Journal of the London Mathematical Society



SCHEMES OF LINE MODULES I


BRAD SHELTON a1 and MICHAELA VANCLIFF a2 1
a1 Department of Mathematics, University of Oregon, Eugene, OR 97403-1222, USA shelton@math.uoregon.edu
a2 Department of Mathematics, Box 19408, University of Texas at Arlington, Arlington, TX 76019-0408, USA vancliff@math.uta.edu

Abstract

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.

(Received September 14 2000)
(Revised August 1 2001)



Footnotes

1 The second author was supported in part by NSF grants DMS-9622765 and DMS-9996056.