Numerical invariants for equivariant cohomology (Q2851197)

Let \(G\) be a finite or compact Lie group, \(p\) be a prime such that \(G\) has at least one element of order \(p\) and \(X\) be a \(G\)-space. Suppose that the Borel equivariant cohomology ring \(H_G^*(X)=H^*(EG\times_GX, k)\) is finitely generated as a module over \(H^*_G=H^*(BG, k)\). In this paper the author proposes an approach for studying primary decompositions of the zero submodule of \(H_G^*(X)\).NEWLINENEWLINELet \(\mathcal{A}(G, X)\) denote a category whose objects are pairs \((A, c)\) where \(A\) is an elementary abelian \(p\)-subgroup of \(G\) and \(c\) is a component of \(X^A\), and whose morphisms \((A, c) \to (B, d)\) are defined by giving \(g \in G\) such that \(gAg^{-1} \subseteq B\) and \(d \subseteq gc\). Let \(C_G(A, c)\) be the subgroup of the centralizer \(C_G(A)\) of \(A\) in \(G\) consisting of elements \(g\) such that \(gc=c\). Then the multiplication map \(m : C_G(A, c) \times A \to G\) induces a homomorphism \(m^*=m^*_{(A, c)}(G, X) : H^*_G(X) \to H^*_{C_G(A, c)}(c)\otimes H^*_A\). The cases of \(p=2\) or \(p\) odd are discussed separately.NEWLINENEWLINEIf \(p\) is odd, then \(H^*_A\) is the tensor product of a polynomial ring over \(k\) in variables \(t_1, \dots, t_a\) of degree 2 and an exterior algebra over \(k\) in variables \(e_1, \dots, e_a\) of degree 1 where \(a=\text{rank} \, A\). So for every homogeneous element \(x \in H^*_G(X)\) we can write \(m^*(x)\) as an expression in \(t_1, \dots, t_a\); \(e_1, \dots, e_a\) with coefficients in \(H^*_{C_G(A, c)}(c)\). We define \(j^{\beta, \alpha}_{(G, X), (A, c)}\) to be the submodule of \(H^\alpha_G(X)\) generated by \(x\) such that \(m_j(x)=0\) for \(j > \alpha-\beta\) where \(m_j(x)\) denotes the \(j\)th term of \(m^*(x)\), and set \(j^{(u)}_{(G, X), (A, c)}=\bigoplus^\infty_{\alpha=0} \;j^{u, \alpha}_{(G, X), (A, c)}\) if \(A\neq 1\) (the case when \(A=1\) omitted). The map \(m\) above also induces a ring homomorphism \(m^*_{\mathcal{A}(G, X), n}\) of \(H^*_G(X)\) into \(\Pi_{(A, c)\in \mathcal{A}(G, X)} \;H^{<n}_{C_G(A, c)}(c)\otimes H^*_A\). The number \(n_0(G, X)\) is then defined as the least \(n\) such that \(\text{ker} \;m^*_{\mathcal{A}(G, X), n}=0\).NEWLINENEWLINEDenote by \({\mathfrak p}_{(G, X), (A, c)}\) the kernel of the composite of the canonical ring homomorphisms \(H^{\text{ev}}_G(X) \to H^{\text{ev}}_A(c) \to H^{\text{ev}}_A \to H^{\text{ev}}_A/\sqrt{0}\). Then the main theorem states that if \((B_1, b_1), \dots , (B_t, b_t)\) is a set of representatives for the isomorphism classes in \(\mathcal{A}(G, X)\) of those \((B, b)\) for which \({\mathfrak p}_{(G, X), (B, b)} \in \text{Ass}_{H^{\text{ev}}_G(X)}H^*_G(X)\), then \(n_0(G, X) < \infty\) and \(0=\bigcap^t_{i=1} \;j^{(u)}_{(G, X), (B_i, b_i)}\) is a reduced primary decomposition of \(0\) in the \(H^{\text{ev}}_G(X)\)-module \(H^*_G(X)\).NEWLINENEWLINEIn the last section the theory developed here is illustrated again via an example.