Animals rely on olfaction to find food, attract mates, and avoid predators. To support these behaviors, they must be able to identify odors across different odorant concentrations. The neural circuit operations that implement this concentration invariance remain unclear. We found that despite concentration-dependence in the olfactory bulb (OB), representations of odor identity were preserved downstream, in the piriform cortex (PCx). The OB cells responding earliest after inhalation drove robust responses in sparse subsets of PCx neurons. Recurrent collateral connections broadcast their activation across the PCx, recruiting global feedback inhibition that rapidly truncated and suppressed cortical activity for the remainder of the sniff, discounting the impact of slower, concentration-dependent OB inputs. Eliminating recurrent collateral output amplified PCx odor responses rendered the cortex steeply concentration-dependent and abolished concentration-invariant identity decoding.