  A chemotaxis system possibly containing rotational components of the
cross-diffusive flux is studied under no-flux boundary conditions in a bounded
domain $\Omega\subset R^n$, $n\ge 1$, with smooth boundary, where the evolution
of the signal is determined by consumption through cells. In contrast to
related Keller-Segel-type problems with scalar sensitivities, in presence of
such tensor-valued sensitivities this system in general apparently does not
possess any useful gradient-like structure. Accordingly, its analysis needs to
be based on new types of a priori bounds.
  Using a spatio-temporal $L^2$ estimate for the gradient of the logarithm of
the cell density as a starting point, we derive a series of compactness
properties of solutions to suitably regularized versions of the system.
Motivated by these, we develop a generalized solution concept which requires
solutions to satisfy very mild regularity hypotheses only.
  On the basis of the above compactness properties, it is finally shown that
within this framework, under a mild growth assumption on the sensitivity matrix
and for all sufficiently regular nonnegative initial data, the corresponding
initial-boundary value problem possesses at least one global generalized
solution.
  This extends known results which in the case of such general matrix-valued
sensitivities provide statements on global existence only in the
two-dimensional setting and under the additional restriction that the initial
signal concentration be suitably small.
