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Schemes of Line Modules I
Author(s) -
Shelton Brad,
Vancliff Michaela
Publication year - 2002
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610702003186
Subject(s) - mathematics , scheme (mathematics) , functor , algebra over a field , incidence algebra , line (geometry) , quadratic equation , point (geometry) , differential graded algebra , commutative property , filtered algebra , algebra representation , pure mathematics , cellular algebra , mathematical analysis , geometry
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.