## Pure Injectives And The Spectrum Of The Cohomology Ring Of A Finite Group (1999)

Venue: | J. reine angew. Math |

Citations: | 11 - 3 self |

### BibTeX

@ARTICLE{Benson99pureinjectives,

author = {David Benson and Henning Krause},

title = {Pure Injectives And The Spectrum Of The Cohomology Ring Of A Finite Group},

journal = {J. reine angew. Math},

year = {1999},

volume = {542},

pages = {23--51}

}

### OpenURL

### Abstract

This paper grew out of an attempt to understand this phenomenon. The purpose of this paper is to investigate a certain functor T from injective modules I over the cohomology ring H

### Citations

486 |
Homological algebra
- Cartan, Eilenberg
- 1956
(Show Context)
Citation Context ...e cohomology ring ^ H (A; k). The elements of positive degree in H (A; k) form a maximal ideal m = H + (A; k) and we have H (A; k)=m = k. By Tate duality (see for example Cartan and Eilenberg [10], section XII.6), the negative cohomology ^ H (A; k) is the k-dual of H (A; k), and is hence isomorphic to the injective 4 DAVID BENSON AND HENNING KRAUSE hull of k as an H (A; k)-module. The nega... |

200 |
Des Catégories Abéliennes
- Gabriel
- 1962
(Show Context)
Citation Context ...egory of additive contravariant functors mod(A) ! Ab into the category of abelian groups. It is an abelian Grothendieck category which as far as we are concerned means that it has injective envelopes =-=-=-[14]. Dene a functor T 0 : Mod(H (A; k)) ! (mod(A) op ; Ab) PURE INJECTIVES AND THE SPECTRUM OF THE COHOMOLOGY RING OF A FINITE GROUP 15 via T 0 (M) = Hom H (A;k) ( ^ H (A; ); M)j mod(A) : One shou... |

171 |
Cohen–Macaulay rings. Cambridge
- Bruns, Herzog
- 1993
(Show Context)
Citation Context ...ES AND THE SPECTRUM OF THE COHOMOLOGY RING OF A FINITE GROUP 5 The classication of injective modules over a graded commutative Noetherian ring 1 R is well known, and can be found in Bruns and Herzog [=-=9]-=-. An arbitrary direct sum of injective modules is injective, and every injective module decomposes essentially uniquely as a direct sum of injective indecomposables. Given a homogeneous prime ideal p,... |

112 |
Axiomatic stable homotopy theory
- Hovey, Palmieri, et al.
- 1997
(Show Context)
Citation Context ...the sense of Benson, Carlson and Rickard [4] of the module T (I) for every indecomposable injective module I. This is essentially a question of translating a proof from Hovey, Palmieri and Strickland =-=[18-=-] into the language of group representation theory. In Section 8, we investigate T as a map from the projective variety Proj(H (A; k)) to the Ziegler spectrum Ind(A) of indecomposable pure injectives... |

109 |
The Grothendieck duality theorem via Bousfield’s techniques and Brown representability
- Neeman
- 1996
(Show Context)
Citation Context ...to Hom H (A;k) ( ^ H (A; M); I) takes triangles to exact sequences and coproducts to products. So by the contravariant version of Brown representability (which always holds, see Brown [8], Neeman [24], whereas covariant Brown representability depends on extra assumptions), there exists a representing A-module T (I) satisfying Hom H (A;k) ( ^ H (A; ); I) = Hom A ( ; T (I)): (3.1) PURE INJECTI... |

42 |
la dimension cohomologique des groupes profinis, Topology 3
- Serre, Sur
- 1965
(Show Context)
Citation Context ...has the property that ( ^ H (H; k)) n = 0. Then every element of ( ^ H (G; k)) 2n is divisible by the Bocksteins(x) for each 0 6= x 2 H 1 (G; F p ), by the Quillen{Venkov lemma. By a theorem of Serre =-=[28]-=-, there exist nonzero elements x 1 ; : : : ; xm 2 H 1 (G; F p ) such thats(x 1 ) : : :s(xm ) = 0. Therefore ( ^ H (G; k)) 2nm = 0. For a ring R we denote by Inj(R) the full subcategory of injective R-... |

37 |
Model theoretic algebra
- Jensen, Lenzing
- 1989
(Show Context)
Citation Context ...s. In other words, Hom( ; M) is exact on pure exact sequences. So for example, every injective module is pure injective. For a more detailed introduction to purity, see for example Jensen and Lenzing =-=[19-=-]. A module is said to be -pure injective if an arbitrary direct sum of copies of the module is pure injective. So for example, a ring is left Noetherian if and only if every injective (left) module i... |

32 |
Spectra and the Steenrod Algebra
- Margolis
- 1983
(Show Context)
Citation Context ...zation functor FN : StMod(A) ! StMod(A) with respect to the localizing subcategory N. Proof. It is true in general that localization with respect to a homology theory exists, see for example Margolis =-=[22]-=-, chapter 7. The homology theory in question is just Tate cohomology. For the convenience of the reader, we also provide a direct proof, for which we are indebted to Jeremy Rickard. Consider the local... |

30 |
Abstract homotopy theory
- Brown
- 1965
(Show Context)
Citation Context ... a module M to Hom H (A;k) ( ^ H (A; M); I) takes triangles to exact sequences and coproducts to products. So by the contravariant version of Brown representability (which always holds, see Brown [8], Neeman [24], whereas covariant Brown representability depends on extra assumptions), there exists a representing A-module T (I) satisfying Hom H (A;k) ( ^ H (A; ); I) = Hom A ( ; T (I)): (3.1)... |

30 |
Idempotent modules in the stable category
- Rickard
- 1997
(Show Context)
Citation Context ... modular representations and the cohomology of G has been a subject of major interest over the last several years. For example, recent study of the stable category of kG-modules StMod(kG) led Rickard =-=[2-=-7] to introduce certain idempotent modules and functors. This has led to a theory of varieties for innitely generated modules [3, 4], and the classication of thick subcategories of the stable category... |

23 |
Tame algebras and generic modules
- Crawley-Boevey
- 1991
(Show Context)
Citation Context ...r some time in representation theory ofsnite dimensional algebras, mostly because certain innitely generated pure injectives (so-called generic modules) control the representation type of an algebra [=-=11]-=-. For example, generic modules have been used to show that the representation type of an algebra is an invariant of the stable module category [20]. More recently, the notion of a phantom map in StMod... |

21 |
Products in negative cohomology
- Benson, Carlson
- 1992
(Show Context)
Citation Context ...r of a nontrivial p-group is necessarily nontrivial, it follows that every maximal elementary abelian p-subgroup of G has rank at least two. If G is elementary abelian, it follows from theorem 3.1 of =-=[2]-=- that ( ^ H (G; k)) 2 = 0, so we may assume that G is not elementary abelian. The rest of the proof is based on an argument of Quillen and Venkov [26]. By induction and Lemma 2.3, we may assume that t... |

15 |
Large modules over artin algebras, Algebra, Topology and Categories", Academie Press
- Auslander
- 1976
(Show Context)
Citation Context ...module T (I) is indecomposable by Lemma 3.8 and a direct summand of a direct product of modules of the form n k, n 2 Z, by Proposition 4.5. It follows from a result of Auslander (see corollary 3.2 in [1]) that T (I) = n k for some n 2 Z. Thus T (I) = T (J) for J = I m [ n + 1], and therefore ^ I = ^ J by Lemma 3.3. We conclude from Lemmas 2.1 and 2.2 that I = J . Theorem 4.7. Suppose that eit... |

14 |
Phantom maps and purity in modular representation theory II, Algebras and
- Gnacadja
(Show Context)
Citation Context ...modules have been used to show that the representation type of an algebra is an invariant of the stable module category [20]. More recently, the notion of a phantom map in StMod(kG) was introduced in =-=[16, 6]-=- as an analogue of a classical concept from stable homotopy theory. In this context, it became apparent as well that pure injective modules play an important role, as the modules which receive no phan... |

11 | Stable equivalence preserves representation type
- Krause
- 1997
(Show Context)
Citation Context ...dules) control the representation type of an algebra [11]. For example, generic modules have been used to show that the representation type of an algebra is an invariant of the stable module category =-=[20]-=-. More recently, the notion of a phantom map in StMod(kG) was introduced in [16, 6] as an analogue of a classical concept from stable homotopy theory. In this context, it became apparent as well that ... |

10 | Rings with a local cohomology theorem and applications to cohomology rings of groups
- Greenlees, Lyubeznik
(Show Context)
Citation Context ... of H (G; k)=p is closely related to the corresponding Rickard idempotent module (V ) introduced in [4]. The proof of this involves a detailed analysis of the Greenlees{Lyubeznik spectral sequence [1=-=7-=-], and will be the subject of a separate paper. Conventions and notation. When we talk about modules and maps over a Z-graded ring R, we always mean graded modules unless otherwise specied. If M and N... |

7 |
Remarks on elementary duality
- Prest
- 1993
(Show Context)
Citation Context ... ) = fM 2 Ind(A) j F (M) = 0g: The sets of the form D(F ) for some coherent F form a basis for the open subsets of the Zariski topology on Ind(A). This topology on Ind(A) has been introduced by Prest =-=[25]-=-; it is the analogue of the Zariski topology on the set of isomorphism classes of indecomposable injective quasi-coherent sheaves on a Noetherian scheme which has been considered by Gabriel (chapitre ... |

5 |
Ph.: Phantom maps in the stable module category
- Gnacadja
- 1998
(Show Context)
Citation Context ...modules have been used to show that the representation type of an algebra is an invariant of the stable module category [20]. More recently, the notion of a phantom map in StMod(kG) was introduced in =-=[16, 6]-=- as an analogue of a classical concept from stable homotopy theory. In this context, it became apparent as well that pure injective modules play an important role, as the modules which receive no phan... |

3 |
Spektralkategorien und regulare Ringe im von Neumannschen
- Gabriel, Oberst
- 1966
(Show Context)
Citation Context ...orphism ring of X. Note that every object X in C has End(X)=rad(End(X)) as endomorphism ring in Spec(C). The spectral category Sp(R) of a ring R is by denition the spectral category Spec(Inj(R)). In [=-=15]-=-, Gabriel and Oberst studied the spectral category of a ring R: it is an abelian Grothendieck category where every object is projective and injective. Moreover, two injective R-modules are isomorphic ... |

3 |
Injective modules over Noetherian rings, Pacific
- Matlis
- 1958
(Show Context)
Citation Context ..., the injective hull I p = E(R=p) of the quotient R=p is indecomposable, and each injective indecomposable is isomorphic to a shifted copy I p [n] for some prime p and some n 2 Z. A theorem of Matlis =-=[23]-=- describes the endomorphism ring of I p [n]. It is isomorphic to the completion R ^ p of the localization R p with respect to the powers of the maximal ideal p p . Each element of R ^ p acts locally n... |

2 |
Complexity and varieties for in generated modules
- Benson, Carlson, et al.
- 1995
(Show Context)
Citation Context ...ecent study of the stable category of kG-modules StMod(kG) led Rickard [27] to introduce certain idempotent modules and functors. This has led to a theory of varieties for innitely generated modules [=-=3, 4-=-], and the classication of thick subcategories of the stable category stmod(kG) ofsnitely generated modules, at least in the case of a p-group [5]. The cohomology of the group G is intimately related ... |

2 |
Generic idempotent modules for a group
- Benson, Krause
- 1999
(Show Context)
Citation Context ...ongly unbounded representation type. We show in Section 9 that in case A = kG, the generic modules corresponding to minimal primes are, up to translation, the modules investigated in our recent paper =-=[-=-7]. More generally, it turns out that for a nonmaximal prime ideal p of H (G; k), the module obtained by applying T to the injective hull of H (G; k)=p is closely related to the corresponding Rickar... |

2 |
Cohomology of group schemes over a Invent
- Friedlander, Suslin
- 1997
(Show Context)
Citation Context .... We denote by k the trivial A-module and the cohomology ring H (A; k) is by denition Ext A (k; k). This is asnitely generated graded commutative k-algebra by a theorem of Friedlander and Suslin [13=-=]-=-; in particular, it is a Noetherian ring. We write Mod(A) for the category of all A-modules and homomorphisms, and mod(A) for the full subcategory whose objects are thesnitely generated A-modules. We ... |

1 |
Venkov, Cohomology of groups and elementary abeilan subgroups, Topology 11
- Quillen, B
- 1972
(Show Context)
Citation Context ...elementary abelian, it follows from theorem 3.1 of [2] that ( ^ H (G; k)) 2 = 0, so we may assume that G is not elementary abelian. The rest of the proof is based on an argument of Quillen and Venkov =-=[26]-=-. By induction and Lemma 2.3, we may assume that there exists n > 0 such that every proper subgroup H of G has the property that ( ^ H (H; k)) n = 0. Then every element of ( ^ H (G; k)) 2n is divisibl... |

1 |
Model theory of modules, Annals of Pure & Applied Logic 26
- Ziegler
- 1984
(Show Context)
Citation Context ...F A FINITE GROUP 21 Proof. The sets of the form V (F ) = Ind(A) n D(F ) for some coherent F form a basis for the open subsets of the Ziegler topology on Ind(A) which has been introduced by Ziegler in =-=[29-=-]; see [21] for the denition of this topology in terms of functors. A Ziegler-open set is quasi-compact if and only if it is of the form V (F ) for some coherent functor F (cf. corollary 4.5 in [21]).... |