  The chemotaxis system \[ \left\{ \begin{array}{l} u_t = \Delta u - \chi\nabla
\cdot (\frac{u}{v}\nabla v), v_t=\Delta v - v+u, \end{array} \right. \] is
considered in a bounded domain $\Omega\subset \mathbb{R}^n$ with smooth
boundary, where $\chi>0$. An apparently novel type of generalized solution
framework is introduced within which an extension of previously known ranges
for the key parameter $\chi$ with regard to global solvability is achieved. In
particular, it is shown that under the hypothesis that\[ \chi < \left\{
\begin{array}{ll} \infty \qquad & \mbox{if } n=2, \sqrt{8} \qquad & \mbox{if }
n=3, \frac{n}{n-2} \qquad & \mbox{if } n\ge 4, \end{array} \right. \] for all
initial data satisfying suitable assumptions on regularity and positivity, an
associated no-flux initial-boundary value problem admits a globally defined
generalized solution. This solution inter alia has the property that \[ u\in
L^1_{loc}(\bar\Omega\times [0,\infty)). \]
