Naïve noncommutative blowups at zero-dimensional schemes
\textit{D. S. Keeler} and the authors [Duke Math. J. 126, No. 3, 491--546 (2005; Zbl 1082.14003)] introduced the concept of naïve blowup of a scheme. The algebras studied in this work were obtained by blowing up in a single point, and they proved that these objects had properties quite unlike their commutative counterparts. In a companion paper [A class of noncommutative projective surfaces, in press], the authors use the concept of naïve blowups to classify a large class of noncommutative algebras. This classification requires one to naïvely blow up any suitably general zero-dimensional subscheme. The aim of the paper is therefore to study this more general case. The underlying data for a naïve blowup is as follows. Fix an integral scheme \(X\), with automorphism \(\sigma\) and a \(\sigma\)-ample sheaf \(\mathcal{L}\). Let \(Z=Z_{\mathcal{I}}\subset X\) be a zero-dimensional subscheme, with defining ideal \(\mathcal I\subset\mathcal{O}_X\). The \textit{bimodule algebra} is given by \(\mathcal{R}=\mathcal{R}(X,Z,\mathcal{L},\sigma)=\mathcal{O}_X \oplus\mathcal{R}_1 \oplus\mathcal{R}_2 \oplus\cdots,\) where \(\mathcal{R}_n=\mathcal{L}_n \otimes_{\mathcal{O}_X}\mathcal{I}_n,\) for \(\mathcal{L}_n=\mathcal{L} \otimes\sigma^\ast\mathcal{L} \otimes\cdots \otimes(\sigma^{n-1})^\ast\mathcal{L}\), and \(\mathcal{I}_n=\mathcal{I}\cdot\sigma^\ast\mathcal{I} \cdots(\sigma^{n-1})^\ast\mathcal{I}.\) \textit{The naïve blowup algebra of \(X\) at \(Z\)} is then the algebra of sections \[ R=R(X,Z,\mathcal{L},\sigma)= \text{H}^0(X,\mathcal{R})=k\oplus \text{H}^0(X,\mathcal{R}_1) \oplus\text{H}^0(X,\mathcal{R}_2) \oplus\cdots. \] If \(Z\) is the empty set, \(R\) is simply the twisted homogeneous coordinate ring \(B(X,Z,\mathcal{L},\sigma)\) which is essential in noncommutative projective geometry. If \(B=B(X,Z,\mathcal{L},\sigma)\) for a \(\sigma\)-ample invertibele sheaf \(\mathcal{L}\), then \(B\) has extremely pleasant properties, among others: (a) \(B\) is \textit{strongly noetherian} (b) \(qgr-B\cong \text{coh} X\) (c) When \(B\) is generated in degree one, the set of point modules for \(B\), both in \(gr-B\) and in \(qgr-B\), is parametrized by the scheme \(X\). (d) \(B\) has balanced dualizing complex. In contrast, the naïve blowup algebra \(R(X,Z,\mathcal{L},\sigma)\) for \(Z\neq\emptyset\) has properties quite unlike those just mentioned. These properties are the main object of this article and form the main theorem. Its proof is the main body of the article, and the rest is concerned with applying this theorem to get a deeper understanding of the properties of \(\mathcal{R}\) and \(R\). First of all, the authors prove that the properties described by the main theorem is not exceptional. As soon as \(Y\) has at least one critically dense \(\sigma\)-orbit each noetherian connected graded (cg) subalgebra of \(k(Y)[t,t^{-1};\sigma]\) that is generated in degree one inherit the properties. Given a naïve blowup algebra \(R=R(X,Z,\mathcal{L},\sigma)\). The \textit{maximal right torsion extension} of \(R\) is defined to be the ring \(T=T(R)=\{x\in Q(R):xR_{\geq n}\subseteq R\text{ for some } n\geq 0\}\). If \(R\) is the naïve blowup algebra at a single point then \(T(R)/R\) is always finite dimensional. In contrast, if one blows up at multiple points then \(T(R)/R\) can be infinite dimensional. This implies that naïve blowup algebras can have very nonsymmetric properties. To prove the main theorem, and to describe the maximal right torsion extension of a naïve blowup algebra, the authors pass to a larger class of algebras, called generalized naïve blowup algebras. The last main result of the article shows that the differences between the categories \(qgr-R\) and \(\text{coh}-X\) are quite subtle. The article depends heavily on the article by Keeler, Stafford and Rogalski where the concept of naïve blowup algebras where introduced. The article is clearly important for the study of noncommutative geometry, and introduces a lot of new concepts useful for applications in this field.
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