1 Introduction

First, we describe in general terms how the results in this paper advance the use of tools from unstable homotopy theory across the field of Toric Topology and how one field informs the other within the context of certain unstable spectral sequences—objects that give insight into the homotopy type of a space. Second, we highlight some specific applications as they relate to cohomological rigidity as described in [7, 14].

Given a space X and a complex-orientable homology theory E, the higher derived functors of the primitive element functor, \(R^iPE_*(X)\), are important homotopy invariants that are used to make certain homotopy calculations for X [5]. In addition, these higher derived functors are a critical part of the input needed to compute the \(E_2\)-term of a certain composite functor spectral sequence (CFSS), which is often used to facilitate unstable homotopy calculations. Bendersky and his collaborators constructed and computed a CFSS which converges to the \(E_2\)-term of the unstable Adams Novikov spectral sequence (UANSS) [5, 6]. Under certain conditions, they were able to derive long exact sequences of \(E_2\)-terms of the UANSS making calculations somewhat tractable for various spaces such as \(\Omega S^{2n+1}\). Moreover, computing differentials in the UANSS becomes more manageable when the CFSS has two rows, which is the case when \(R^iPE_*(X)\) vanish for \( i \ge 2\). That is, the primitive dimension of the coalgebra is less than two. In such cases, Bousfield referred to these coalgebras as nice homology coalgebras [9].

One of the objectives of the research initiated in [1] was to set-up the unstable machinery and extend homotopy calculations made by Buchstaber and his collaborators for a wide variety of toric spaces including quasitoric manifolds [8]. There is an interesting interplay between the combinatorics of the orbit—a simple convex polytope P and the methods of unstable homotopy theory. Let us recall the definition of the Borel space-\(B_TP\); let T be the topological torus of the appropriate dimension acting on an n-dimensional manifold M, which for the purposes of this paper we shall refer to as a quasitoric manifold [8, 15]. The Borel space is \(ET \times _T M = B_TP\) and it fits into a fibration \(M \rightarrow B_TP \rightarrow \prod _n {\mathbb {C}}P^{\infty }\) [8].

The Stanley–Reisner ring, which in this context is the cohomology ring of the Borel space, plays a critical role in explicitly writing down the generators of the higher derived functors of the primitive element functor, \(R^iP(E_*(B_TP))\). This became apparent in [1], where listing both stable and unstable classes in certain unstable coalgebras [5] required relating the combinatorics of the orbit to Hopf Rings. In fact, through a range where the unstable machinery works [1], the square free monomial generators of the ideal I in the Stanley–Reisner ring corresponded to generators of \(R^1P(E_*(B_TP))\). From a spectral sequence perspective, the towers in the UANSS, through a range that depended on relations among relations of minimal degree, among the monomials, were indexed by the cardinality of the generating set of the ideal. It also became clear from this work that the combinatorics of the orbit obstructed the existence of an induced long exact sequence of \(R^iP(E_*(B_TP))\)—a critical tool used in making specific calculations. Such obstructions make it very difficult to extract useful information from the CFSS and without some additional structure or information this spectral sequence offers little hope of giving insight into the homotopy type of \(B_TP\).

Recall, for a given n-dimensional simple convex polytope P with m facets there is a quasitoric manifold that exhibits a very rich mathematical structure [15]. The expression of its cohomology ring as the quotient of the Stanley–Reisner ring modulo the ideal generated by a certain regular sequence of elements \(\lambda _1,\ldots ,\lambda _n\) coming from the map of lattices \({\mathbb {Z}}^m \rightarrow {\mathbb {Z}}^n\) links the algebra to the combinatorics of the orbit. This additional structure is part of the datum that allows one to relate the higher right derived functors of the primitive element functor of \(E_*(M)\) with those of \(E_*(B_TP)\) for a complex orientable theory \(E_*\). One of the main results in this paper is the proof that \(R^i(E_*(M)) \cong R^i(E_*(B_TP))\) holds for \(i >1\). We leverage the lambda map to bridge the gap between torus actions and the combinatorics of the orbit by way of these higher right derived functors. How this line of inquiry provides a new method to view rigidity [7, 14, 17, 18] problems will be elaborated upon below.

In certain cases, it can be shown, using the methods described in [6, 9], that \(R^iPE_*(M)\) vanish for \( i \ge 2\), expanding the library of nice homology coalgebras. This result hinged on the existence of a certain ESP-sequence that allowed for commutative algebra to be related to the unstable machinery via [9, 15]. We are able to relax the condition on the Krull Dimension of a certain regular sequence that is related to the torus action (i.e., \(\lambda _1,\ldots ,\lambda _n\)) to produce the short exact sequence in the category of interest, facilitating calculations using machinery from [9]. Using the sequence \(\lambda _1,\ldots ,\lambda _n\), the interplay between the Homological Algebra and Combinatorics is further exposed in the general context by using faithful systems as described in [13].

The methods in this paper provide a methodology that can be used to attack certain combinatorial rigidity problems from a homotopy-theoretic perspective. We recall a key definition from [14] (c.f., Pg 13, Definition 6.1). A simplicial complex \(K_P\) (the complex dual to P) is rigid if \(H^*(M_P) \cong H^*(M_Q)\) implies \(K_P\) is isomorphic to \(K_Q\) for any manifold \(M_P\) associated to P and any manifold \(M_Q\) associated to Q.

In previous work [2] it was shown that there is a rational isomorphism \(R^iP(-)\) between the Stanley–Reisner rings associated to K and \(K'\) (with dual polytopes P and \(P'\)) if there exists a rational isomorphism between the higher derived functors of the primitive element functor of the homology of the corresponding quasitoric manifolds. By the results of the current paper this isomorphism has been extended to certain complex orientable theories the reasons for which will be elaborated upon below. Combining these results and observations implies that these higher right derived functors are independent of the torus action. Equally important is the observation that this isomorphism imposes an orbit-dependent combinatorial constraint on the torus action. Hence, we observe that these derived functors can be used to test cohomological rigidity to produce non-existence results [4]. For instance, to test if a simplicial complex is cohomologically rigid, one could start by comparing the second derived functor of the Stanley–Reisner rings corresponding to both simplicial complexes or comparing \(R^2P(E_*(B_TP))\) to \(R^2P(E_*(M_{P'}))\) for a quasitoric manifold corresponding to the polytope \(P'\).

The extent to which torsion in these higher derived functors manifests and what that means in terms of the ideal J and the torus action is a question that deserves further exploration within the framework of the unstable homotopy machinery used throughout the paper. One can derive a research thread, perhaps a general principle from these notions. More explicitly, given a torus action on M that produces \(H^*(M)\) with a certain algebraic structure, determine if such a structure is permissible under the combinatorial constraints of the orbit. We re-iterate that the higher derived functors—\(R^iP(-)\) are used to interpolate between the Stanley–Reisner ring of the orbit and \(H^*(M)\), making it possible to link the algebra to the combinatorics using homotopy-theoretic methods. For instance, from this perspective, it was shown in [4] that if the ideal I in the Stanley–Reisner has generating set of monomials that produce relations among relations [1], then there does not exist a torus action that endows \(H^*(M)\) with a structure that is dual to a nice homology coalgebra in the sense of [9]. This of course, has spectral sequence consequences. The types of torus actions that can occur is in a broad sense controlled by the orbit and determining the size of the space of torus actions given a fixed algebraic structure on \(H^*(M)\) as a function of the combinatorics is an interesting line of inquiry.

To detect torsion, the isomorphism described in [2] needed to be extended from the rationals to more general rings and we close by commenting that these arguments can be augmented to work for other non-additive functors defined over the appropriate category.

In an effort to make the paper self-contained we have listed key definitions from toric topology in Sect. 1 and those from Unstable Homotopy Theory in Sect. 2 for readers who may be unfamiliar with certain results from either field.

Notational conventions Throughout the paper, we assume R is a commutative ring with unit and that \({\mathcal {M}}_{R}\) is the category of free, graded, connected R-modules. Tensor products over the integers will be denoted by \(\otimes \). Otherwise, the ring will be specified explicitly. The primitive element functor of R-coalgebras will be denoted by P and its higher right derived functors will be symbolized by \(R^iP(-;R)\). When the context is clear, the orbit of the torus action, a simple convex polytope, will also be denoted by P. All cohomology theories E are assumed to be complex orientable such that the coefficient ring \(E_*\) is graded with unit.

Main results The main results concerning the higher derived functors of the primitive element functor are:

Theorem 5.10

Let R be a ring. Suppose \(C_{1}\) and \(C_{2}\) are \({\mathbb {Z}}\)-coalgebras. If \(R^{i}P(C_{1})\cong R^{i}P(C_{2})\) for \(i>k\), then

$$\begin{aligned} R^{i}P(C_{1}\otimes R;R)\cong R^{i}P(C_{2}\otimes R;R) \end{aligned}$$

for \(i>k\).

The following theorem generalizes Theorem 4.5 [2] which held over \({\mathbb {Q}}\).

Theorem 7.3

Let E be a complex orientable theory with coefficients concentrated in even degrees. Suppose \(M_1\) and \(M_2\) are two quasitoric manifolds with orbit a simple convex polytope P, then \(R^kP(E_*(M_1);E_{*}) \cong R^kP(E_*(M_2);E_{*})\) for \(k > 1\).

Generalizing the results of [2] to more generalized homology theories is necessary because the analysis of certain spectral sequences related to homotopy computations require such. For example, there are composite functor spectral sequences and the Bousfield–Kan spectral sequence based on a theory \(E_*\) and they require a careful analysis of the higher derived functors mentioned above as well as certain coaction formulae. Having the flexibility to consider various theories such as the Morava K-theories may offer additional insight into computing differentials.

One goal of the research is to understand the relationship between the torus action and the right higher derived functors of the primitive element functor. The fibration \(M \rightarrow B_TP \rightarrow BT^n\) originally motivated the question since for two manifolds \(M_1\) and \(M_2\) with orbit P, there is an isomorphism \(\pi _*(M_1) \cong \pi _*(M_2)\) for \(* > 2\). It seemed plausible, although not obvious, that a similar type of statement held for the right higher derived functors of the primitive element functor for these manifolds.

We observe that Theorem 7.3 has implications in regards to cohomological rigidity as described in the introduction. We recall a critical result from [2] that relates these higher derived functors of the homology coalgebra of a quasitoric manifold M to those of the Borel space of the orbit. There it was shown that \(R^iP( {\mathbb {Q}}(P)) \cong R^iP(H_*(M; {\mathbb {Q}}))\) holds for \(i > 1\). Combining these results gives a necessary condition for cohomological rigidity. Specifically, a simplicial complex K is cohomologically rigid if there exists a simplicial complex \(K'\) such that there is an isomorphism of the higher right derived functors of the primitive element functor of the corresponding Stanley–Reisner rings.

By restricting to the quasitoric manifolds we obtain an abstract isomorphism which provides additional flexibility in attacking the cohomological rigidity problem. For instance, in the paper [7] the notion of C-rigid (Definition 3.3 pg. 15) is related to transferring structure from one orbit to another through an isomorphism of cohomology rings of quasitoric manifolds. By the results in this paper, a certain flexibility is obtained and that is, the transfer of structure can be done without having a map of orbits a priori. Additional details regarding rigidity-type questions and their relations to the higher derived functors can be found in Sect. 3.

There are cohomological analogues with specific calculations of \(L_iQ(-)\)—the left higher derived functors of the indecomposable functor and certain details regarding the dual can be found in Sect. 2. In the paper [4] the following are determined explicitly for a class of general algebras for which the Stanley–Reisner ring is one such example: \(L_0Q(-)\), \(L_1Q(-)\) and \(L_2Q(-)\). In the special case of the Stanley–Reisner ring the first derived functors are generated by the relations and the second derived functor is generated by certain relations among relations. These notions are then related back to the combinatorics of the orbit to obtain a classification result for torus actions. Specific calculations can be found in [4].

For the purposes of this paper, the main thrust of the applications are geared toward setting-up and analyzing various unstable spectral sequences. However, there is an application currently under development regarding diffeomorphisms that highlights the utility of the approach that we shall discuss briefly. Details related to this example and related methods will appear in another work, but we mention them here. The testbed for the application related to diffeomorphism can be found in [7] Example 3.2 (pg. 15). Here, one begins with a triangular prism and conducts two subsequent vertex truncations. Three polyhedra result from this procedure and we label them as \(P_1\), \(P_2\) and \(P_3\). There is a quasitoric manifold and it is: \({\mathbb {C}}P ^{3} \sharp {\mathbb {C}}P ^{3} \sharp {\mathbb {C}}P ^{3} \sharp {\mathbb {C}}P ^{3} \) and this manifold has orbit one of the \(P_j\) as a result of varying the torus action. Each manifold over the \(P_j\) is diffeomorphic to the other. If one computes the \(L_iQ(-)\) of the Face ring of each orbit, then using the results in this paper and [4] paired with a careful calculation of the higher derived functors shows that the second derived functor of the Face ring of the \(P_j\), i.e., \(L_2Q(H^*(B_TP_j))\) have rank eight on six dimensional generators after accounting for linear independence. However, the higher derived functors do vary over the faces. Hence, one needs more sophisticated methods that allows for the isomorphism to extend from the faces of the polytopes to the isomorphism highlighted in this paper. Stated slightly differently, one would require moving from each polytope to its faces while preserving in some fashion the isomorphism that has been proven thus far. This is current work that applies the calculations in this paper in a more general setting so they can be used to differentiate diffeomorphisms, such as those mentioned above, for a variety of orientable theories.

2 Toric topology

For the sake of completeness we will highlight some well-known constructions in Toric Topology. Some excellent references for this material would include the conference proceedings [8, 11]. The material in this section can be found in [2].

For the applications needed in this paper, it is assumed that P is a simple convex polytope with m facets \(F_1,\ldots ,F_m\). Furthermore, \(T^n\) will denote the n-dimensional topological torus and \(BT^n\) its classifying space ([8] pgs. 9–11).

Using the perspective of derived forms the notion of a quasitoric manifold and related spaces can be described. In fact, derived forms are formulated in a much more general setting, where sets more general than polyhedra are used [11]. Following Buschstaber and Ray’s exposition, there is a map

$$\begin{aligned} \lambda : P \rightarrow T(T^n) \end{aligned}$$

where \(T(T^n)\) is the lattice of subtori of the torus ordered by inclusion and the topological structure is induced from the lower limit topology. \(\lambda \) sends \(q \in P\) to a certain subtorus, \(\lambda (q)\). The derived space is by definition the following quotient space:

$$\begin{aligned} D(\lambda ) = (T^n \times P)/ \sim \end{aligned}$$

where \((g,q) \sim (h,q)\) if and only if \(g^{-1}h \in \lambda (q)\). It can be shown that \(\sim \) is an equivalence relation. The elements in \(D(\lambda )\) are equivalence classes [g, q] for which there is a canonical action of the torus on \(D(\lambda )\) via multiplication on the first coordinate. The orbit space of this action is P; using the atlas \(\{U_v\}\) given by [8] page 63, construction 5.8, it can be shown that the torus action is locally standard [15] page 420 (here locally standard is referred to as locally isomorphic to the standard representation).

To obtain a quasitoric manifold (or toric manifold in the language of [15]), one must impose conditions on \(\lambda \) and a smooth structure on \(D(\lambda )\) cf. [15]. First, \(\lambda \) associates a circle to the interior of a facet. Hence, if \(F_j\) is a facet, then \(T_{\lambda }(F_j)\) is a circle. For additional details, see page 64, (5.3) [8]. Second, if F is a codimension k-face, that is, the intersection of k facets, then the torus subgroup associated to it is the product of those coordinate tori coming from each of the facets whose intersection is the face F. The \(F_i\) are the facets defining F. More succinctly, the following map is an isomorphism:

$$\begin{aligned} Im \left( \prod _{1 \le j \le m} T_{\lambda }(F_j)\right) \longrightarrow T(F) \end{aligned}$$

Finally, it is required that the kernel of \(\lambda \) partitions P by the interior of the faces. By a quasitoric manifold \(M^{2n}(\lambda )\) one means the derived space \(D(\lambda )\) where \(\lambda \) is subject to the conditions described above.

Remark

Davis and Januszkiewicz [15] refer to the subgroup \(G_F\) of the lattice \({\mathbb {Z}}^m\) “determined by \(T^n\) and \(\lambda \)” cf., page 423, 1.5. In addition, \(\lambda \) determines a map between integer lattices \({\mathbb {Z}}^m \rightarrow {\mathbb {Z}}^n\). Note that this map is also referred to as “lambda” in many research articles. Recall, to each facet \(F_j\) of P, one associates a certain circle subgroup, say \(T_{\lambda }(F_j)\). In [8] these subtori were made explicit, see page 64, (5.3). Each such torus subgroup gives rise to a facet vector \(\lambda _{ij} \in {\mathbb {Z}}^n\), for \(1 \le i \le n\), \(1 \le j \le m\). These vectors are indexed by the facets and from this and the discussion above, a function can be defined from the set of facets of P, \(\mathfrak {I}\), to the integer lattice whose dimension depends on the dimension of P. This is how the characteristic function \(\mathfrak {I}\rightarrow {\mathbb {Z}}^n\) is obtained and by a simple identification, the function \({\mathbb {Z}}^m \rightarrow {\mathbb {Z}}^n\) can be derived. By condition (*) in [15], one requires that the free \({\mathbb {Z}}\)-module spanned by those facet vectors coming from the subtori that are the images of those facets whose intersection is the appropriate face of P, be a direct summand of \({\mathbb {Z}}^n\). Davis and Januszkiewicz [15] refer to such modules as unimodular.

Examples of quasitoric manifolds would include the following: \({\mathbb {C}}P^n\) with orbit \(\Delta ^n\). Buchstaber and Ray [10] shows that the 2n-dimensional manifold \(B_n\) of all bounded flags in \({\mathbb {C}}^{n+1}\) is a quasitoric manifold over \(I^n\). Buchstaber and Ray [10] shows that \(CP^n \sharp CP^n\) is a quasitoric manifold over \(\Delta ^1 \times \Delta ^{n-1}\) by defining a connected sum operation on the level of the polytopes. Orlik et al. [20] classified four dimensional quasitoric manifolds that sit over polygons and showed that they are connected sum of the Hirzebruch surface with connected sums of \({\mathbb {C}}P^2\).

Given a set X endowed with an action of a group G, the Borel Construction can be used to replace the orbit space X / G, by a space \(EG \times _{G} X\) and if the action is free, then it is homotopy equivalent to the orbit. In the case of a quasitoric manifold M, we have the following:

Definition 2.1

Let M be a quasitoric manifold. The Borel space \(B_TM\) is the identification space

$$\begin{aligned} ET^n \times M/\sim = ET^n \times _{T^n} M \end{aligned}$$

where the equivalence relation is defined by: \((e,x)\sim (eg,g^{-1}x)\) for any \(e\in ET^n\) and \(x\in M\) , \(g\in T^n\).

The following notation is often used to denote the Borel space in our context: \(B_TP\) where P is the orbit of the torus action on M and this space does not depend on \(\lambda \). We amplify this by stating that the Borel space (i.e., the Borel construction of a quasitoric manifold depends only on the orbit ) whereas the map \(B_TP \rightarrow BT^n\) does depend on \(\lambda \). The following fibration appears in [8, 15]:

$$\begin{aligned} M^{2n}(\lambda ) \longrightarrow B_TP \longrightarrow BT^n \end{aligned}$$

The Face ring is an invariant that will be useful in the sections that follow. Recall,

Definition 2.2

Let \(F_1,\ldots ,F_m\) be the facets of P. For a fixed commutative ring R with unit we have

$$\begin{aligned} R(P) =R[v_1,\ldots ,v_m]/ (v_{i_1} \cdots v_{i_k}| F_{i_1}\cap \cdots \cap F_{i_k} = \emptyset ) \end{aligned}$$

where \(|v_i| = 2\) are indexed by the facets and the ideal I is generated by square free monomials coming from trivial intersection of facets.

If \(P = \partial \Delta ^2\), then \({\mathbb {Z}}(P) ={\mathbb {Z}}[v_1,v_2,v_3]/ (v_1v_2v_3)\). Sometimes we refer to R(P) as the Stanley–Reisner algebra (or the Stanley–Reisner ring or the Face ring. For the purposes of this paper they are all synonymous) and I the Stanley–Reisner ideal. It is shown in [15] that \(H^*(B_TP) \cong {\mathbb {Z}}(P)\) and that the Borel Construction is a contravariant functor from the category of simplicial complexes to the category of homotopy types of spaces (refer to pages 436 and 437).

When P is an n-dimensional simple convex polytope, the dual complex \(K_P\) [15], page 425 has a face structure encoded by the f-vector: \((f_0,\ldots ,f_{n-1})\) where \(f_i\) is the number of i-simplicies in K. Following [15], there is a polynomial, \((t-1)^n + \sum _{i=0}^{n-1} f_i(t-1)^{n-1 -i}\). The coefficients of \(t^{n-i}\) in the polynomial are denoted by \(h_i\) and the h-vector of P is: \((h_0,\ldots ,h_n)\). The homology groups of a quasitoric manifold are a function of the h-vector of P [15].

Theorem 2.3

If M is a quasitoric manifold with orbit P, then \(H_{2i}(M)\) is a free abelian group of rank \(h_i(P)\) and \(H_{2i-1}(M)\cong 0\) for all i.

Following [8, 15] we define certain linear forms that are important in the study of ESP-sequences in what follows. Let \(\lambda _i \in H^*(B_TP)\) and they are defined as follows: \(\lambda _i = \lambda _{i1}v_1 + \cdot \cdot \cdot + \lambda _{im}v_m\). Recall, the map \(H^*(BT^n) \rightarrow H^*(B_TP)\) is induced by a map of spaces and it sends \(t_i \mapsto \lambda _i\) which can be written as a polynomial in the generators \(v_i\) as described above. Let J be the ideal generated by \(\lambda _i\) in the Face ring.

Theorem 2.4

If M is a quasitoric manifold, then \(H^*(M) \cong H^*(B_TP) / J\).

Appealing to the references listed above and Definition 2.2, the cohomology ring is the quotient \({\mathbb {Z}}[v_1,\ldots ,v_m]/ (I +J)\).

Notation 2.5

Throughout the paper the dimension of a quasitoric manifold may be dropped as well as any reference to \(\lambda \) when the context is clear. For example, we may write M instead of \(M^{2n}(\lambda )\). In addition, if M is a quasitoric manifold with orbit P, then it is common to say that M is a quasitoric manifold over P or M sits over P. The orbit of the \(T^n\)-action on a quasitoric manifold M can be identified with P and this should give some insight into the notational convention \(B_TP\).

3 The higher derived functors of the primitive element functor

3.1 Preliminaries on homological algebra of nonadditive functors

We review the theory of derived functors of nonadditive functors. For more details the reader can see the appendix of [9]. Let \({\mathcal {R}}\) and \({\mathcal {A}}\) be categories where we assume \({\mathcal {A}}\) is abelian and \(T:{\mathcal {R}}\rightarrow {\mathcal {A}}\) is a functor. Let \({\mathcal {R}}^{+}\) be the category such that \(Obj({\mathcal {R}}^{+})=Obj({\mathcal {R}})\) but \(Hom_{{\mathcal {R}}^{+}}(-,-)\) is the free abelian group on \(Hom_{{\mathcal {R}}}(-,-)\). Let \(T^{+}:{\mathcal {R}}^{+}\rightarrow {\mathcal {A}}\) be the unique additive functor which is the extension of T to \({\mathcal {R}}^{+}\).

Definition 3.1

Let \({\mathcal {P}}\) be a class of objects in \({\mathcal {R}}\). A \({\mathcal {P}}\)-resolution for \(N\in Obj({\mathcal {R}})\) is an augmented chain complex

$$\begin{aligned} \mathbf N :N\rightarrow M^{0}\rightarrow M^{1}\rightarrow \cdots \end{aligned}$$

such that:

  1. (1)

    \(M^{i}\in {\mathcal {P}}\)

  2. (2)

    For each \(M \in Obj({\mathcal {P}}), Hom_{{\mathcal {R}}^{+}}(\mathbf N ,M)\) is an acyclic complex of abelian groups.

A class of injective models for \({\mathcal {R}}\) consist of a class \({\mathcal {P}}\) such that each \(N\in Obj({\mathcal {R}})\) has a \({\mathcal {P}}\)-resolution.

Let \({\mathbf {N}}_{u}\) be the un-augmented \({\mathcal {P}}\)-resolution of N. Define the higher right derived functors of T as

$$\begin{aligned} R^{i}T(N)=H^{i}(T^{+}{\mathbf {N}}_{u}) \end{aligned}$$

Suppose there are two categories \({\mathcal {C}},{\mathcal {D}}\) and functors \(F:{\mathcal {C}}\rightarrow {\mathcal {D}}\) and \(G:{\mathcal {D}}\rightarrow {\mathcal {C}}\) such that F is left adjoint to G. Let \(\varphi :id\rightarrow GF\) and \(\phi :FG\rightarrow id\) be the adjunction maps. For notational reasons we will not write “id” in what follows unless it is necessary. For instance, GidF will simply be written GF. Let \(K=GF\); for \(N\in Obj({\mathcal {C}})\) define an augmented cosimplicial object \({\mathbf {K}}(N)\) as follows:

  1. (1)

    \({\mathbf {K}}(N)^{n}=K^{n+1}(N)\) for \(n\ge -1\).

  2. (2)

    \(K^{i}(\varphi _{K^{n-i}})=d^{i}:{\mathbf {K}}(N)^{n}\rightarrow {\mathbf {K}}(N)^{n+1}\) for \(0\le i \le n\)

  3. (3)

    \(s^{i}=K^{i}(\mu _{K^{n-i}}):{\mathbf {K}}(N)^{n+1}\rightarrow {\mathbf {K}}(N)^{n}\) for \(0\le i \le n\) where the map \(\mu \) is defined as \(id \phi id:G(FG)F\rightarrow GF\)

Let \(ch({\mathbf {K}}(N))\) be the augmented chain complex such that \(ch({\mathbf {K}}(N))^{n}={\mathbf {K}}(N)^{n}\) and \(\delta ^{n}:ch({\mathbf {K}}(N))^{n}\rightarrow ch({\mathbf {K}}(N))^{n+1}\) be defined by \(\delta ^{n}=\sum _{i=0}^{n-1}(-1)^{i}d^{i}\).

There is the following result [9]

Theorem 3.2

Let \({\mathcal {P}}\) be the class of all \(M\in Obj({\mathcal {C}})\) such that \(M=G(D)\) for some \(D\in Obj({\mathcal {C}})\). Then \({\mathcal {P}}\) is a class of injective objects for \({\mathcal {C}}\) and \(ch({\mathbf {K}}(N))\) is a right \({\mathcal {P}}\)-resolution for N.

This means that if \(ch_{u}({\mathbf {K}}(N))\) is the un-augmented chain complex obtained from \(ch({\mathbf {K}}(N))\), then for any functor T, the higher right derived functors of T are given by: \(R^{i}T(N)=H^{i}(ch_{u}T^{+}{\mathbf {K}}(N))\).

3.2 Coalgebras and the derived functors of the primitive element functor

We review a particular higher derived functor that appears in the \(E_2\)-term of a composite functor spectral sequence (CFSS) that appeared in [5]. Since these are non-additive functors, we make use of the previous section. Some parts of this section were taken from the survey paper [3]. We recall that R is a commutative ring with unit. Material from [5, 6, 9] will be used throughout and the interested reader can refer to these papers for additional information and arguments.

Definition 3.3

A (graded) R-coalgebra C is a (graded) R-module with a comultiplication map \(\Delta :C\rightarrow C\otimes _{R} C\) and a counit \(\epsilon :C\rightarrow R\) such that the following diagrams commute:

An R-coalgebra C with unit is connected if there is a map \(\eta :R\rightarrow C\) such that \(\epsilon \circ \eta =id\).

It is easy to see that if C is a connected R-coalgebra, then it is isomorphic to \(R\oplus \overline{C}\) where \(\overline{C}=coker(\eta )\). From this point forward we only consider connected coalgebras. Let \({\mathcal {M}}_{R}\) be the category of free, graded, connected R-modules and \({\mathcal {C}}_{R}\) the category of connected coalgebras over R. We take note of the inclusion: \({\mathcal {C}}_{R}\subset {\mathcal {M}}_{R}\) so \(C_{R}\) are free R-modules which are R-coalgebras.

Definition 3.4

The module of primitives of C is defined as follows:

$$\begin{aligned} P(C) = ker\left( \overline{C}\overset{\overline{\psi }}{\longrightarrow } \overline{C}\otimes _{R} \overline{C}\right) \end{aligned}$$

Remark 3.5

If we consider coalgebras without counit, then the module of primitives is defined as the kernel of the diagonal map. In [6] the authors show that working with either type of coalgebra does not affect the discussion that follows if one adds a particular summand in degree zero c.f, page 378 last paragraph.

For any element \(M\in Obj({\mathcal {M}}_{R})\), [9] defines the connected, cofree, cocommutative, coalgebra functor with counit \(S(M)\in Obj(\mathcal {C_R})\). This functor comes equipped with a natural transformation \(s:S\rightarrow 1\) of R-modules such that if \(C\in Obj({\mathcal {C}}_{R})\) and there is a map of R-modules \(f:C\rightarrow M\), then there is a unique map in \({\mathcal {C}}_{R}\), \(\bar{f}:C\rightarrow S(M)\) such that the following diagram commutes:

We also have the forgetful functor \(J:{\mathcal {C}}_{R}\rightarrow {\mathcal {M}}_{R}\) giving

Lemma 3.6

For any \(C \in Obj({\mathcal {C}}_{R})\) and \(M\in Obj({\mathcal {M}}_{R})\)

$$\begin{aligned} Hom_{{\mathcal {C}}_{R}}(C,S(M))\cong Hom_{{\mathcal {M}}_{R}}(J(C),M) \end{aligned}$$

Using this result we can apply the methods of the previous section. By abuse of notation we write S for SJ. If \(C\in Obj({\mathcal {C}}_{R})\), then there is the (augmented) cosimplicial resolution \({\mathbf {S}}(C)\):

$$\begin{aligned} \begin{array}{ccccc} C \overset{d^{0}}{\rightarrow }&{}S(C) \begin{array}{c} \overset{d^{0}}{\rightarrow }\\ \overset{d^{1}}{\rightarrow } \end{array} &{} S^{2}(C) \begin{array}{c} \overset{d^{0}}{\rightarrow } \\ \overset{d^{1}}{\rightarrow }\\ \overset{d^{2}}{\rightarrow } \end{array} \cdots \end{array} \end{aligned}$$
(1)

The higher right derived functors of the primitive functor P are defined as

$$\begin{aligned} R^{i}P(C;R)=H^{i}(ch_{u}P^{+}{\mathbf {S}}(C)) \end{aligned}$$

The S-resolution is not the only way to obtain these functors [6]. There has been a lot of effort devoted to applying the general framework i.e., cosimplicial methods and category theory, in a way that provides homotopy theoretic information e.g., [1, 2, 5, 6, 12].

Knowing how to construct \(R^iP(C;R)\) and being able to compute them are notions that are separated by a degree of difficulty that is hard to quantify. [9] makes calculations that show that if C is the coalgebra of a Hopf Algebra, then there is a vanishing of its higher derived functors of P. He also discusses other examples that relate these ideas to Borel ideals. In [6] the authors compute \(R^iP(BP_*(\Omega S^{2n+1}))\) as well as an assortment of other coalgebras in the category of Unstable \({\mathcal {G}}\) -coalgebras. [1] computes these derived functors through a range for a wide variety of toric spaces and in [2] the authors relate such functors for quasitoric manifolds in the same family. Many of the calculations of the higher derived functors of the primitive element mentioned above rely on the notion of injective extension sequence. Roughly speaking, such a sequence is a short exact sequence in the category of coalgebras. Calculations by direct appeal to the (simplicial resolution) are made in [4].

Definition 3.7

A sequence of R-coalgebras \(R\rightarrow C^{\prime }\overset{f^{\prime }}{\rightarrow }C\overset{f}{\rightarrow } C^{\prime \prime }\rightarrow R\) is an injective extension sequence if:

  1. (1)

    f is a an epimorphism of R-modules.

  2. (2)

    C is an injective \(C^{\prime \prime }\)-comodule.

  3. (3)

    \(f'\) is the inclusion \(C\square _{C^{\prime \prime }} R\rightarrow C\).

This gives the following [9].

Theorem 3.8

If \(R\rightarrow C^{\prime } \rightarrow C\rightarrow C^{\prime \prime }\rightarrow R\) is a injective extension sequence then there is a long exact sequence

$$\begin{aligned} 0\rightarrow R^{0}P(C^{\prime };R)\rightarrow R^{0}P(C;R)\rightarrow R^{0}P(C^{\prime \prime };R)\rightarrow R^{1}P(C^{\prime };R)\rightarrow \cdots \end{aligned}$$

We are also interested in the dual concept of a projective extension sequence. To provide a more self-contained account of the relation between the Primitive element functor and the Indecomposable functor we list results and commentary from [4] for the convenience of the reader. These results and additional details can be found there.

Let A be a commutative ring with unit and we will assume that all A-algebras are graded, free A-modules endowed with a product map m and a unit. Let B be such an algebra with unit \(\eta :A\rightarrow B\). Furthermore, we require the restriction \(\eta |_{A_{0}}:A_{0}\rightarrow B\) to be an isomorphism and we let \(\overline{B}=Coker(A\overset{\eta }{\rightarrow }B)\).

Definition 3.9

A sequence of R-algebras \(R\rightarrow A^{\prime }\overset{f^{\prime }}{\rightarrow }A\overset{f}{\rightarrow } A^{\prime \prime }\rightarrow R\) is a projective extension sequence if:

  1. (1)

    \(f^{\prime }\) is injective as an R-module map.

  2. (2)

    A is a projective \(A^{\prime }\)-module.

  3. (3)

    f is the surjection \(A\rightarrow A\otimes _{A^{\prime }} R\).

The module of indecomposables will be useful and it is the R-module generated by elements in the algebra that can not be decomposed as a product. If C is an R-coalgebra, then \(A=C^{*}\) is an R-algebra such that Q(A) is dual to P(C).

Definition 3.10

The module of indecomposables of B is defined and denoted by

$$\begin{aligned} Q(B)= \overline{B}/\overline{B}^2 \end{aligned}$$

Q defines a non-additive functor from the category of A-algebras to the category of A-modules.

Let F be the free, commutative algebra functor with unit over A. It is a functor from the category of free A-modules to the category of A-algebras and comes equipped with a natural transformation \(s:1\rightarrow F\). There is a diagram that describes the universal property: if M is a free A-module and B is an A-algebra with an A-module map \(f:M\rightarrow B\), then there is a unique A-algebra map \(\bar{f}:F(M)\rightarrow B\) such that the following diagram commutes:

(2)

Let \(M=B\), \(f=id\), \(s_{-1}=s\) and \(d_0=\overline{id}\), then we obtain an augmented simplicial object over the category of A-algebras: \({\mathbf {F}}^\bullet (B)\)

The \(d_i=F^{n}((d_{0})_{F^{n-i}(B)}):F^n(B)\rightarrow F^{n-1}(B)\) for \(0\le i\le n\) and for \(0\le i \le n\) we have \(s_i=F^{n}((s_{-1})_{F^{n-i}(B)}):F^{n}(B)\rightarrow F^n(B)\). Applying the functor Q we obtain an un-augmented chain complex, \(ch_u (Q{\mathbf {F}}^\bullet (B))\) with \(\delta _n=\sum _{i=0}^n (-1)^i Q(d_i)\).

Definition 3.11

$$\begin{aligned} L_iQ(B;A)=H_i(ch_u (Q{\mathbf {F}}^\bullet (B))) \end{aligned}$$

Dualizing and generalizing results from [9] we have

Theorem 3.12

For any algebras B, D and A

  1. (i)

    \(L_{0}Q(B;A)=QB\)

  2. (ii)

    If B is a free algebra, then \(L_{i}Q(B;A)=0\) for \(i>0\)

  3. (iii)

    \(L_{i}Q(B\otimes _A D;A)\cong L_{i}Q(B;A)\oplus L_{i}Q(D;A)\)

Proof

See [4] \(\square \)

Remark 3.13

If we assume that the algebra B is of finite type and C is a coalgebra of finite type, then \(B^{*}\) is a coalgebra with coproduct \(m^{*}\) and \(C^{*}\) is an algebra with product \(\nabla ^{*}\). We refer the reader to [19] for additional details. Also, by the universality of S and F, we have \(F(B)^*=S(B^*)\) and \(S(C)^*=F(C^*)\). From this it follows that \(F^n(B)^*=S^n(B^*)\) and \(S^n(C)^*=F^n(C^*)\).

Remark 3.14

In [19], the authors define, for an augmented algebra B, the module of indecomposables as \(Q(B)=A\otimes _{B}\overline{B}\) and for an augmented coalgebra C, the module of primitives as \(P(C)=A\square _{C}\overline{C}\). Because our algebras and coalgebras are augmented, it is easy to see that the definitions given by the authors for these modules coincide with the definition in [19]. The interested reader can refer to Proposition 3.2 (2) of [19] and 3.7 (pg 224). From this, \(Q(B)^{*}=P(B^{*})\) follows.

For the relation between these two functors we have the following results whose proofs can be found in [4].

Theorem 3.15

If B is an A-algebra of finite type then \(({\mathbf {F}}^\bullet (B))^*={\mathbf {S}}^\bullet (B^*)\).

The relation between the Primitive element functor and the Indecomposable functor are elucidated in

Lemma 3.16

If B is an A-algebra of finite type which is free as an A-module, then \(Q(F(B))^{*}=P(S(B^{*}))\)

4 Applications of the higher derived functors toward cohomological rigidity

One of the original motivations behind studying the relation between torus actions and the higher derived functors mentioned in the pervious section was to determine if it were possible to use the torus action to augment the cohomology ring of a quasitoric manifold in such a way as to ensure there is a vanishing line in a certain composite functor spectral sequence [5, 6]. In [4] this was referred to as “Bendersky’s” question. From this paper we list additional details and definitions.

More formally, a torus action on the quasitoric manifold \(M(\lambda )\) is nice if for \(i > 1\) \(R^iPE_*(M(\lambda )) = 0 \). By a nice quasitoric manifold we mean a quasitoric manifold endowed with a nice torus action. That is, there is a \(\lambda \) satisfying condition \((*)\) as described in [15] such that the manifold \(M(\lambda )\) is nice. We describe Bendersky’s question more formally: “Is it possible to find a \(\lambda \) satisfying condition \((*)\) of [15] such that \(E_*(M(\lambda ))\) is nice as a homology coalgebra?”. The content of [4] was focused on answering this question (see Corollary 4.1 below).

A simple example was described in [4] and we list it here too to provide insight and a certain level of intuition regarding this critical question. When \(P = \Delta ^1 \times \Delta ^1\), the \(E_*\)-face ring is \(E_*[v_1,v_2,v_3,v_4]/ (v_1v_3, v_2v_4)\). The cohomology ring of the quasitoric manifold \(M(\lambda )\) is the \(E_*\)-face ring modulo the ideal generated by the \(\lambda _i\). In particular, \((\lambda _1, \lambda _2)\) where

$$\begin{aligned} \lambda _1 =&\lambda _{11}v_1 + \lambda _{12}v_2 + \lambda _{13}v_3 + \lambda _{14}v_4 \\ \lambda _2 =&\lambda _{21}v_1 + \lambda _{22}v_2 + \lambda _{23}v_3 + \lambda _{24}v_4 \end{aligned}$$

Following [15] there is a matrix

$$\begin{aligned} \begin{pmatrix} \lambda _{11} &{}\quad \lambda _{12} &{}\quad \lambda _{13} &{}\quad \lambda _{14} \\ \lambda _{21} &{}\quad \lambda _{22} &{}\quad \lambda _{23} &{}\quad \lambda _{24} \end{pmatrix} \end{aligned}$$

This matrix must satisfy condition \((*)\) as described in [15]. This of course, provides ample flexibility in choosing \(\lambda _{ij}\) that might be computationally useful from a spectral sequence perspective. In this motivating example, we can see that the following matrix, coming from certain choices of \(\lambda _{ij}\) produce a nice homology coalgebra in the sense of [9]. That is, a coalgebra dual to a free algebra.

$$\begin{aligned} \begin{pmatrix} 1 &{}\quad 0 &{}\quad -1 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad -1 \end{pmatrix} \end{aligned}$$

This matrix gives rise to \(\lambda _1 = v_1 -v_3\) and \(\lambda _2 = v_2 -v_4\), from which it follows that \(H^*(M) \cong H^*(S^2 \times S^2)\). The latter ring being dual to a nice homology coalgebra. We observe that certain choices of \(\lambda \) facilitate spectral sequence calculations. However, as the next result Corollary shows (see [4]) it is not always possible to find such.

Corollary 4.1

Let P be a fixed simple convex polytope. If for some indices i and j, there exists relations \(r_i\) and \(r_j\) such that \(r_i \cap r_j \ne \emptyset \) in \(E^{*}(B_TP)\), then there does not exist a nice torus action on \(M(\lambda )\).

Summarizing, the orbit limits the types of quasitoric manifolds and the link between the two is provided by the isomorphism

Proposition 4.2

\(L_iQE^*(M(\lambda )) \cong L_iQE^*(B_TP)\) for \(i\ge 1\).

Proof

[4] \(\square \)

In terms of rigidity we recall Definition 3.3 (pg. 15) from [7]. The authors declare a simple polytope P to be C -rigid if either of the conditions hold:

  1. (1)

    there does not exists a Quasitoric manifold M with orbit P or

  2. (2)

    assuming the quasitoric manifolds M and \(M'\) exist with orbits P and \(P'\) with a ring isomorphism \(H^*(M) \cong H^*(M')\), then there is a combinatorial equivalence \(P \simeq P'\).

Corollary 4.1 shows that if there are two relations that have common terms (intersection), then the polytope is C-rigid in the sense that it is not possible to find a quasitoric manifold whose right higher derived functors vanish in degrees greater than one.

There is also the notion of C -rigid properties of polytopes which can be analyzed using the higher derived functors since the isomorphism does not depend on a map of spaces induced by a map of polytopes.

5 Preliminary results

The results in this section are of a general nature; however, the examples of interest are the homology coalgebras that arise from the \(E_*\)-homology of a quasitoric manifold for a suitable complex orientable theory. This line of inquiry was motivated by the desire to generalize the rational isomorphism of the higher derived functors of the primitive element functor [2] to an isomorphism that depends on more general rings. Ultimately, these results will be part of the input needed to analyze certain generalized composite functor spectral sequences.

We begin by recording various categories of interest; from there, a functor from R-modules to R-coalgebras (with some additional structure) is used to construct general cosimplicial resolutions.

5.1 The functor \(\widehat{S}\)

For the convenience of the reader we list the notational conventions stated earlier and those that will be used from this point forward. We assume throughout that R is a commutative ring with unit.

  1. (1)

    \({\mathcal {M}}_{R}\) is the category of free, graded, connected R-modules

  2. (2)

    \({\mathcal {M}}={\mathcal {M}}_{{\mathbb {Z}}}\)

  3. (3)

    \({\mathcal {C}}_{R}\subset {\mathcal {M}}_{R}\) is the subcategory of R-coalgebras

  4. (4)

    \({\mathcal {C}}={\mathcal {C}}_{{\mathbb {Z}}}\)

Let \({\mathcal {M}}^{\prime }_R\subset {\mathcal {M}}_R\) be the subcategory with objects that are R-modules of the form \(M\otimes R\) where \(M\in Obj({\mathcal {M}})\) and morphisms are of the form \(f\otimes 1:M_1\otimes R \rightarrow M_2\otimes R\), where \(f\in Hom_{{\mathcal {M}}}(M_1,M_2)\). Let I be the functor \({\mathcal {C}} \rightarrow \mathcal {C_R}\) defined by \(I(C) = C \otimes R\) (C is a \({\mathbb {Z}}\)-coalgebra). For a map of \({\mathbb {Z}}\)-coalgebras \(f: D \rightarrow E\), \(I(f) = f \otimes 1: D \otimes R \rightarrow E\otimes R\). This will be elucidated below.

Lemma 5.1

If \(C\in Obj({\mathcal {C}})\) then \(C\otimes R \in Obj({\mathcal {C}}_{R})\).

Proof

Let \(\psi _C :C\rightarrow C\otimes _{R} C\) be the coalgebra map for C and \(\psi _{R}: R \rightarrow R\otimes R\) the R-module map defined by \(1\mapsto 1\otimes 1\). Define the R-coalgebra structure on \(C \otimes R\) as follows:

where \(\tau \) is the switching map and the last map is induced by the natural map from \({\mathbb {Z}}\) to R. \(\square \)

Lemma 5.1 allows us to define a functor \(I:{\mathcal {C}}\rightarrow {\mathcal {C}}_{R}\) as follows: \(I(C)=C\otimes R\) and for a map of coalgebras \(f: C_1 \rightarrow C_2\), \(I(f)= f\otimes 1\). Let \(\mathcal {C'}\subset {\mathcal {C}}_R\) be the category such that \(Obj(\mathcal {C'})\) are coalgebras that are the image of the functor I highlighted above. The maps in \({\mathcal {C'}}\) are those that can be written as a tensor product. More specifically, if \(h': C_1' \rightarrow C_2'\) is a morphism in \({\mathcal {C'}}\), then \(h'\) can be written as \(h \otimes 1: C_1 \otimes R \rightarrow C_2 \otimes R\) for \({\mathbb {Z}}\)-coalgebras \(C_1\) and \(C_2\).

Let \(J:{\mathcal {C}}^{\prime }\rightarrow {\mathcal {M}}^{\prime }_{R}\) be the functor that forgets the coalgebra structure. More precisely, \(J(C') =J(C \otimes R) = C \otimes R\) as a free R-module.

The cofree coalgebra functor S gives rise to a canonical resolution [9]. In the definition that follows and the remaining part of the paper, it is an essential ingredient in constructing resolutions with coefficients of interest.

Definition 5.2

Let \(\widehat{S}:{\mathcal {M}}^{\prime }_{R}\rightarrow {\mathcal {C}}^{\prime }\) be given by \(\widehat{S}(M^{\prime })=\widehat{S}(M\otimes R)=S(M)\otimes R\).

For illustrative purposes, we comment on the effect \(\widehat{S}\) has on maps in \({\mathcal {M}}^{\prime }_{R}\). Suppose \(g': D' \rightarrow E'\) is a map in \({\mathcal {M}}^{\prime }_{R}\). By the definition of this category, these R-modules can be written as a tensor product \(g' = g \otimes 1: D \otimes R \rightarrow E \otimes R\) where \(g: D \rightarrow E\) is a map of \({\mathbb {Z}}\)-modules. We have a commutative diagram:

Continuing with the illustration, observe that \(\widehat{S}(g') = \widehat{S}(g \otimes 1)\) giving

In terms of preserving composition, suppose the following composition, \(\phi ' \circ \tau ' \) is defined in \(\mathcal {M'_R}\).

By the remarks above there is a commutative diagram

We wish to verify \(\widehat{S}{(\phi ' \circ \tau ')} = \widehat{S}{(\phi ')} \circ \widehat{S}{(\tau ')}\). There is a map \(\tau : D \rightarrow E\), so there is a \({\mathbb {Z}}\)-coalgebra map \(S(\tau ): S(D) \rightarrow S(E)\). By Lemma 5.1 there is an R-coalgebra map \(S(\tau ) \otimes 1: S(D) \otimes R \rightarrow S(E) \otimes R\). We obtain a commutative diagram:

We conclude that \(\widehat{S}(\tau ) = S(\tau ) \otimes 1\). By the same argument, we have \(\widehat{S}(\phi ) = S(\phi ) \otimes 1\). The composition \(\widehat{S}(\phi ') \circ \widehat{S}(\tau ')\) can be written as \((S(\phi ) \otimes 1) \circ (S(\tau ) \otimes 1)\) which by properties of the tensor product can be written as \([S(\phi ) \circ S(\tau )] \otimes (1 \circ 1) = S(\phi \circ \tau ) \otimes 1\). We conclude by observing that \(S(\phi \circ \tau ) \otimes 1\) can be obtained by applying the previous argument to the map \(\phi \circ \tau : D \rightarrow G\).

Let \(id_{D'}: D' \rightarrow D'\) be the identity map in the category \(\mathcal {M'_R}\). We observe that \(id_{D'}\) is induced from a map \(id_{D} \otimes 1: D \otimes R \rightarrow D \otimes R\) where the map of \({\mathbb {Z}}\)-modules \(id_{D}: D \rightarrow D\) is the identity. Once again, since S is a functor, we observe, in this case, that the coalgebra map \(S(D) \otimes R \rightarrow S(D) \otimes R\) is the identity giving \(\widehat{S}(id_{D'}) = id_{\widehat{S}{(D')}}\). For the sake of completeness we comment on the relation between the decomposition of an object in the category \(\mathcal {M'_R}\) (ie., an R-module \(D'\) that can be written as a tensor \(D \otimes R\) for a \({\mathbb {Z}}\)-module D, with additional structure on the maps) and \(\widehat{S}\).

Suppose, there are two \({\mathbb {Z}}\)-modules, say F and G such that \(F \otimes R \cong D' \cong G \otimes R\). We assert that in such a case \(\widehat{S}(F \otimes R) = \widehat{S}(G \otimes R)\). It will suffice to show that there is a map of \({\mathbb {Z}}\)-modules \(F \rightarrow G\) that is an isomorphism. There exists an isomorphism of R-modules \(g \otimes 1: F \otimes R \rightarrow G \otimes R\) which is induced by a map of \({\mathbb {Z}}\)-modules \(g: F \rightarrow G\) which is also an isomorphism. By relying on the functorality of S, it follows that there is a \({\mathbb {Z}}\)-coalgebra isomorphism \(S(F) \cong S(G)\). Hence, \(\widehat{S}(F \otimes R) = \widehat{S}(G \otimes R)\).

To ensure that a canonically acyclic resolution arises, the following Lemma gives the explicit formulation of an adjunction.

Lemma 5.3

For any \(D' \in Obj({\mathcal {C}}^{\prime })\) and \(M'\in Obj({\mathcal {M}}^{\prime }_{R})\)

$$\begin{aligned} Hom_{{\mathcal {C}}^{\prime }}(D',\widehat{S}(M'))\cong Hom_{{\mathcal {M}}^{\prime }_{R}}(J(D'),M') \end{aligned}$$

Proof

Let \(f^{\prime }:D^{\prime }\rightarrow \widehat{S}(M^{\prime })\). Then there is a coalgebra \(D\in Obj({\mathcal {C}})\) and an \(f \in Hom_{{\mathcal {C}}}(D,S(M))\) such that the following diagram commutes:

Observe that the category \({\mathcal {C}}^{\prime }\) is a subcategory \({\mathcal {M}}^{\prime }_{R}\) and let \(\bar{f}\in Hom_{{\mathcal {M}}^{\prime }_{R}}(J(D^{\prime }),M^{\prime })\) be the composition:

where \(s:S(M)\rightarrow M\) is the natural transformation of S described earlier. Suppose \(f^{\prime }\in Hom_{{\mathcal {M}}^{\prime }_{R}}(J(D^{\prime }),M^{\prime })\). Using the definition of the functor J and the category \({\mathcal {M}}^{\prime }_{R}\), this map can be written as \(f^{\prime }=f\otimes 1:D\otimes R \rightarrow M\otimes R\). Note that \(D\otimes R\) is a coalgebra, but \(f^{\prime }\) is a module map. By the universal property of the cofree coalgebra functor discussed earlier there is a commutative diagram:

(3)

This gives a map \(\bar{f}\otimes 1:D\otimes R \rightarrow S(M)\otimes R = \widehat{S}(M^{\prime })\). To complete the proof we observe that each process is inverse to the other. \(\square \)

If \(C^\prime \in Obj({\mathcal {C}}^{\prime }_{R})\), then we can form the cosimplicial resolution \(\widehat{{\mathbf {S}}}(C^\prime )\) defined in Sect. 3.1.

Lemma 5.4

Let \(C^\prime \in Obj({\mathcal {C'}})\) be an R-coalgebra and C a \({\mathbb {Z}}\)-coalgebra such that \(C^\prime =C\otimes R\). Then

$$\begin{aligned} {\mathbf {S}}(C)\otimes R=\widehat{{\mathbf {S}}}(C^\prime ) \end{aligned}$$

Proof

We have

Applying \(-\otimes R\) to this resolution gives:

(4)

The resolution in the bottom row is \(\widehat{{\mathbf {S}}}(C)\) . \(\square \)

Theorem 3.2 yields the following:

Lemma 5.5

If \(C^\prime \in Obj({\mathcal {C}}^{\prime })\) then

$$\begin{aligned} R^{i}P(C^\prime ;R)\cong H^{i}(P^{+}ch_{u}\widehat{\mathbf{S }}(C^\prime )) \end{aligned}$$

where \(\widehat{\mathbf{S }}_{u}\) is the un-augmented resolution obtained from resolution (4).

Theorem 5.6

For any \(C\in Obj({\mathcal {C}})\) there is a natural injective map of R-modules

$$\begin{aligned} 0\rightarrow P(C)\otimes R \overset{\iota _C}{\longrightarrow } P(C\otimes R) \end{aligned}$$

Proof

Since C is connected it is easy to see that \(\overline{C\otimes R}\cong \overline{C}\otimes R\). Consider the following diagram:

where \(\iota : P(C) \rightarrow C\) is the inclusion. This gives \((\iota \otimes 1)(x \otimes 1) = x \otimes 1\) and this element maps to zero when we apply \(\overline{\psi _C} \otimes \psi _{R}\). This implies that it pulls back to \(P(C \otimes R)\). By the universal property of the kernel, there exists a unique map of R-modules \(\iota _C:P(C)\otimes R \rightarrow P(C \otimes R)\). Finally, the injectivity follows from the commutativity of the diagram. \(\square \)

Remark 5.7

In general, the map \(\iota _C\) in Theorem 5.6 is not an isomorphism. For example, let \(C={\mathbb {Z}}(x_2,x_4,\ldots )\) with \(\psi (x_{2n})=\sum _{i+j=n}{i+j \atopwithdelims ()i}x_{2i}\otimes x_{2j}\). It can be shown that \(P(C)\otimes {\mathbb {Z}}_q={\mathbb {Z}}_q(x_2)\), but \(P(C\otimes {\mathbb {Z}}_q)={\mathbb {Z}}_q(x_2,x_{2q},x_{4q},\ldots )\).

Lemma 5.8

If C is a cofree \({\mathbb {Z}}\)-coalgebra then \(\iota _C\) is an isomorphism.

Proof

Start with the term \(P(C \otimes R)\). Since the underlying algebra/coalgebra are projective R-modules, the functor P is dual to the indecomposable functor Q. Therefore, we obtain the module \(Q(F\otimes R)\) where F is the dual free \({\mathbb {Z}}\)-algebra generated by the free \({\mathbb {Z}}\)-module \(M={\mathbb {Z}}\{J\}\) for some set J. This implies that \(Q(F\otimes R) = R\{J\} = {\mathbb {Z}}\{J\}\otimes R\). The term \(P(C)\otimes R\) is dual to \(Q(F)\otimes R\) where \(Q(F) = {\mathbb {Z}}\{J\}\) giving the desired result. \(\square \)

Notation 5.9

If C is a coalgebra over R, let \(R^{i}P(C;R)\) denote the higher right derived functors of the primitive element functor over R. If \(R={\mathbb {Z}}\) then, \(R^{i}P(C;{\mathbb {Z}})=R^{i}P(C)\).

Theorem 5.10

Let R be a commutative ring with unit. Suppose \(C_{1}\) and \(C_{2}\) are \({\mathbb {Z}}\)-coalgebras. If \(R^{i}P(C_{1})\cong R^{i}P(C_{2})\) for \(i>k\), then there is a non-canonical isomorphism

$$\begin{aligned} R^{i}P(C_{1}\otimes R;R)\cong R^{i}P(C_{2}\otimes R;R) \end{aligned}$$

for \(i>k\).

Proof

Applying the primitive element functor P to the complex \(\widehat{{\mathbf {S}}}(C_{j})\) for \(j=1,2\) gives the following:

$$\begin{aligned} P(\mathbf {\widehat{S}}(C_{j}\otimes R))&=P({\mathbf {S}}(C_{j})\otimes R)\qquad \text {by Lemma } 4.4\\&=P({\mathbf {S}}(C_{j}))\otimes R \qquad \text {by Lemma }4.8 \end{aligned}$$

We obtain the following isomorphisms:

$$\begin{aligned} R^iP(C_{j} \otimes R;R)&=H^i(ch_{u}P^{+}(\mathbf {\widehat{S}}(C_{j}\otimes R)))\qquad \text {by Lemma }4.5\\&= H^{i}(ch_{u}P^{+}(\mathbf S (C_{j}))\otimes R))\qquad \text {by the previous paragraph}\\&\cong H^{i}(ch_{u}P^{+}(\mathbf S (C_{j})))\otimes R \\&\quad \oplus Tor(H^{i+1}(ch_{u}P^{+}(\mathbf S (C_{j})), R) \ {\mathrm{Universal~Coef.~Thm}}\\&\cong R^{i}P(C_{j})\otimes R \oplus Tor(R^{i+1}P(C_{j}), R) \end{aligned}$$

Since \(R^{i}P(C_{1})\cong R^{i}P(C_{2})\) for \(i>k\), we obtain:

$$\begin{aligned} R^iP(C_{1}\otimes R;R)&\cong R^{i}P(C_{1})\otimes R \oplus Tor(R^{i+1}P(C_{1}), R)\\&\cong R^{i}P(C_{2})\otimes R \oplus Tor(R^{i+1}P(C_{2}), R)\\&\cong R^iP(C_{2}\otimes R;R) \end{aligned}$$

\(\square \)

6 ESP sequences

In this section it is shown that ESP-sequences in the rational Face ring can be pulled back to ESP-sequences in the integral Face ring and subrings of the rationals. In addition, theorems from [16, 21] are generalized from a field k to free R-modules. As an application, the main theorem (Theorem 4.5 in [2]) concerning the isomorphism of \(R^iP(-)\) of quasitoric manifolds is extended from \({\mathbb {Q}}\) to modules over subrings of the rationals and ultimately to complex orientable theories. The precise relationship between the ESP sequence coming from the \(\lambda \)-map (i.e., \({\mathbb {Z}}^m \rightarrow {\mathbb {Z}}^n\) when there is a quasitoric manifold) and more general rings is exposed. Recall, Theorem 5.10 depended on the coalgebras being free modules over a ring R and the resulting isomorphism relied on the Universal Coefficient Theorem; in this section the ESP sequence plays a more central role.

Definition 6.1

Let A be an algebra over R. A sequence \(a_{1},\ldots , a_{n}\in A\) is an ESP-sequence if \(a_{j}\) is not a zero divisor in \(A/(a_{1},\ldots , a_{j-1})\) for \(j\le n\).

Let H be a subring of \({\mathbb {Q}}\) and suppose A is an H-algebra which is free as an H-module. Then \(\widetilde{A}=A\otimes {\mathbb {Q}}\) is an H-module and there is a map \(\iota :A\rightarrow \widetilde{A}\) induced by the inclusion \(H \rightarrow {\mathbb {Q}}\).

Lemma 6.2

Let \(x_{1},\ldots ,x_{n}\in A\) be a non-negatively graded algebra such that \(A/(x_{1},\ldots ,x_{n})\) is a free H-module. Suppose the image of \(x_{1},\ldots ,x_{n}\) forms an ESP-sequence in \(\widetilde{A}\). Then \(x_{1},\ldots ,x_{n}\) is an ESP-sequence in A.

Proof

By abuse of notation we write \(x_{1},\ldots ,x_{n}\) for the image of \(x_{1},\ldots ,x_{n}\) in \(\widetilde{A}\). Let \(A_{i}=A/(x_{1},\ldots ,x_{i})\) and \(\widetilde{A}_{i}=\widetilde{A}/(x_{1},\ldots ,x_{i})\). First we prove that the map \(\iota _i:A_{i}\rightarrow \widetilde{A}_{i}\) is injective for \(1\le i\le n\) as an H-module.

Suppose that \(A_{n-1}\) has p-torsion. Then there exist \(a_{0}\in A_{n-1}\) such that \(pa_{0}=0\) in \(A_{n-1}\). Since \(A_{n}\) is a free H-module then \(a_{0}\) is zero in \(A_{n}\), there are \(a_{11},\ldots ,a_{1n} \in A\) such that \(a_{0}=a_{11}x_{1}+\cdots +a_{1n}x_{n}\in A\). Hence, \(a_{0}=a_{1n}x_{n}\) in \(A_{n-1}\). Let \(a_{1}=a_{1n}\). Since \(pa_{1}x_{n}=0\) and \(x_{n}\) is part of an ESP-sequence then \(a_{1}=0\) in \(A_{n}\). There exists \(a_{21},\ldots ,a_{2n}\) such that \(a_{1}=a_{21}x_{1}+\cdots +a_{2n}x_{n}\). So \(a_{1}=a_{2n}x_{n}\in A_{n-1}\). Following this procedure let \(a_{j} = a_{jn}\) and each product \(a_{j-1} = a_{jn}x_n\) has the property that \(|a_{jn}| < |a_{j-1}|\). We obtain a sequence of elements \(a_{0},a_{1},\ldots \) such that \(|a_{i}|>|a_{i+1}|>0\) for all \(i\ge 0\). Since A is non-negatively graded, this sequence has to terminate for some positive integer k, implying \(pa_{k}x_{n} \ne 0\) in \(A_{n-1}\). Suppose this were not the case, then the construction could continue until the elements \(a_k\) (i.e., \(a_{kn}\)) were contained in negatively graded modules contradicting the gradation on A. Therefore, \(A_{n-1}\) has no torsion. By the same argument we can deduce that \(A_{i}\) has no torsion for all \(0\le i\le n\). Then there is a short exact sequence:

$$\begin{aligned} 0\rightarrow A_{i}^{\prime } \otimes {\mathbb {Z}}_{p}\rightarrow A_{i}\otimes {\mathbb {Z}}_{p} \rightarrow A_{n}\otimes {\mathbb {Z}}_{p}\rightarrow 0 \end{aligned}$$

Assume further that \(a\in A_i\) such that \(\iota _i(a)=0\), then \(a=a_1x_1+\cdots +a_ix_i\) where \(a_1,\ldots ,a_i\in {\mathbb {Q}}\). There is an integer m such that \(ma_1,\ldots ,ma_i\in H\). Then ma is in the image of \(\iota \) and is zero in \(A_i\). Since \(A_i\) is torsion free, one concludes that \( a = 0\). The statement concerning the ESP-sequence follows from the following commutative diagram:

\(\square \)

Assume that A is a graded, commutative, connected algebra over the ring R and suppose that M is an A-module. Following the treatment in [13], there is a short exact sequence \(0\rightarrow I \rightarrow A\rightarrow R\rightarrow 0\). Furthermore, there is an isomorphism of R-modules: \(M\otimes _A R\cong M/IM\). If \(N \subseteq M\), then let \(F_N\) be the free A-module generated by N and \(R_N=ker (F_N\rightarrow A)\), \(L_N=Coker (F_N\rightarrow A)\). There is a short exact sequence

$$\begin{aligned} 0\rightarrow R_N\rightarrow F_N\rightarrow A\rightarrow L_N\rightarrow 0 \end{aligned}$$

Definition 6.3

Suppose that M is an A-module. Then

  1. (1)

    M is proper if \(M\otimes _A R\ne 0\)

  2. (2)

    A set \(N\subseteq M\) is faithful if for all \(N^\prime \subseteq N\) the A-modules \(R_{N^\prime }\) and \(L_{N^\prime }\) are proper.

Remark 6.4

By proposition 5.1 of §VIII of [13], if A is a graded ring then every graded A-module is proper and every set of homogeneous elements is faithful.

Theorem 6.5

Suppose that M is an A-module such that \(M\otimes _{A} R\) is a free R-module. If \(Tor^{1}_{A}(M,R)=0\), then every faithful subset of M that generates M over A contains an A-base for M.

Proof

We follow the proof of Theorem 5.3 [13] modifying when necessary. Suppose N is a faithful set that generates M. Then the image of the elements of N generates \(M\otimes _A R\) as an R-module. Since \(M\otimes _{A} R\) is a free R-module there is a \(N^\prime \subseteq N\) such that the image of the elements of \(N^{\prime }\) in \(M\otimes _A R\) is a base over R. Then \(N^{\prime }\) is faithful and by Theorem 5.2, §VIII [13], \(N^{\prime }\) is a base for M over A. \(\square \)

Corollary 6.6

Let M be a graded A-module such that \(M\otimes _{A} R\) is a free R-module. If \(Tor^1_A(M,R)=0\), then M is a free A-module.

Proof

This follows from Remark 6.4 and Theorem 6.5. \(\square \)

We now recall the construction of the Koszul complex of a graded commutative R-algebra A. Let \(x_1,\ldots ,x_n\in A\) and \(\Lambda (u_1,\ldots ,u_n)\) be the exterior algebra generated by \(x_1,\ldots ,x_n\). We define a complex \({\mathcal {E}}(x_{1},\ldots ,x_{n})=A\otimes \Lambda (u_1,\ldots ,u_n)\) with \(d:{\mathcal {E}}(x_{1},\ldots ,x_{n})\rightarrow {\mathcal {E}}(x_{1},\ldots ,x_{n})\) defined by \(d(a\otimes u_i)=ax_i\otimes 1 \) and \(d(a)=0\) for \(a \in A\) and requiring that d be a map of algebras. Then \(d\circ d=0\) and if we let \(bideg(a)=(0,deg(a))\) and \(bideg(u_i)=(-1,deg(u_i))\) for \(1\le i\le n\) we have a bigraded structure on \({\mathcal {E}}(x_1,\ldots ,x_n)\), then \(H^0({\mathcal {E}}(x_{1},\ldots , x_{n}))=A/(x_1,\ldots ,x_n)\).

Theorem 6.7

If A is free as an R-module and \(x_1,\ldots x_n\in A\) is an ESP-sequence, then \({\mathcal {E}}(x_{1},\ldots , x_{n})\) is acyclic.

Proof

Since A is free as an R-module, the proof follows by mimicking the proof of Proposition 7.1 in [16]. \(\square \)

We now generalize Corollary 1.2 in [21].

Corollary 6.8

Suppose \(B=R[z_1,\ldots ,z_n]\) and M is a B-module that is free as an R-module. Then

$$\begin{aligned} Tor_{B}(M,R)=H(M\otimes _{B} {\mathcal {E}}(z_{1},\ldots ,z_{n}))\cong H(M\otimes \Lambda (u_{1},\ldots ,u_{n})) \end{aligned}$$

with \(d(a\otimes u_i)=az_i\otimes 1\).

Proof

Since \(z_{1},\ldots ,z_{n}\) is an ESP-sequence and is R-free, then by Theorem 6.7 the result follows. \(\square \)

Let \(x_{1},\ldots , x_{n}\) be a set of elements (not necessarily an ESP sequence) of A and \(R[z_{1},\ldots , z_{n}]\rightarrow A\) be defined by \(z_{i}\mapsto x_{i}\) with \(|z_{i}|=|x_{i}|\). Using this map we have an \(R[z_{1},\ldots , z_{n}]\)-module structure on A. With this notation we have the following

Corollary 6.9

If \(x_{1},\ldots , x_{n}\) is an ESP-sequence in A with M and \(M \otimes _{R[z_{1},\ldots , z_{n}]} R\) free R-modules, then M is a free \(R[z_{1},\ldots , z_{n}]\)-module.

Proof

Corollary 6.8 and Theorem 6.7 together imply \(Tor^{- 1,*}_{R[z_{1},\ldots , z_{n}]}(M,R)=0\). The result follows by final appeal to Corollary 6.6. \(\square \)

7 Applications to quasitoric manifolds

Recall the following short exact sequence of graded rings from [15]:

(5)

Lemma 7.1

\(H^{*}(M)\) and \(H^{*}(B_{T}P)\) are free \({\mathbb {Z}}\)-modules.

Proof

Suppose there is an \(x\in H^{2k}(M)\) such that \(px=0\) for some prime p. Then by the Universal Coefficient Theorem \(H_{2n-1}(M)\) has torsion. However, by [15] ([Theorem 3.1]), \(H_{2k+1}(M)\) is trivial for all \(k\ge 0\). Therefore, \(x=0\).

We also observe that there is a basis for \(H^*(B_TP)\) as a \({\mathbb {Z}}\)-module which consists of all those products of \(v_i\) that are not divisible by any of the relations in the ideal. More explicitly, for a multi-index, \(I = \{\iota _1,\ldots ,\iota _k\}\), let \(v^I = v_{\iota _1} \cdot \cdot \cdot v_{\iota _k}\). If \(r_j\) is a relation in \(H^*(B_TP)\), then we say that I is allowable if the image of the relation \(r_j\) in the exact sequence \({\mathbb {Z}}[r_1,\ldots ,r_k] \rightarrow {\mathbb {Z}}[v_1,\ldots ,v_m]\rightarrow H^*(B_TP)\) does not divide \(v^{I}\in H^*(B_TP)\) [1]. There is a basis for \(H^*(B_TP)\) as a \({\mathbb {Z}}\)-module: \(\{v^I |\ \text {I allowable} \}\). It follows that \(H^*(B_TP)\) is also a free \({\mathbb {Z}}\)-module. Finally, for two allowable multi-indices, I and \(I'\), the algebra structure over \({\mathbb {Z}}\) comes from \(v^I \otimes v^{I'} \mapsto v^Iv^{I'}\). \(\square \)

Let \(t_1,..,t_n\) be the generators of \(H^*(BT^n)\) and \(p^*(t_i) = \lambda _i\). Consider the following commutative diagram:

For \(\lambda _i \in H^*(B_TP)\) let \(\widetilde{\lambda }_i = \iota (\lambda _i)\). That is, \(\widetilde{\lambda }_i\) is an element in the rational Face ring. It is clear from the diagram that \(\widetilde{\lambda }_1,\ldots , \widetilde{\lambda }_n\) are elements of \(Ker (j^*_{{\mathbb {Q}}})\) and form a rational ESP sequence. By Lemma 6.2 they pullback to an ESP-sequence \(\lambda _1,\ldots ,\lambda _n\in H^{*}(B_{T}P)\).

Lemma 7.2

The sequence (5) is a projective extension sequence.

Proof

The result will follow from Corollary 6.9 once we prove that the \({\mathbb {Z}}\)-module \(H^{*}(B_{T}P)\otimes _{{\mathbb {Z}}[t_{1},\ldots , t_{n}]}{\mathbb {Z}}\) is free. Observe that \(H^{*}(BT^{n})\cong {\mathbb {Z}}[t_{1},\ldots , t_{n}]\) with \(t_{i}\mapsto \lambda _{i}\) and \(\lambda _{1},\ldots , \lambda _{n}\) form an ESP-sequence in \(H^*(B_TP)\). Using the tensor product of modules over an algebra and the universal property of the cokernel, it follows that

$$\begin{aligned} H^{*}(B_{T}P)\otimes _{{\mathbb {Z}}[t_{1},\ldots , t_{n}]}{\mathbb {Z}}\cong H^{*}(M) \end{aligned}$$

and we obtain the result by virtue of Lemma 7.1. \(\square \)

We remark that \(H^*(M)\) is free by [15] and the proof of Lemma 7.2 is an alternative using different methods.

Theorem 7.3

Let E be a complex orientable theory with coefficients concentrated in even degrees. Suppose \(M_1\) and \(M_2\) are two quasitoric manifolds with orbit a simple convex polytope P, then \(R^kP(E_*(M_1);E_{*}) \cong R^kP(E_*(M_2);E_{*})\) for \(k > 1\).

Proof

By Lemma 7.2 sequence (5) is a projective extension sequence. Since all of the algebras are of finite type, then the sequence dual to (5) is an injective extension sequence. Following the proof of Theorem 4.5 of [1] we have an isomorphism \(R^kP(H_*(M_1)) \cong R^kP(H_*(M_2))\) for \(k>1\). Since \(H_*(M_{i})\) is a free \({\mathbb {Z}}\)-module generated by even dimensional elements, the Atiyah–Hirzebruch spectral sequence collapses and gives \(E_*(M_i)\cong H_*(M_i;E_*)\cong H_*(M_i)\otimes E_*\) where \(i=1,2\). From this and Theorem 5.10 we obtain \(R^kP(E_*(M_1);E_{*}) \cong R^kP(H_*(M_1)\otimes E_{*}) \cong R^kP(E_*(M_2)\otimes E_{*};E_{*}) \cong R^kP(E_*(M_1);E_{*})\) for \(k>1\). \(\square \)

Theorem 7.3 does not depend on a map of spaces. That is, the assumption is that there is an abstract isomorphism of higher derived functors of the primitive element functor for i large enough, not necessarily induced from a map of quasitoric manifolds.

Suppose the simple convex polytopes P and Q are the orbits of the quasitoric manifolds \(M_P\) and \(M_Q\). The dual complex to P, \(K_P\), is rigid if \(H^*(M_P) \cong H^*(M_Q)\) implies \(K_P\) is isomorphic to \(K_Q\).

Corollary 7.4

Let P be a simple convex polytope. If the dual complex \(K_P\) is rigid then for \(i >1\), \(R^iPE_*(B_TP) \cong R^iPE_*(B_TQ)\) for any Q such that \(H^*(M_P) \cong H^*(M_Q)\).

Proof

The result follows by using the argument in the proof of Theorem 7.3. \(\square \)

Corollary 7.4 gives a necessary condition for a complex to be rigid and provides another perspective to analyze C-rigid properties as described in [7]. It also frames the rigidity question in the simplicial setting. An interesting application geared toward torus actions is discussed in [4].

There has been research in determining to what extent an isomorphism of cohomology rings of two quasitoric manifolds determine the homeomorphism and the diffeomorphism type of the manifolds [18]. It is well established that an isomorphism of the equivariant cohomology rings of quasitoric manifolds induces an isomorphism of the underlying simplicial complexes and Corollary 7.4 brings to bear other methods that can be used to study the problem.