negative K-theory
The positive K-groups K_0, K_1, K_2, ... climb upward, but there is also a basement: K-groups in negative degrees K_{-1}, K_{-2}, and so on. These were defined by Bass before the higher theory existed, and they act as detectors of singularities. For a smooth or regular ring the basement is empty — all negative K-groups vanish — so a nonzero negative K-group is a precise algebraic signal that the ring (or scheme) is not regular.
Bass defined the negative K-groups by a descending recursion using the ring R[t, t^{-1}] of Laurent polynomials. The fundamental theorem of K-theory gives K_n(R[t, t^{-1}]) = K_n(R) ⊕ K_{n-1}(R) ⊕ (two nil terms), and Bass used the analogous splitting to define K_{-1}(R) as the cokernel of K_0(R[t]) ⊕ K_0(R[t^{-1}]) -> K_0(R[t, t^{-1}]), then iterated: K_{-n}(R) = K_{-n+1}(R[t, t^{-1}]) / (the contracted part). Equivalently, in modern terms, K(R) extends to a spectrum with homotopy groups in all integer degrees, and the negative homotopy groups are the negative K-groups.
The key structural fact is regularity: if R is a regular Noetherian ring then K_{-n}(R) = 0 for all n greater than 0, and moreover K_n(R[t]) = K_n(R) (homotopy invariance). When these fail, negative K-theory is nonzero and measures the defect. The vanishing, finiteness, and behavior of negative K-groups are governed by Weibel's conjecture (now a theorem of Kerz-Strunk-Tamme): for a Noetherian scheme of dimension d, K_{-n} vanishes for n greater than d, so the basement has depth bounded by the dimension.
The nodal cubic coordinate ring R = k[x, y]/(y^2 - x^3 - x^2), a one-dimensional ring with a singular point, has K_{-1}(R) = Z, nonzero because the curve is not regular at the node. Its normalization (the regular model) has K_{-1} = 0, so K_{-1} precisely sees the singularity.
K_{-1} of the nodal cubic is Z, detecting the singularity.
The full Bass fundamental theorem packages all of this: K_n(R[t, t^{-1}]) splits as K_n(R) ⊕ K_{n-1}(R) ⊕ NK_n(R) ⊕ NK_n(R), where the NK terms are the nil K-groups, vanishing exactly when R is K_n-regular. Negative K-theory, nil K-theory, and the failure of homotopy invariance are thus three faces of non-regularity.