Hochschild Cohomology Ring of an Order of a Simple Component of the Rational Group Ring of the Generalized Quaternion Group

Abstract

We will give an efficient bimodule projective resolution of an order Γ, where Γ is an order of a simple component of the rational group ring ℚ Q _{2 ^r} of the generalized quaternion 2-group Q _{2 ^r} of order 2 ^r+2. Moreover, we will determine the ring structure of the Hochschild cohomology HH*(Γ) by calculating the Yoneda products using this bimodule projective resolution.

Key Words:

• Bimodule projective resolution
• Hochschild cohomology ring
• Order
• Quaternion algebra
• Yoneda product

2000 Mathematics Subject Classification:

• 16E40

ACKNOWLEDGMENTS

The author would like to express his gratitude to Professor K. Sanada for valuable comments and suggestions. The author would also like to express his gratitude to the referee for valuable comments and helpful suggestions towards the improvement of the article.

Notes

Communicated by D. Zacharia.