Noncommutative ampleness from finite endomorphisms
A homogeneous coordinate ring \(R\) on a projective integral scheme \(X\) can be made from global sections of \(\mathcal L^{\otimes n}\) with \(\mathcal L\) an ample invertible sheaf. The sheaf \(\mathcal L\) has been used to prove the Serre correspondence Theorem which shows an equivalence of categories between the tails of finitely generated graded \(R\)-modules and coherent sheaves on \(X\). A twisted coordinate ring \(B\) was constructed by \textit{M. Artin} and \textit{M. Van den Bergh} [J. Algebra 133, No. 2, 249--271 (1990; Zbl 0717.14001)] by using an automorphism \(\sigma\) of \(X\) to twist the usual multiplication. This leads to a noncommutative definition of ampleness, and they proved an equivalence of categories between the category of tails of graded (right) \(B\)- modules and the category of coherent sheaves on \(X\). It was not proved by Van den Bergh or Artin that the definition of ampleness is equivalent on the left and the right, but it was proved by the first author [\textit{D. S. Keeler}, J. Am. Math. Soc. 13, No. 3, 517--532 (2000; Zbl 0952.14002)] by studying the behaviour of the automorphism \(\sigma\) on the numerical equivalence classes of divisors. \textit{M. Van den Bergh} [J. Algebra 184, No. 2, 435--490 (1996; Zbl 0876.17011)] then generalized the definitions to the case where \(\sigma\) is a general finite endomorphism. In this article, the authors examine the noncommutative ampleness in the finite endomorphism case. In this case, one never has ampleness on the left, while ampleness on the right is possible: Let \(X\) be a regular projective integral scheme with a finite endomorphism \(\sigma\) and an invertible sheaf \(\mathcal L\). If \(X\) is not an automorphism, the sequence of \(\mathcal O_X\)-bimodules defined by \(\sigma\) and \(\mathcal L\) is not left ample. If \(\mathcal L\) is ample in the commutative sense and \(\sigma^\ast\mathcal L\simeq\mathcal L^{\otimes r}\) for some \(r\in\mathbb Z\), then the sequence is right ample. The authors prove that in case the sequence is right ample, the twisted homogeneous coordinate ring and the bimodule algebra defining the ring are non-noetherian. The article gives the necessary properties of finite endomorphisms and follows up with an investigation of bimodule algebras: The following is verbatim, and is one of the main constructions in the article: Let \(k\) be a field. Let \(\mathcal L\) be a quasi-coherent sheaf on a Noetherian scheme \(X\) over \(S=\mathrm{Spec }k\) and \(\sigma\) a finite \(S\)-homomorphism \(X\rightarrow X\). Let \(\mathcal L_0=\mathcal O_X\). Then for all integers \(j\geq 1\), define \[ \mathcal L_j:=\mathcal L\otimes_{\mathcal O_X}\sigma^\ast\mathcal L\otimes_{\mathcal O_X}\cdots\otimes_{\mathcal O_X}(\sigma^{j-1})^\ast\mathcal L. \] Define \(\mathcal B_0=\mathcal O_X\) and, for all \(j\geq 1\), define \(\mathcal B_j\) to be the bimodule \({}_1(\mathcal L_j)_{\sigma^j}.\) Then let \(\mathcal B=\bigoplus_{j=0}^{\infty}\mathcal B_j\) (alternatively, let \(\mathcal B_j=0\) for all \(j<0\) and let \(\mathcal B=\bigoplus_{j\in\mathbb Z}\mathcal B_j\)). Then let \(B_j=\Gamma(\mathcal B_j)=H^0(X,\mathcal B_j)\) and define \(B=\bigoplus_{n=0}^\infty B_j=\Gamma(\mathcal B).\) The authors prove that this gives an \(\mathbb N\)-graded \(k\)-algebra \(B\) and a graded bimodule algebra \(\mathcal B\). This review should also include the definitions of right and left ampleness: A sequence \((\mathcal B_n)_n\) of coherent \(\mathcal O_X\) bimodules is \textit{left ample} if for any coherent \(\mathcal O_X\)-module \(\mathcal M\), one has that for \(n\gg 0,\;H^q(X,\mathcal B_n\otimes_{\mathcal O_X}\mathcal M)=0\) for \(q>0\). The sequence is \textit{right ample} if for any coherent \(\mathcal O_X\)-module \(\mathcal M\), one has that for \(n\gg 0,\;H^q(X,\mathcal M\otimes_{\mathcal O_X}\mathcal B_n)=0\) for \(q>0\). The article gives an interesting example. The case with \(\sigma\) the relative Frobenius endomorphism on \(X=\mathbb P^m\) is described. More or less verbatim: For \(X=\mathbb P^m_k\) with \(k\) a perfect field of characteristic \(p>0\) and for \(f\) the relative Frobenious morphsims, the twisted homogeneous coordinate ring \(F=\bigoplus\Gamma(\mathcal O(\frac{p^n-1}{p-1}))\) is not finitely generated unless \(p=2\) and \(p=1\). In that case, \(F\) is generated by \(F_1=\Gamma(\mathcal O((1))\), but \(F\) is not Noetherian. As the category equivalences of Artin and Van den Bergh rely on the Noetherianity, a main conclusion of the article is that the equivalences given cannot be directly generalized to the finite endomorphism situation. All in all a nice article, illustrating the techniques and applications of noncommutative algebraic geometry.
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