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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 centralseries, L_i = L_i(A), defined inductively by L_1 = A, L_{i+1} = [A,L_{i}], andtheir 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 Feiginand 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. Newphenomena arise, namely, torsion in B_i over Z, and jumps in dimension overF_p. We describe the torsion in the reduced quotient RB_1 and B_2 geometricallyin terms of the De Rham cohomology of Z^n. As a corollary we obtain a completedescription of \bar{B}_1(A_n(Z)) and \bar{B}_1(A_n(F_p)), as well as ofB_2(A_n(Z[1/2])) and B_2(A_n(F_p)), p>2. We also give theoretical andexperimental results for B_i with i>2, formulating a number of conjectures andquestions based on them. Finally, we discuss the supercase, when some of thegenerators are odd (fermionic) and some are even (bosonic), and provide sometheoretical results and experimental data in this case.

Lower central series of a free associative algebra over the integers and finite fields