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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.

MSC:
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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