Noncommutative blowups of elliptic algebras
The article studies the noncommutative analogue of the ordinary blowing up in (commutative) algebraic geometry, and it chooses the graded rings approach to noncommutative projective algebraic geometry. The authors show that there is a noncommutative analogue of the blowing-up process, and they give a lot of properties for the resulting algebras. Let \(\mathbb K\) be an algebraically closed ground field and let \(T=\mathbb K\oplus T_1\oplus T_2\oplus\dots\) be a connected graded (cg) domain. \(T\) is called an \textit{elliptic algebra} if there exists a central element \(g\in T_1\) s.t. \(T/gT\) is a \textit{twisted homogeneous coordinate ring} (TCR) \(B(E,\mathcal M,\tau)\) on an elliptic curve \(E\) with \(\mathcal M\) ample and \(\tau\in\text{Aut}(E)\) of infinite order. A TCR is \(B(E,\mathcal M,\tau)=\bigoplus_{n\geq 0}H^0(E,\mathcal L_n)\), \(\mathcal L_n=\mathcal L\oplus\mathcal L^\tau\otimes\cdots\otimes\mathcal L^{\tau^{n-1}}\), and multiplication given by \(f\star g=f\otimes(\tau^m)^\ast(g)\) for \(f\in B_m\) and \(g\in B_n\). The elliptic algebras are the noncommutative analogues of anti-canonical coordinate rings of projective (Fano- or del Pezzo) surfaces. The degree of \(T\) is defined to be \(\deg T=\mu=\deg\mathcal M=\dim T_1-1\). When \(R\) is a noetherian graded domain, its \textit{graded quotient ring} \(Q_{gr}(R)\) is given by inverting all nonzero homogeneous elements. \(R\) is called a \textit{maximal order} of there is no order \(R\subsetneqq R^\prime\subseteq Q_{gr}(R)\) with \(a R^\prime b\subseteq R\) for nonzero \(a,b\in R\). (\(S^{(r)}\) denotes the homogeneous elements of degree \(r\)). The main result in the article is that one can blow up \(T\) in divisors on the elliptic curve \(E\) to obtain new elliptic algebras: Let \(\mathbf d\) be an effective divisor on \(E\) with \(\deg\mathbf d\leq\mu\). Then \(T(\mathbf d)\) is a graded subalgebra of \(T\) with properties given in the main theorem as follows. With \(T,\mu,\mathbf d\) as above, \(d=\deg\mathbf d\), \(R=T(\mathbf d)\). Then (1) \(R\) is an elliptic algebra of degree \(\mu-d\), with \(R/gR\simeq B(E,\mathcal M(-\mathbf d),\tau).\) As such, \(R\) is strongly noetherian, Auslander-Gorenstein, Cohen-Macaulay, and a maximal order. \(R\) satisfies the Artin-Zhang \(\chi\) conditions and possesses a balanced dualizing complex. (2) \(R\) has Hilbert series \(h_R(t)=\sum(\dim_{\mathbb K}R_i)t^i=h_T(t)-d/(1-t)^3\), and \(Q_{gr}(R)=Q_{gr}(T)\). (3) \(R(\mathbf d)\) is generated in degree \(1\) if \(\mu-d\geq 2\). If \(\mu-d=1\), then \(R\) is generated in degrees \(\leq 2\), and \(R^{(2)}\) is generated in degree \(1\). (4) \(T(\mathbf d)\) can be obtained by a series of one-point blowups. For one-point blowups, the authors prove that if \(T\) is as above, with \(\deg T=\mu>1\), \(p\in E\), \(R=T(p)\), then there is a finitely generated right \(R\)-module \(L\) so that \(T/R\simeq\bigoplus_{i\geq 1}L[-i]\) as right \(R\)-modules. Further, \(L\) is \(g\)-torsion-free and has Hilbert series \(h_L(t)=\frac{1}{(1-t)^2}.\) The article concentrates in particular on blowups of Veronese subalgebras of generic Sklyanin algebras: Let \(S\) be a generic cubic or quadratic Sklyanin algebra with central element \(g\in S_\gamma.\) Letting \(T=S^{(\gamma)}\), \(\mu=\deg T\), let \(\mathbf d\) be an effective divisor on \(E\) with \(\deg\mathbf d<\mu\). Then \(T(\mathbf d)\) has a minimal sporadic ideal \(I\) , meaning that \(T/I\) has GK-dimension \(\leq 1\), and for any graded ideal \(J\subseteq T\) with \(\text{GK}\dim T/J=1,\) \(J_n\supseteq I_n\) for \(n\gg 0.\) The motivation of the article is to answer the question if one can classify all maximal orders \(R\) inside a generic (quadratic) Sklyanin algebra \(S\) with \(Q_{gr}(R)=Q_{gr}(S^{(n)})\) for some \(n\). Here a maximal order is the noncommutative version of the integral closure of an algebra, and the question corresponds to the classification of such inside its quotient field. An application of this theory, is classification of noncommutative surfaces. Here the blowups and exceptional curves are essential. The authors study a special class of rings. An \(\mathbb N\)-graded \(\mathbb K\)-algebra \(R\) is \textit{connected graded} (cg) if \(R_0=\mathbb K \) and \(\dim_k R_n<\infty\) for all \(n\). If nothing else is said, the following hypothesis is supposed valid: \(T\) is a cg \(\mathbb K\)-algebra which is a domain with a central element \(g\in T_1\) such that \(T/gT\simeq B=B(E,\mathcal M,\tau)\) for an elliptic curve \(E\), invertible sheaf \(\mathcal M\) of degree \(\mu\geq 2\) and infinite order automorphism \(\tau\). Then \(T\) is finitely generated in degree 1. Given \(X\subseteq T\) the authors write \(\overline X=(X+gt)/gT\subseteq B(E,\mathcal M,\tau).\) The article contains the basic terminology and results, making it sufficiently self contained. A categorical equivalence giving a formally useful way to describe and work with cyclic \(T\)-modules of GK-dimension 1 and their defining ideals is given. This is used to associate some special right ideals on a ring \(T\) to lists of divisors on \(E\), defined as \textit{allowable divisor layering}. Each divisor in this layering tracks which points that occur in the \(j^{\text{th}}\) layer \(Mg^j/Mg^{j+1}\) of the module \(M=T/J(\mathbf d^\bullet)\). The article considers certain functors on \(X=\text{qgr}-T\) and their actions on the ideals \(\mathcal I(\mathbf d^\bullet)\). Certain right ideals are studied by means of their Hilbert series. The algebras \(T(\mathbf d)\) corresponding to blowing up \(T\) at the points on an effective divisor \(\mathbf d\) are given with their properties, and left and right ideals are compared and proven to have good relations. The authors construct the exceptional line modules of a one-point blowup \(T(p)\subseteq T\) as part of the main result. Finally, the concept of \textit{sporadic ideals} is consider as the original version of divisors of a module. The article is explicit and detailed and it contains important results on noncommutative projective schemes which is essential for the development of the field.
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