# Homological transcendence degree Academic Article

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### abstract

• Abstract Let $D$ be a division algebra over a base field $k$. The homological transcendence degree of $D$, denoted by $\text {Htr}\; D$, is defined to be the injective dimension of the algebra $D\otimes_k D^{\circ}$. We show that $\text {Htr}$ has several useful properties which the classical transcendence degree has. We extend some results of Resco, Rosenberg, Schofield and Stafford, and compute $\text {Htr}$ for several classes of division algebras. The main tool for the computation is Van den Bergh's rigid dualizing complex.

### publication date

• January 1, 2006