicial map, which leads to a contradiction. Hence, the right hand side of (2.1) is in . ∎
2.2. Cohen–Macaulay rings and complexes
Cohen–Macaulay rings and modules play an important role in homological commutative algebra and algebraic geometry. A standard reference for the subject is [6], where the reader may find proofs of the basic facts about Cohen–Macaulay rings and regular sequences mentioned in this subsection. In the case of simplicial complexes, the Cohen–Macaulay property of the corresponding face rings leads to important combinatorial and topological consequences.
Let be a finitelygenerated commutative graded algebra over . We assume that is connected () and has only evendegree graded components, so that we do not need to distinguish between graded and nongraded commutativity. We denote by the positivedegree part of and by the set of homogeneous elements in .
A sequence of algebraically independent homogeneous elements of is called an hsop (homogeneous system of parameters) if is a finitelygenerated module (equivalently, has finite dimension as a vector space).
Lemma 2.3 (Nöther normalisation lemma).
Any finitelygenerated graded algebra over a field admits an hsop. If has characteristic zero and is generated by degreetwo elements, then a degreetwo hsop can be chosen.
A degreetwo hsop is called an lsop (linear system of parameters).
A sequence of elements of is called a regular sequence if is not a zero divisor in for . A regular sequence consists of algebraically independent elements, so it generates a polynomial subring in . It can be shown that t is a regular sequence if and only if is a free module.
An algebra is called Cohen–Macaulay if it admits a regular hsop t. It follows that is Cohen–Macaulay if and only if it is a free and finitely generated module over its polynomial subring. If is a field of zero characteristic and is generated by degreetwo elements, then one can choose t to be an lsop. A simplicial complex is called Cohen–Macaulay (over ) if its face ring is Cohen–Macaulay.
Example 2.4.
Let be the boundary of a 2simplex. Then
The elements are algebraically independent, but do not form an hsop, since is not finitedimensional as a space. On the other hand, the elements , of form an hsop, since . It is easy to see that is a free module with one 0dimensional generator 1, one 1dimensional generator , and one 2dimensional generator . Thus, is Cohen–Macaulay and is a regular sequence.
For an arbitrary simplex define its link and star as the subcomplexes
If is a vertex, then is the subcomplex consisting of all simplices of containing , and all their subsimplices. Note also that is the cone over .
The following fundamental theorem characterises Cohen–Macaulay complexes combinatorially.
Theorem 2.5 (Reisner).
A simplicial complex is Cohen–Macaulay over if and only if for any simplex (including ) and , it holds that .
Using standard techniques of topology the previous theorem may be reformulated in purely topological terms.
Proposition 2.6 (Munkres).
is Cohen–Macaulay over if and only if for an arbitrary point , it holds that
Thus any triangulation of a sphere is a Cohen–Macaulay complex.
2.3. Resolutions and algebras
Let be a finitelygenerated graded module. A free resolution of is an exact sequence
(2.2) 
where the are finitelygenerated graded free modules and the maps are degreepreserving. By the Hilbert syzygy theorem, there is a free resolution of with for . A resolution (2.2) determines a bigraded differential module , where , and . The bigraded cohomology module has for and . Let be the bigraded module with for , , and zero differential. Then the resolution (2.2) determines a bigraded map inducing an isomorphism in cohomology.
Let be another module; then applying the functor to a resolution we get a homomorphism of differential modules
which in general does not induce an isomorphism in cohomology. The th cohomology module of the cochain complex
is denoted by . Thus,
Since all the and are graded modules, we actually have a bigraded module
The following properties of are well known.
Proposition 2.7.
(a) the module does not depend on a choice of resolution in (2.2);
(b) and are covariant functors;
(c) ;
(d) .
Now put and . Since , we have
Define the bigraded Betti numbers of by
(2.3) 
We also set
Example 2.8.
Let be the boundary of a square. Then
Let us construct a resolution of and calculate the corresponding bigraded Betti numbers. The module has one generator 1 (of degree 0), and the map is the quotient projection. Its kernel is the ideal , generated by two monomials and . Take to be a free module on two 4dimensional generators, denoted and , and define by sending to and to . Its kernel is generated by one element . Hence, has one generator of degree 8, say , and the map is injective and sends to . Thus, we have a resolution
where , and .
The Betti numbers are important combinatorial invariants of the simplicial complex . The following result expresses them in terms of homology groups of subcomplexes of .
Given a subset , we may restrict to and consider the full subcomplex .
Theorem 2.9 (Hochster).
We have
where denotes the reduced cohomology groups and we assume that .
Hochster’s original proof of this theorem uses rather complicated combinatorial and commutative algebra techniques. Later in subsection 5.1 we give a topological interpretation of the numbers as the bigraded Betti numbers of a topological space, and prove a generalisation of Hochster’s theorem.
Example 2.10 (Koszul resolution).
Let with the module structure defined via the map sending each to 0. Let denote the exterior algebra on generators. The tensor product (here and below we use for ) may be turned to a differential bigraded algebra by setting
(2.4) 
and requiring to be a derivation of algebras. An explicit construction of a cochain homotopy shows that for and . Since is a free module, it determines a free resolution of . It is known as the Koszul resolution and its expanded form (2.2) is as follows:
where is the subspace of spanned by monomials of length .
Now let us consider the differential bigraded algebra with defined as in (2.4).
Lemma 2.11.
There is an isomorphism of bigraded modules:
which endows with a bigraded algebra structure in a canonical way.
Proof.
Using the Koszul resolution in the definition of , we calculate
The cohomology in the right hand side is a bigraded algebra, providing an algebra structure for . ∎
The bigraded algebra is called the algebra of the simplicial complex .
Lemma 2.12.
A simplicial map between two simplicial complexes on the vertex sets and respectively induces a homomorphism
(2.5) 
of the corresponding algebras.
3. Toric spaces
Momentangle complexes provide a functor from the category of simplicial complexes and simplicial maps to the category of spaces with torus action and equivariant maps. This functor allows us to use the techniques of equivariant topology in the study of combinatorics of simplicial complexes and commutative algebra of their face rings; in a way, it breathes a geometrical life into Stanley’s ‘combinatorial commutative algebra’. In particular, the calculation of the cohomology of opens a way to a topological treatment of homological invariants of face rings.
The space was introduced for arbitrary finite simplicial complex by Davis and Januszkiewicz [11] as a technical tool in their study of (quasi)toric manifolds, a topological generalisation of smooth algebraic toric varieties. Later this space turned out to be of great independent interest. For the subsequent study of , its place within ‘toric topology’, and connections with combinatorial problems we refer to [8] and its extended Russian version [9]. Here we review the most important aspects of this study related to the cohomology of face rings.
3.1. Momentangle complexes
The torus is a product of circles; we usually regard it as embedded in in the standard way:
It is contained in the unit polydisk
For an arbitrary subset , define
The subspace is homeomorphic to .
Given a simplicial complex on , we define the momentangle complex by
(3.1) 
The torus acts on coordinatewise and each subspace is invariant under this action. Therefore, the space inherits a torus action. The quotient can be identified with the unit cube:
The quotient is then the following dimensional face of :
Thus the whole quotient is identified with a certain cubical subcomplex in , which we denote by .
Lemma 3.1.
The cubical complex is homeomorphic to .
Proof.
Let denote the barycentric subdivision of (the vertices of correspond to nonempty simplices of ). We define a embedding by mapping each vertex to the vertex where if and otherwise, the cone vertex to , and then extending linearly on the simplices of . The barycentric subdivision of a face is a subcomplex in , which we denote . Under the map the subcomplex maps onto the face . Thus the whole complex maps homeomorphically onto , which concludes the proof. ∎
It follows that the momentangle complex can be defined by the pullback diagram
where is the projection onto the orbit space.
Example 3.2.
The embedding for two simple cases when is a three point complex and the boundary of a triangle is shown on Figure 2. If is the whole simplex on vertices, then is the whole cube , and the above constructed homeomorphism between and defines the standard triangulation of .
The next lemma shows that the space is particularly nice for certain geometrically important classes of triangulations.
Lemma 3.3.
Suppose that is a triangulation of an dimensional sphere. Then is a closed dimensional manifold.
In general, if is a triangulated manifold then is a noncompact manifold, where is the cone vertex and .
Proof.
We only prove the first statement here; the proof of the second is similar and can be found in [9]. Each vertex of corresponds to a vertex of the barycentric subdivision , which we continue to denote . Let be the star of in , that is, the subcomplex consisting of all simplices of containing , and all their subsimplices. The space has a canonical face structure whose facets (codimensionone faces) are
(3.2) 
and whose faces are nonempty intersections of tuples of facets. In particular, the vertices (0faces) in this face structure are the barycentres of dimensional simplices of .
For every such barycentre we denote by the subset of obtained by removing all faces not containing . Since is a triangulation of a sphere, is an ball, hence each is homeomorphic to an open subset in via a homeomorphism preserving the dimension of faces. Since each point of is contained in some , this displays as a manifold with corners. Having identified with and further with , we see that every point in lies in a neighbourhood homeomorphic to an open subset in and thus in . ∎
A particularly important class of examples of sphere triangulations arise from boundary triangulations of convex polytopes. Suppose is a simple dimensional convex polytope, i.e. one where every vertex is contained in exactly facets. Then the dual (or polar) polytope is simplicial, and we denote its boundary complex by . is then a triangulation of an sphere. The faces of introduced in the previous proof coincide with those of .
Example 3.4.
Let . Then . In particular, for from (3.1) we get the familiar decomposition
of a 3sphere into a union of two solid tori.
Using faces (3.2) we can identify the isotropy subgroups of the action on . Namely, the isotropy subgroup of a point in the orbit space is the coordinate subtorus
In particular, the action is free over the interior (that is, near the cone point) of .
It follows that the momentangle complex can be identified with the quotient
where if and only if and . In the case when is the dual triangulation of a simple polytope we may write instead. The latter manifold is the one introduced by Davis and Januszkiewicz [11], which thereby coincides with our momentangle complex.
3.2. Homotopy fibre construction
The classifying space for the circle can be identified with the infinitedimensional projective space . The classifying space of the torus is a product of copies of . The cohomology of is the polynomial ring , (the cohomology is taken with integer coefficients, unless another coefficient ring is explicitly specified). The total space of the universal principal bundle over can be identified with the product of infinitedimensional spheres.
In [11] Davis and Januszkiewicz considered the homotopy quotient of by the action (also known as the Borel construction). We refer to it as the Davis–Januszkiewicz space:
where . There is a a fibration with fibre . The cohomology of the Borel construction of a space is called the equivariant cohomology and denoted by .
A theorem of [11] states that the cohomology ring of (or the equivariant cohomology of ) is isomorphic to . This result can be clarified by an alternative construction of [8], which we review below.
The space has the canonical cell decomposition in which each factor has one cell in every even dimension. Given a subset , define the subproduct
where is the basepoint (zerocell) of . Now for a simplicial complex on define the following cellular subcomplex:
(3.3) 
Proposition 3.5.
The cohomology of is isomorphic to the Stanley–Reisner ring . Moreover, the inclusion of cellular complexes induces the quotient epimorphism
in the cohomology.
Proof.
Let denote the dimensional cell in the th factor of , and the cellular cochain module. A monomial represents the cellular cochain in . Under the cochain homomorphism induced by the inclusion the cochain maps identically if and to zero otherwise, whence the statement follows. ∎
Theorem 3.6.
There is a deformation retraction such that the diagram
is commutative.
Proof.
We have , and each is invariant. Hence, there is the corresponding decomposition of the Borel construction:
Suppose . Then , so we have
The space is the total space of a bundle over , and is contractible. It follows that there is a deformation retraction . These homotopy equivalences corresponding to different simplices fit together to yield the required homotopy equivalence between and . ∎
Corollary 3.7.
The space is the homotopy fibre of the cellular inclusion . Hence [11] there are ring isomorphisms
In view of the last two statements we shall also use the notation for , and refer to the whole class of spaces homotopy equivalent to as the Davis–Januszkiewicz homotopy type.
An important question arises: to what extent does the isomorphism of the cohomology ring of a space with the face ring determine the homotopy type of ? In other words, for given , does there exist a ‘fake’ Davis–Januszkiewicz space, whose cohomology is isomorphic to , but which is not homotopy equivalent to ? This question is addressed in [21]. It is shown there [21, Prop. 5.11] that if is a complete intersection ring and is a nilpotent cell complex of finite type whose rational cohomology is isomorphic to , then is rationally homotopy equivalent to . Using the formality of , this can be rephrased by saying that the complete intersection face rings are intrinsically formal in the sense of Sullivan.
Note that the class of simplicial complexes for which the face ring is a complete intersection has a transparent geometrical interpretation: such is a join of simplices and boundaries of simplices.
3.3. Coordinate subspace arrangements
Yet another interpretation of the momentangle complex comes from its identification up to homotopy with the complement of the complex coordinate subspace arrangement corresponding to . This leads to an application of toric topology in the theory of arrangements, and allows us to describe and effectively calculate the cohomology rings of coordinate subspace arrangement complements and in certain cases identify their homotopy types.
A coordinate subspace in can be written as
(3.4) 
for some subset . Given a simplicial complex , we may define the corresponding coordinate subspace arrangement and its complement
Note that if is a subcomplex, then . It is easy to see [8, Prop. 8.6] that the assignment defines a onetoone order preserving correspondence between the set of simplicial complexes on and the set of coordinate subspace arrangement complements in .
The subset is invariant with respect to the coordinatewise action. It follows from (3.1) that .
Proposition 3.8.
There is a equivariant deformation retraction
Proof.
In analogy with (3.3), we may write
(3.5) 
where
Then there are obvious homotopy equivalences (deformation retractions)
These patch together to get the required map . ∎
Example 3.9.

Let . Then (recall that in this case).

Let ( points). Then
the complement to the set of all codimension 2 coordinate planes.

More generally, if