### Abstract

Let k be a field and let G be a finite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology gamma(G) is an element of HH3,-1 (H) over cap*(G,k) with the following property. Given a graded (H) over cap*(G,k)-module X, the image of gamma(G) in Ext((H) over cap*(G,k))(3,-1) (X,X) vanishes if and only if X is isomorphic to a direct summand of (H) over cap*(G, M) for some kG-module M.

The description of the realizability obstruction works in any triangulated category with direct sums. We show that in the case of the derived category of a differential graded algebra A, there is also a canonical element of Hochschild cohomology HH3,-1 H*(A) which is a predecessor for these obstructions.

Original language | English |
---|---|

Pages (from-to) | 3621-3668 |

Number of pages | 47 |

Journal | Transactions of the American Mathematical Society |

Volume | 356 |

Issue number | 9 |

DOIs | |

Publication status | Published - Jan 2004 |

### Keywords

- HOMOLOGICAL FINITENESS CONDITIONS
- UNIVERSAL TODA BRACKETS
- BROWN REPRESENTABILITY
- INFINITE GROUPS
- HOMOTOPY PAIRS
- CATEGORIES
- COMPLEXITY
- VARIETIES
- ALGEBRAS

### Cite this

*Transactions of the American Mathematical Society*,

*356*(9), 3621-3668. https://doi.org/10.1090/S0002-9947-03-03373-7

**Realizability of modules over Tate cohomology.** / Benson, David John; Krause, H.; Schwede, S.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 356, no. 9, pp. 3621-3668. https://doi.org/10.1090/S0002-9947-03-03373-7

}

TY - JOUR

T1 - Realizability of modules over Tate cohomology

AU - Benson, David John

AU - Krause, H.

AU - Schwede, S.

PY - 2004/1

Y1 - 2004/1

N2 - Let k be a field and let G be a finite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology gamma(G) is an element of HH3,-1 (H) over cap*(G,k) with the following property. Given a graded (H) over cap*(G,k)-module X, the image of gamma(G) in Ext((H) over cap*(G,k))(3,-1) (X,X) vanishes if and only if X is isomorphic to a direct summand of (H) over cap*(G, M) for some kG-module M.The description of the realizability obstruction works in any triangulated category with direct sums. We show that in the case of the derived category of a differential graded algebra A, there is also a canonical element of Hochschild cohomology HH3,-1 H*(A) which is a predecessor for these obstructions.

AB - Let k be a field and let G be a finite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology gamma(G) is an element of HH3,-1 (H) over cap*(G,k) with the following property. Given a graded (H) over cap*(G,k)-module X, the image of gamma(G) in Ext((H) over cap*(G,k))(3,-1) (X,X) vanishes if and only if X is isomorphic to a direct summand of (H) over cap*(G, M) for some kG-module M.The description of the realizability obstruction works in any triangulated category with direct sums. We show that in the case of the derived category of a differential graded algebra A, there is also a canonical element of Hochschild cohomology HH3,-1 H*(A) which is a predecessor for these obstructions.

KW - HOMOLOGICAL FINITENESS CONDITIONS

KW - UNIVERSAL TODA BRACKETS

KW - BROWN REPRESENTABILITY

KW - INFINITE GROUPS

KW - HOMOTOPY PAIRS

KW - CATEGORIES

KW - COMPLEXITY

KW - VARIETIES

KW - ALGEBRAS

U2 - 10.1090/S0002-9947-03-03373-7

DO - 10.1090/S0002-9947-03-03373-7

M3 - Article

VL - 356

SP - 3621

EP - 3668

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 9

ER -