Proper identities, Lie identities and exponential codimension growth

The exponent $\mbox{exp}(A)$ of a PI-algebra $A$ in characteristic zero is an integer measuring the exponential rate of growth of the sequence of codimensions of $A$ (\cite{gz1,gz2}). In this paper we study the exponential rate of growth of the sequences of proper codimensions and Lie codimensions of an associative PI-algebra. We prove that the corresponding proper exponent exists for all PI-algebras, except for some algebras of exponent two strictly related to the Grassmann algebra. We also prove that the Lie exponent exists for any finitely generated PI-algebra. The value of both exponents is always equal to $\mbox{exp}(A)$ or $\mbox{exp}(A)-1$.