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Journal of the London Mathematical Society 2002 65(3):575-590; doi:10.1112/S0024610702003186
© 2002 by London Mathematical Society
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© The London Mathematical Society

Schemes of Line Modules I

Brad Shelton and Michaela Vancliff

Department of Mathematics, University of Oregon Eugene, OR 97403-1222, USA, shelton{at}math.uoregon.edu
Department of Mathematics Box 19408, University of Texas at Arlington, Arlington, TX 76019-0408, USA vancliff{at}math.uta.edu

Received 14 September 2000. Revision received 1 August 2001.

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.


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