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The quantum Serre problem. (English. Russian original) Zbl 0931.16002
Russ. Math. Surv. 53, No. 4, 657-730 (1998); translation from Usp. Mat. Nauk 53, No. 4, 3-76 (1998).
This article is a survey, with essentially full proofs, of results concerning the structure of projective modules over quantum polynomial algebras. The starting point is the well-known theorem of Quillen and Suslin stating that projective modules over ordinary commutative polynomial algebras \(k[X_1,\ldots,X_n]\) are free (“Serre’s Problem”). This is no longer true when the field \(k\) is replaced by a noncommutative division ring \(D\) and \(n\geq 2\) [M. Ojanguren and R. Sridharan, J. Algebra 18, 501-505 (1971; Zbl 0223.16006)]. However, if \(D\) has infinite center then all projective \(D[X_1,\ldots,X_n]\)-modules of rank at least \(2\) are indeed free [J. T. Stafford, Invent. Math. 59, 105-117 (1980; Zbl 0432.16011)].
The present article surveys corresponding results obtained, for the most part by the author, for the quantum polynomial algebras \(A=D_{Q,\alpha}[X_1^{\pm 1},\ldots,X_m^{\pm 1},X_{m+1},\ldots,X_n]\). Here, \(Q=\{q_{ij}\mid i,j=1,\ldots,n\}\subset D^\ast\) and \(\alpha=\{\alpha_i\mid i=1,\ldots,n\}\subset\operatorname{Aut}(D)\) are given data satisfying certain conditions so that the rules \(X_id=\alpha(d)X_i\) for \(d\in D\) and \(X_iX_j=q_{ij}X_jX_i\) define an associative multiplication on \(A\). Several open problems are mentioned.

16D40 Free, projective, and flat modules and ideals in associative algebras
16S36 Ordinary and skew polynomial rings and semigroup rings
16-02 Research exposition (monographs, survey articles) pertaining to associative rings and algebras
16E20 Grothendieck groups, \(K\)-theory, etc.
18F30 Grothendieck groups (category-theoretic aspects)
19A13 Stability for projective modules
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