Noncommutative Algebraic Varieties
For a natural number \(r\) , we define the free \(r\,\times \, r\) matrix polynomial algebras and their quotients. We define algebraic sets and tangent spaces between different points. We then study their naive geometry by deformation theory, and prove th
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Abstract For a natural number r , we define the free r × r matrix polynomial algebras and their quotients. We define algebraic sets and tangent spaces between different points. We then study their naive geometry by deformation theory, and prove that this defines noncommutative varieties in a natural way.
1 Introduction Algebraic geometry has a long tradition, and in fact comes from a natural place. Then after making algebraic geometry to a categorical theme, it is possible to define noncommutative algebraic geometry. In this text we try to take noncommutative algebraic geometry back to the natives. We will use deformation theory to define higher order derivatives between points, and then use this to construct a noncommutative variety. Our main commutative reference is Hartshorne’s classical book [2]. Through this notes, k is an algebraically closed field of characteristic 0.
2 Polynomial Matrix Algebras Let r ∈ N and let (di j ) be an r × r -matrix with entries di j ∈ N. We start by defining the free r × r matrix polynomial algebra generated by the matrix variables ti j (l), 1 ≤ l ≤ di j , in entry 1 ≤ i, j ≤ r . To get into the language, consider the following (in which r = 2 and di j = 1, 1 ≤ i, j ≤ 2):
A. Siqveland (B) Faculty of Technology, Buskerud Vestfold University College, Kongsberg, Norway e-mail: [email protected] A. Makhlouf et al. (eds.), Algebra, Geometry and Mathematical Physics, 275 Springer Proceedings in Mathematics & Statistics 85, DOI: 10.1007/978-3-642-55361-5_16, © Springer-Verlag Berlin Heidelberg 2014
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Example 2.1 Let the matrices
x0 0y 00 00 X= ,Y = ,Z = , and W = 00 00 z0 0w
10 00 and e2 = these matrix be given. Together with the idempotents e1 = 00 01 variables generates a k 2 -algebra which is denoted kx ky S= . kz kw By the notation Si j where Sii is a k-algebra for each i, 1 ≤ i ≤ r , and Si j is a k-vector space, we mean the k r -algebra generated by the matrices M = (m i j ) with m i j ∈ Si j , 1 ≤ i, j ≤ r . Definition 2.1 For a positive integer r , for each pair (i, j), 1 ≤ i, j ≤ r , let di j ∈ N. Then the free polynomial algebra in the matrix variables ti j (l), 1 ≤ i, j ≤ di j , is the k r -algebra generated by the matrix elements in ⎛ ⎞ d1r kt11 (1), . . . , t11 (d11 ) · · · v=1 kt1r (v) ⎜ ⎟ .. .. .. ⎝ ⎠ . . . . dr 1 · · · ktrr (1), . . . , trr (drr ) v=1 ktr 1 (v) Alternatively, we consider the k r -module V generated by ti j (l), and let S be the tensor algebra S = Tk r (V ) . Definition 2.2 For a positive integer r , a finitely generated r × r matrix polynomial algebra is a quotient of a free r × r matrix polynomial algebra. Lemma 2.1 Let π : R S be a surjective k-algebra homomorphism sending non-units to non-units, and let m ⊂ R be a maximal ideal. Then π(m) is maximal in S. Proof First of all, as sπ(m) = π(r )π(m) = π(r m) ∈ π(m) for some r ∈ R, π(m) is an ideal. Assume π(m) a and let a = π(r ) ∈ a \ π(m). Then r ∈ π −1 (a) \ m so that m π −1 (a) so that π −1 (a) = R. But then 1 = π(1) ∈ a implying
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