Syzygies in the theory of invariants

A method is developed for finding all $G$-modules (where $G$ is a connected and simply connected semisimple algebraic group over an algebraically closed field of characteristic zero) whose algebra of invariants has prescribed homological dimension. The main theorem says that the number of such $G$-modules, considered to within isomorphism and addition of a trivial direct summand, is finite. The same result is proved for finite groups $G$. All algebras of invariants of homological dimension $\leqslant10$ of a single binary form are found, as well as all algebras of invariants of a system of binary forms that are hypersurfaces. It is shown that the exceptional simple groups have no irreducible modules with an algebra of invariants of small nonzero homological dimension.
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