Varieties and local cohomology for chromatic group cohomology rings.

*(English)*Zbl 0959.55005This important paper lies at the confluence of two major tributaries to the main stream of algebraic topology.

The central importance of complex oriented multiplicative cohomology theories, in particular the \(2\)-periodic ones, has been evident for almost three decades. Their systematic study and applications have made significant bodies of algebra, number theory and algebraic geometry seemingly an inescapable part of the standard topologist’s toolkits, particularly in the structural theory of the stable homotopy category. This may have something to do with the non-geometric nature of many of the central examples such Morava \(K\)-theories (only rational cohomology, \(K\)-theory and perhaps elliptic cohomology are known to have truly geometric constructions). However, even \(K\)-theory leads via its stable operations to many subtle number theoretic phenomena which are manifestations of this kind of structure.

Equivariant topology (for finite and compact Lie groups) is now a well developed domain which casts its shadows into non-equivariant stable homotopy theory (see for example the relationships between Tate cohomology and periodicity).

In this paper an interesting topic is considered which naturally uses ideas from both of these areas, namely periodic cohomology of classifying spaces of finite groups.

It is worth remarking that the influential work of M. J. Hopkins, N. J. Kuhn, and D. C. Ravenel [Generalized group characters and complex oriented cohomology theories, J. Am. Math. Soc. 13, No. 3, 553-594 (2000)] has already considered similar questions about the \(E\)-cohomology of the classifying space a finite group, \(E^*(BG)\), but their results tend to be most useful when \(E^*\) contains an inverse for the order of \(G\), \(|G|\). The present paper addresses the more delicate issues occurring when for example \(E^*\) is \(p\)-local for a prime \(p\) dividing \(|G|\). Thus these two bodies of work are in a certain sense complementary although there is also some overlap.

In this paper, the main approach to studying \(E^*(BG)\) (where \(E^*(\cdot)\) is a \(2\)-periodic complex oriented multiplicative cohomology theory whose formal group has height \(n\) after reduction modulo the maximal ideal which contains the prime \(p\)) makes use of the algebraic geometry of \(E^*(BG)\) or rather the associated scheme represented by it. This brings in technology developed by the second author [N. P. Strickland, Formal schemes and formal groups, Homotopy invariant algebraic structures, Contemp. Math. 239, 263-352 (1999)] which is a detailed working out of algebraic geometric ideas suggested by Morava and later workers but until now poorly or incompletely documented by topologists. It also follows that of Quillen in his work on the ordinary cohomology of \(BG\) [D. G. Quillen, Ann. Math. (2) 94, 549-572, 573-602 (1971; Zbl 0247.57013)], and reduces to the study of the abelian \(p\)-subgroups of \(G\). More generally, Quillen’s work generalizes to the Borel cohomology \(E^*(EG\times_G Z)\) of a finite \(G\)-space \(Z\). The most important aspect of the analysis given is the chromatic filtration of the formal spectrum \(X(G)=\text{spf}(E^0(BG))\) in the case where \(E^0\) is a complete local ring; this is exactly parallel to the well-known chromatic filtration in stable homotopy theory and has its origins in similar algebra. To mention an example of what is proved, the authors show that \(E^0(BG)/I_k\) and \((E/I_k)^0(BG)\) determine the same varieties, where \(I_k\triangleleft E^0\) is the ideal generated by the first \(p^k\) coefficients of the formal group associated to the given complex orientation of \(E^*(\cdot)\). Using the idea of a level structure from [N. P. Strickland, J. Pure Appl. Algebra 121, No. 2, 161-208 (1997; Zbl 0916.14025)] a decomposition of the \(k\)-th chromatic stratum \(X_k(G)\) of \(X(G)\) into irreducible components is given; this shows that the minimal primes of \(E^0(BG)/I_k\) correspond bijectively to the abelian \(p\)-subgroups of \(G\) with rank at most \(n-k\).

One major application of this is to the study of \(E_*(BG)\) which is accomplished using the first author’s local cohomology machinery. A number of other applications that flow easily from this approach are described in the appendices.

To sum up, this paper provides an excellent illustration of the power of the algebraic geometric techniques being developed by topologists working in stable homotopy theory and equivariant topology, and should convince others of the need to learn more about them.

The central importance of complex oriented multiplicative cohomology theories, in particular the \(2\)-periodic ones, has been evident for almost three decades. Their systematic study and applications have made significant bodies of algebra, number theory and algebraic geometry seemingly an inescapable part of the standard topologist’s toolkits, particularly in the structural theory of the stable homotopy category. This may have something to do with the non-geometric nature of many of the central examples such Morava \(K\)-theories (only rational cohomology, \(K\)-theory and perhaps elliptic cohomology are known to have truly geometric constructions). However, even \(K\)-theory leads via its stable operations to many subtle number theoretic phenomena which are manifestations of this kind of structure.

Equivariant topology (for finite and compact Lie groups) is now a well developed domain which casts its shadows into non-equivariant stable homotopy theory (see for example the relationships between Tate cohomology and periodicity).

In this paper an interesting topic is considered which naturally uses ideas from both of these areas, namely periodic cohomology of classifying spaces of finite groups.

It is worth remarking that the influential work of M. J. Hopkins, N. J. Kuhn, and D. C. Ravenel [Generalized group characters and complex oriented cohomology theories, J. Am. Math. Soc. 13, No. 3, 553-594 (2000)] has already considered similar questions about the \(E\)-cohomology of the classifying space a finite group, \(E^*(BG)\), but their results tend to be most useful when \(E^*\) contains an inverse for the order of \(G\), \(|G|\). The present paper addresses the more delicate issues occurring when for example \(E^*\) is \(p\)-local for a prime \(p\) dividing \(|G|\). Thus these two bodies of work are in a certain sense complementary although there is also some overlap.

In this paper, the main approach to studying \(E^*(BG)\) (where \(E^*(\cdot)\) is a \(2\)-periodic complex oriented multiplicative cohomology theory whose formal group has height \(n\) after reduction modulo the maximal ideal which contains the prime \(p\)) makes use of the algebraic geometry of \(E^*(BG)\) or rather the associated scheme represented by it. This brings in technology developed by the second author [N. P. Strickland, Formal schemes and formal groups, Homotopy invariant algebraic structures, Contemp. Math. 239, 263-352 (1999)] which is a detailed working out of algebraic geometric ideas suggested by Morava and later workers but until now poorly or incompletely documented by topologists. It also follows that of Quillen in his work on the ordinary cohomology of \(BG\) [D. G. Quillen, Ann. Math. (2) 94, 549-572, 573-602 (1971; Zbl 0247.57013)], and reduces to the study of the abelian \(p\)-subgroups of \(G\). More generally, Quillen’s work generalizes to the Borel cohomology \(E^*(EG\times_G Z)\) of a finite \(G\)-space \(Z\). The most important aspect of the analysis given is the chromatic filtration of the formal spectrum \(X(G)=\text{spf}(E^0(BG))\) in the case where \(E^0\) is a complete local ring; this is exactly parallel to the well-known chromatic filtration in stable homotopy theory and has its origins in similar algebra. To mention an example of what is proved, the authors show that \(E^0(BG)/I_k\) and \((E/I_k)^0(BG)\) determine the same varieties, where \(I_k\triangleleft E^0\) is the ideal generated by the first \(p^k\) coefficients of the formal group associated to the given complex orientation of \(E^*(\cdot)\). Using the idea of a level structure from [N. P. Strickland, J. Pure Appl. Algebra 121, No. 2, 161-208 (1997; Zbl 0916.14025)] a decomposition of the \(k\)-th chromatic stratum \(X_k(G)\) of \(X(G)\) into irreducible components is given; this shows that the minimal primes of \(E^0(BG)/I_k\) correspond bijectively to the abelian \(p\)-subgroups of \(G\) with rank at most \(n-k\).

One major application of this is to the study of \(E_*(BG)\) which is accomplished using the first author’s local cohomology machinery. A number of other applications that flow easily from this approach are described in the appendices.

To sum up, this paper provides an excellent illustration of the power of the algebraic geometric techniques being developed by topologists working in stable homotopy theory and equivariant topology, and should convince others of the need to learn more about them.

Reviewer: A.J.Baker (Glasgow)