## 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.

### MSC:

 14A22 Noncommutative algebraic geometry 14H52 Elliptic curves 16E10 Homological dimension in associative algebras 16W50 Graded rings and modules (associative rings and algebras)

### Biographic References:

Grothendieck, Alexander

Zbl 0717.00008