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Register a new userCompletions of quantum coordinate rings
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Authors
Linhong Wang
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Given an iterated skew polynomial ring C[y_1;t_1,d_1]ldots [y_n;t_n,d_n] over a complete local ring C with maximal ideal m, we prove, under suitable assumptions, that the completion at the ideal m + < y_1,y_2,ldots,y_n> is an iterated skew power series ring. Under further conditions, this completion is a local, noetherian, Auslander regular domain. Applicable examples include quantum matrices, quantum symplectic spaces, and quantum Euclidean space.
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A theory of monoids in the category of bicomodules of a coalgebra $C$ or $C$-rings is developed. This can be viewed as a dual version of the coring theory. The notion of a matrix ring context consisting of two bicomodules and two maps is introduced and the corresponding example of a $C$-ring (termed a {em matrix $C$-ring}) is constructed. It is shown that a matrix ring context can be associated to any bicomodule which is a one-sided quasi-finite injector. Based on this, the notion of a {em Galois module} is introduced and the structure theorem, generalising Schneiders Theorem II [H.-J. Schneider, Israel J. Math., 72 (1990), 167--195], is proven. This is then applied to the $C$-ring associated to a weak entwining structure and a structure theorem for a weak $A$-Galois coextension is derived. The theory of matrix ring contexts for a firm coalgebra (or {em infinite matrix ring contexts}) is outlined. A Galois connection associated to a matrix $C$-ring is constructed.
We study the q-commutative power series ring R:=k_q[[x_1,...,x_n]], defined by the relations x_ix_j = q_{ij}x_j x_i, for multiplicatively antisymmetric scalars q_{ij} in a field k. Our results provide a detailed account of prime ideal structure for a class of noncommutative, complete, local, noetherian domains having arbitrarily high (but finite) Krull, global, and classical Krull dimension. In particular, we prove that the prime spectrum of R is normally separated and is finitely stratified by commutative noetherian spectra. Combining this normal separation with results of Chan, Wu, Yekutieli, and Zhang, we are able to conclude that R is catenary. Following the approach of Brown and Goodearl, we also show that links between prime ideals are provided by canonical automorphisms. Moreover, for sufficiently generic q_{ij}, we find that R has only finitely many prime ideals and is a UFD (in the sense of Chatters).
In support variety theory, representations of a finite dimensional (Hopf) algebra $A$ can be studied geometrically by associating any representation of $A$ to an algebraic variety using the cohomology ring of $A$. An essential assumption in this theory is the finite generation condition for the cohomology ring of $A$ and that for the corresponding modules. In this paper, we introduce various approaches to study the finite generation condition. First, for any finite dimensional Hopf algebra $A$, we show that the finite generation condition on $A$-modules can be replaced by a condition on any affine commutative $A$-module algebra $R$ under the assumption that $R$ is integral over its invariant subring $R^A$. Next, we use a spectral sequence argument to show that a finite generation condition holds for certain filtered, smash and crossed product algebras in positive characteristic if the related spectral sequences collapse. Finally, if $A$ is defined over a number field over the rationals, we construct another finite dimensional Hopf algebra $A$ over a finite field, where $A$ can be viewed as a deformation of $A$, and prove that if the finite generation condition holds for $A$, then the same condition holds for $A$.
We continue the first and second authors study of $q$-commutative power series rings $R=k_q[[x_1,ldots,x_n]]$ and Laurent series rings $L=k_q[[x^{pm 1}_1,ldots,x^{pm 1}_n]]$, specializing to the case in which the commutation parameters $q_{ij}$ are all roots of unity. In this setting, $R$ is a PI algebra, and we can apply results of De Concini, Kac, and Procesi to show that $L$ is an Azumaya algebra whose degree can be inferred from the $q_{ij}$. Our main result establishes an exact criterion (dependent on the $q_{ij}$) for determining when the centers of $L$ and $R$ are commutative Laurent series and commutative power series rings, respectively. In the event this criterion is satisfied, it follows that $L$ is a unique factorization ring in the sense of Chatters and Jordan, and it further follows, by results of Dumas, Launois, Lenagan, and Rigal, that $R$ is a unique factorization ring. We thus produce new examples of complete, local, noetherian, noncommutative, unique factorization rings (that are PI domains).
The elliptic algebras in the title are connected graded $mathbb{C}$-algebras, denoted $Q_{n,k}(E,tau)$, depending on a pair of relatively prime integers $n>kge 1$, an elliptic curve $E$, and a point $tauin E$. This paper examines a canonical homomorphism from $Q_{n,k}(E,tau)$ to the twisted homogeneous coordinate ring $B(X_{n/k},sigma,mathcal{L}_{n/k})$ on the characteristic variety $X_{n/k}$ for $Q_{n,k}(E,tau)$. When $X_{n/k}$ is isomorphic to $E^g$ or the symmetric power $S^gE$ we show the homomorphism $Q_{n,k}(E,tau) to B(X_{n/k},sigma,mathcal{L}_{n/k})$ is surjective, that the relations for $B(X_{n/k},sigma,mathcal{L}_{n/k})$ are generated in degrees $le 3$, and the non-commutative scheme $mathrm{Proj}_{nc}(Q_{n,k}(E,tau))$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$, respectively. When $X_{n/k}=E^g$ and $tau=0$, the results about $B(X_{n/k},sigma,mathcal{L}_{n/k})$ show that the morphism $Phi_{|mathcal{L}_{n/k}|}:E^g to mathbb{P}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.
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