Some algebras associated to automorphisms of elliptic curves.

*(English)*Zbl 0744.14024
The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. I, Prog. Math. 86, 33-85 (1990).

[For the entire collection see Zbl 0717.00008.]

For \(A\) be a Gorenstein \(k\)-algebra of dimension 3, generated in degree 1, let \(\{x_ i\}\) a set of generators for \(A\), \(\{f_ j\}\) a generating set for the relations. \(A\) is ‘standard’ if, for all \(j\), \(f_ j=\sum m_{ij}x_ i\) implies that also \(\sum x_ jm_{ij}\) is a relation; \(A\) is ‘regular’ if its growth is polynomial. Regular algebras of dimension 3 must have either 3 generators and 3 quadric relations, or 2 generators and 2 cubic relations; an example is the enveloping algebra of the Heisenberg Lie algebra. It is known that all regular algebras are standard; standard algebras with 2 or 3 generators are classified by a finite number of irreducible varieties and the general element of any such variety is regular.

The aim of this paper is the construction of an effective criterion to decide whether a given standard algebra of dimension 3 is regular. The criterion is obtained associating to the algebra \(A\) a map from a projective variety \(E\) to itself; in most cases, \(E\) turns out to be an elliptic curve.

The construction goes as follows. To any relation \(f_ j\) of degree \(n\), one may associate a multilinear function \(f_ j^*\) on the dual of \(A_ 1\) (the degree 1 piece of \(A)\) hence a hypersurfaces in \((\mathbb{P}^{r-1})^ n\) \((r=\dim A_ 1)\); so it is natural to associate to \(A\) the subvariety \({\mathfrak A}\) of \((\mathbb P^{r-1})^ n\) defined by all the \(f_ j^*\)’s. Let \(E\) be the projection of \({\mathfrak A}\) to the first \(n-1\) factors and let \(E'\) be the projection of \({\mathfrak A}\) to the last \(n-1\) factors. \(E\) is either a cubic plane curve or \(E=\mathbb P^ 2\), when \(r=3\); if \(r=2\), then \(E\) is a divisor of type (2,2) in \(\mathbb P^ 1\times\mathbb P^ 1\) or it coincides with \(\mathbb P^ 1\times\mathbb P^ 1\). In any case, \(E\) and \(E'\) are isomorphic and the construction also defines a map \(s:E\to E'\) and an invertible sheaf \(L\) on \(E\), given by the map \(E\to\mathbb P^ 2\) or \(E\to\mathbb P^ 1\). The authors give an explicit description of all triples \((E,s,L)\) which arise from a regular algebra of dimension 3; the description allows to prove the following criterion: a standard 3-dimensional algebra \(A\) is regular if and only if the map \(s:E\to E'\) induced by the previous construction is an isomorphism.

For \(A\) be a Gorenstein \(k\)-algebra of dimension 3, generated in degree 1, let \(\{x_ i\}\) a set of generators for \(A\), \(\{f_ j\}\) a generating set for the relations. \(A\) is ‘standard’ if, for all \(j\), \(f_ j=\sum m_{ij}x_ i\) implies that also \(\sum x_ jm_{ij}\) is a relation; \(A\) is ‘regular’ if its growth is polynomial. Regular algebras of dimension 3 must have either 3 generators and 3 quadric relations, or 2 generators and 2 cubic relations; an example is the enveloping algebra of the Heisenberg Lie algebra. It is known that all regular algebras are standard; standard algebras with 2 or 3 generators are classified by a finite number of irreducible varieties and the general element of any such variety is regular.

The aim of this paper is the construction of an effective criterion to decide whether a given standard algebra of dimension 3 is regular. The criterion is obtained associating to the algebra \(A\) a map from a projective variety \(E\) to itself; in most cases, \(E\) turns out to be an elliptic curve.

The construction goes as follows. To any relation \(f_ j\) of degree \(n\), one may associate a multilinear function \(f_ j^*\) on the dual of \(A_ 1\) (the degree 1 piece of \(A)\) hence a hypersurfaces in \((\mathbb{P}^{r-1})^ n\) \((r=\dim A_ 1)\); so it is natural to associate to \(A\) the subvariety \({\mathfrak A}\) of \((\mathbb P^{r-1})^ n\) defined by all the \(f_ j^*\)’s. Let \(E\) be the projection of \({\mathfrak A}\) to the first \(n-1\) factors and let \(E'\) be the projection of \({\mathfrak A}\) to the last \(n-1\) factors. \(E\) is either a cubic plane curve or \(E=\mathbb P^ 2\), when \(r=3\); if \(r=2\), then \(E\) is a divisor of type (2,2) in \(\mathbb P^ 1\times\mathbb P^ 1\) or it coincides with \(\mathbb P^ 1\times\mathbb P^ 1\). In any case, \(E\) and \(E'\) are isomorphic and the construction also defines a map \(s:E\to E'\) and an invertible sheaf \(L\) on \(E\), given by the map \(E\to\mathbb P^ 2\) or \(E\to\mathbb P^ 1\). The authors give an explicit description of all triples \((E,s,L)\) which arise from a regular algebra of dimension 3; the description allows to prove the following criterion: a standard 3-dimensional algebra \(A\) is regular if and only if the map \(s:E\to E'\) induced by the previous construction is an isomorphism.

Reviewer: Luca Chiantini (Napoli)

##### MSC:

14A22 | Noncommutative algebraic geometry |

14H52 | Elliptic curves |

16E10 | Homological dimension in associative algebras |

16W50 | Graded rings and modules (associative rings and algebras) |