Animals rely on olfaction to find food, attract mates and avoid predators. To support these behaviors, animals must reliably identify odors across different odorant concentrations. The neural circuit operations that implement this concentration invariance remain unclear. Here we demonstrate that, despite concentration-dependence in olfactory bulb (OB), representations of odor identity are preserved downstream, in piriform cortex (PCx). The OB cells responding earliest after inhalation drive robust responses in a sparse subset of PCx neurons. Recurrent collateral connections broadcast their activation across PCx, recruiting strong, global feedback inhibition that rapidly suppresses cortical activity for the remainder of the sniff, thereby discounting the impact of slower, concentration-dependent OB inputs. Eliminating recurrent collateral output dramatically amplifies PCx odor responses, renders cortex steeply concentration-dependent, and abolishes concentration-invariant identity decoding.