PI-varieties associated to full quivers of representations of algebras

In a series of papers [6], [7], [8] we introduced full quivers and pseudoquivers of representations of algebras, and used them as tools in describing PI-varieties of algebras. In this paper we apply them to obtain a complete proof of Belov's solution of Specht's problem for affine algebras over an arbitrary Noetherian ring. The inductive step relies on a theorem that enables one to find a "q-characteristic coefficient-absorbing polynomial in each T-ideal Γ," i.e., a non-identity of the representable algebra A arising from Γ, whose ideal of evaluations in A is closed under multiplication byq-powers of the characteristic coefficients of matrices corresponding to the generators of A, whereq is a suitably large power of the order of the base field. The passage to an arbitrary Noetherian base ring C involves localizing at finitely many elements a kind of C, and reducing to the field case by a local-global principle.

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Representability of relatively free affine algebras over a Noetherian ring

Kanel-Belov,

Rowen,

Vishne

2023

J. Algebra Appl.

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Over the years questions have arisen about T-ideals of (noncommutative) polynomials. But when evaluating a noncentral polynomial in subalgebras of matrices, one often has little control in determining the specific evaluations of the polynomial. One way of overcoming this difficulty in characteristic 0, is to reduce to multilinear polynomials and to utilize the representation theory of the symmetric group. But this technique is unavailable in characteristic [Formula: see text]. An alternative method, which succeeds, is the process of “hiking” a polynomial, in which one specializes its indeterminates in several stages, to obtain a polynomial in which Capelli polynomial is embedded, in order to get control on its evaluations. This method was utilized on homogeneous polynomials in the proof of Specht’s conjecture for affine algebras over fields of positive characteristic. In this paper, we develop hiking further to nonhomogeneous polynomials, to apply to the “representability question.” Kemer proved in 1988 that every affine relatively free PI algebra over an infinite field, is representable. In 2010, the first author of this paper proved more generally that every affine relatively free PI algebra over any commutative Noetherian unital ring is representable [A. Belov, Local finite basis property and local representability of varieties of associative rings, Izv. Russian Acad. Sci. (1) (2010) 3–134. English Translation Izv. Math. 74(1) (2010) 1–126]. We present a different, complete, proof, based on hiking nonhomogeneous polynomials, over finite fields. We then obtain the full result over a Noetherian commutative ring, using Noetherian induction on T-ideals. The bulk of the proof is for the case of a base field of positive characteristic. Here, whereas the usage of hiking is more direct than in proving Specht’s conjecture, one must consider nonhomogeneous polynomials when the base ring is finite, which entails certain difficulties to be overcome. In Appendix A, we show how hiking can be adapted to prove the involutory versions, as well as various graded and nonassociative theorems.

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PI-varieties associated to full quivers of representations of algebras

Cited by 3 publications

References 19 publications

On generic G-graded Azumaya algebras

On generic G-graded Azumaya algebras

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

Representability of relatively free affine algebras over a Noetherian ring

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