Michaela Vancliff

Schemes of line modules I

It is proved that there exists a scheme that represents the functor of line modules over a graded algebra, and it is called the line scheme of the algebra. Its properties and its relationship to the point scheme are studied. If the line scheme of a quadratic, Auslander-regular algebra of global dimension 4 has dimension 1, then it determines the defining relations of the algebra. Moreover, the following counter-intuitive result is proved. If the zero locus of the defining relations of a quadratic (not necessarily regular) algebra on four generators with six defining relations is finite, then it determines the defining relations of the algebra. Although this result is non-commutative in nature, its proof uses only commutative theory. The structure of the line scheme and the point scheme of a 4-dimensional regular algebra is also used to determine basic incidence relations between line modules and point modules.

Four dimensional regular algebras with point scheme, anonsingular quadric in P3

In [22], a class of four-dimensional, quadratic, Artin-Schelter regular algebras was introduced, whose point scheme is the graph of an automorphism of a nonsingular quadric in P3. These algebras are the first examples of quadratic Artin-Schelter regular algebras whose defining relations are not determined by the point scheme and, hence, not determined by the algebraic data obtained from the point modules. In this paper, we study these algebras via their line modules. In particular, the set of lines in P3 that correspond to left line modules is not the set of lines in P3 that correspond to right line modules. Our analysis focuses on a distinguished member R λ of this class of algebras, where R λ is a twist by a twisting system of the other algebras. We prove that R λ is a finite module over its center and that its central Proj is a smooth quadric inP4.

Schemes of line modules. II

ABSTRACT In this sequel to[1], we study the scheme of line modules for several classes of quantum s, including Clifford algebras, homogenized and algebras associated to smooth quadrics in . We also prove that a quantum with enough symmetry in its defining relations has a line scheme of dimension at least two, with infinitely many line modules incident to any point module.

Classifying quadratic quantum P2s by using graded skew Clifford algebras

Abstract We prove that quadratic regular algebras of global dimension three on degree-one generators are related to graded skew Clifford algebras. In particular, we prove that almost all such algebras may be constructed as a twist of either a regular graded skew Clifford algebra or of an Ore extension of a regular graded skew Clifford algebra of global dimension two. In so doing, we classify all quadratic regular algebras of global dimension three that have point scheme either a nodal cubic curve or a cuspidal cubic curve in P 2 .

The one-dimensional line scheme of a certain family of quantum P3s

Abstract A quantum P 3 is a noncommutative analogue of a polynomial ring on four variables, and, herein, it is taken to be a regular algebra of global dimension four. It is well known that if a generic quadratic quantum P 3 exists, then it has a point scheme consisting of exactly twenty distinct points and a one-dimensional line scheme. In this article, we compute the line scheme of a family of algebras whose generic member is a candidate for a generic quadratic quantum P 3 . We find that, as a closed subscheme of P 5 , the line scheme of the generic member is the union of seven curves; namely, a nonplanar elliptic curve in a P 3 , four planar elliptic curves and two nonsingular conics.

Corrigendum. Generalizations of graded Clifford algebras and of complete intersections

A correction is provided for Proposition 3.5 in the article “Generalizations of Graded Clifford Algebras and of Complete Intersections”. The correction is: if S is a skew polynomial ring on finitely many generators of degree one that are normal elements in S, and if I is a homogeneous ideal of S that is generated by a normalizing sequence, then dimk(S/I) is finite if and only if S/I has no point modules and no fat point modules. A similar correction is provided for Corollary 3.6 of the same article. The proof of Proposition 3.5 in [4] contains an error, so that [4, Proposition 3.5 and Corollary 3.6] need to be modified (see Proposition 10 and Corollary 11 below). The authors would like to thank J. T. Stafford for alerting them to this issue, which occurs in the paragraph in [4] immediately preceding Proposition 3.5. The main result of [4], namely Theorem 4.2, is correct as stated, provided that the definitions of base point and base-point free are changed from those given in [4, Definition 1.7] to those given in Definition 2 below. The examples and other results in the remaining sections of [4] are unchanged. Additionally, the reader should note that the results in [7] are unchanged. We recall the notation of [4]: k denotes an algebraically closed field; k = k \ {0} and similarly for modules and other rings; Mc(k) is the ring of c× c matrices over k; μ = (μij) ∈ Mn(k ), where μijμji = 1 for all i, j; S = k〈z1, . . . , zn〉/〈U〉, where U = span{zjzi − μijzizj : 2010 Mathematics Subject Classification: . 16S38, 16S37, 16S36.

Graded Skew Clifford Algebras That Are Twists of Graded Clifford Algebras

In 2010, a quantized analog of a graded Clifford algebra (GCA), called a graded skew Clifford algebra (GSCA), was proposed by Cassidy and Vancliff, and many properties of GCAs were found to have counterparts for GSCAs. In particular, a GCA is a finite module over a certain commutative subalgebra C, while a GSCA is a finite module over a (typically noncommutative) analogous subalgebra R. We consider the case that a regular GSCA is a twist of a GCA by an automorphism, and we prove, in this case, R is a skew polynomial ring and a twist of C by an automorphism.

On the Notion of Complete Intersection Outside the Setting of Skew Polynomial Rings

In recent work of T. Cassidy and the author, a notion of complete intersection was defined for (noncommutative) regular skew polynomial rings, defining it using both algebraic and geometric tools, where the commutative definition is a special case. In this article, we extend the definition to a larger class of algebras that contains regular graded skew Clifford algebras, the coordinate ring of quantum matrices, and homogenizations of universal enveloping algebras. Regular algebras are often considered to be noncommutative analogues of polynomial rings, so the results herein support that viewpoint.

Non-commutative Spaces for Graded Quantum Groups and Graded Clifford Algebras

We propose that the notion of “quantum space” from Artin, Tate and Van den Bergh’s non-commutative algebraic geometry be considered the “non-commutative space” of a quantum group.