Local finite basis property and local representability for varieties of associative rings

PI-varieties associated to full quivers of representations of algebras

“…F satisfies the identity λ 16 − λ. On the other hand, since the radical of C is ε 2 , we see that C satisfies the identity (λ 4 −λ) 2…”

“…Given the variety V of a representable algebra, we can take the Zariski closure of its relatively free algebra U and construct its full quiver Γ. (In fact, every variety is the variety of a representable algebra, but this is a deep theorem, due to Kemer [18] in characteristic 0 [17] over arbitrary infinite fields and to Belov [2] over finite fields and arbitrary commutative rings.) Then id(V ) = id(U ) = id(A(Γ)), so results about the relatively free algebra U determine the identities of V .…”

“…Unfortunately, when a path B has glued vertices, the polynomial given in Example 4.13 might turn out to be an identity of the algebra of B. Fortunately, this only happens when consecutive vertices are glued. The extreme case of gluing is when all vertices are glued together; i.e., in (16) I n(t) 2,2 ] must be to the radical, above the diagonal, which means that x 1 must go to a diagonal element. In order for the polynomial z[y 1,1 , y 1,2 ]x 1 [y 2,1 , y 2,2 ]x 2 not to vanish, the positioning of the empty block forces a radical substitution of z. Consequently, the polynomial ALEXEI BELOV-KANEL, LOUIS H. ROWEN, AND UZI VISHNE…”

Specht’s problem for associative affine algebras over commutative Noetherian rings

“…When char(F ) > 0 there are non-affine counterexamples [2,13], with a straightforward exposition given in [9], so the best one could hope for is a positive result for affine PI-algebras. Kemer [18] proved this result for affine PI-algebras over infinite fields, and Belov extended the theorem to affine PI-algebras over arbitrary commutative Noetherian rings, in his second dissertation, with the main ideas given in [3]. We give full details of the proof (over arbitrary commutative Noetherian rings), cutting through combinatoric complications by utilizing the full strength of the theory of full quivers as expounded in [6], [7], and [8].…”

“…The main difficulty in this approach is to discern whether the algebras we are working with actually are representable. When the base ring is an infinite field F , Kemer [17] proved that any relatively free affine F -algebra is representable; this is also treated in [3] for F finite, but the proof is rather difficult. Consequently, we plan to treat the representability theorem in a separate paper.…”