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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