In commutative algebra, the **deviations of a local ring** *R* are certain invariants ε_{i}(*R*) that measure how far the ring is from being regular.

The deviations ε_{n} of a local ring *R* with residue field *k* are non-negative integers defined in terms of its Poincaré series *P*(*x*) by

P
(
x
)
=
∑
n
≥
0
x
n
Tor
n
R
(
k
,
k
)
=
∏
n
≥
0
(
1
+
t
2
n
+
1
)
ε
2
n
(
1
−
t
2
n
+
2
)
ε
2
n
+
1
.
The zeroth deviation ε_{0} is the embedding dimension of *R* (the dimension of its tangent space). The first deviation ε_{1} vanishes exactly when the ring *R* is a regular local ring, in which case all the higher deviations also vanish. The second deviation ε_{2} vanishes exactly when the ring *R* is a complete intersection ring, in which case all the higher deviations vanish.