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Maps between non-commutative spaces. (English) Zbl 1076.14003
Trans. Am. Math. Soc. 356, No. 7, 2927-2944 (2004); erratum ibid. 368, No. 11, 8295-8302 (2016).
Let \(A = \oplus_{n\geq 0} A_n\) be a graded algebra over a field \(k\) with \(\dim A_n < \infty\) for all \(n\); in what follows, a “graded algebra” for short. Let \(\text{Proj}_{\text{nc}} A\) be the “non-commutative” projective scheme associated to \(A\) as in [M. van den Bergh, “Blowing up of non-commutative smooth surfaces”, Mem. Am. Math. Soc. 734, 140 p. (2001; Zbl 0998.14002)]. In the present paper, maps between such projective spaces are studied. The definition of closed immersions is formally spelled out. Let \(J\) be a graded ideal of \(A\). It is shown that the projection \(\pi: A\to A/J\) induces a closed immersion \(\text{Proj}_{\text{nc}} (A/J) \to \text{Proj}_{\text{nc}} A\). Next, it is shown that a homomorphism \(\phi: A\to B\) of graded algebras induces an affine map \(U \to \text{Proj}_{\text{nc}} A\), where \(U \subseteq \text{Proj}_{\text{nc}} B\) is a non-empty open subset. Finally, a detailed analysis of a “non-commutative” Veronese map (for an arbitrary noetherian graded algebra) is presented. As an application, “non-commutative” resolutions of weighted projective spaces are obtained.

MSC:
14A22 Noncommutative algebraic geometry
16S38 Rings arising from noncommutative algebraic geometry
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