  Consider the free algebra A_n generated over Q by n generators x_1, ..., x_n.
Interesting objects attached to A = A_n are members of its lower central
series, L_i = L_i(A), defined inductively by L_1 = A, L_{i+1} = [A,L_{i}], and
their associated graded components B_i = B_i(A) defined as B_i=L_i/L_{i+1}.
These quotients B_i, for i at least 2, as well as the reduced quotient
\bar{B}_1=A/(L_2+A L_3), exhibit a rich geometric structure, as shown by Feigin
and Shoikhet and later authors,
(Dobrovolska-Kim-Ma,Dobrovolska-Etingof,Arbesfeld-Jordan,Bapat-Jordan).
  We study the same problem over the integers Z and finite fields F_p. New
phenomena arise, namely, torsion in B_i over Z, and jumps in dimension over
F_p. We describe the torsion in the reduced quotient RB_1 and B_2 geometrically
in terms of the De Rham cohomology of Z^n. As a corollary we obtain a complete
description of \bar{B}_1(A_n(Z)) and \bar{B}_1(A_n(F_p)), as well as of
B_2(A_n(Z[1/2])) and B_2(A_n(F_p)), p>2. We also give theoretical and
experimental results for B_i with i>2, formulating a number of conjectures and
questions based on them. Finally, we discuss the supercase, when some of the
generators are odd (fermionic) and some are even (bosonic), and provide some
theoretical results and experimental data in this case.
