Multiplicativity and nonrealizable equivariant chain complexes

Automorphisms of the Koszul homology of a local ring

“…The homology of K, denoted H(K), has a structure of a graded-commutative k-algebra, where k is the residue field of R. It is by know well understood that the structure of K and of H(K) capture interesting information about the ring R. So it is natural to study dg R-algebra automorphisms of K, and the automorphisms of H(K) induced by them. But in fact we were led to study these because of recent work [RS19] of the second and third authors on transformation groups, answering a question of Walker and the first author [IW18]. In [RS19] the problem arose of lifting automorphisms of the ring R to those of H(K); it was proved that this is possible when R is the group algebra over F p of an elementary abelian p-group.…”

“…But in fact we were led to study these because of recent work [RS19] of the second and third authors on transformation groups, answering a question of Walker and the first author [IW18]. In [RS19] the problem arose of lifting automorphisms of the ring R to those of H(K); it was proved that this is possible when R is the group algebra over F p of an elementary abelian p-group. As explained in Remark 2.3, the obstructions to lifting are precisely automorphisms of the dg R-algebra K that are nontrivial on H(K).…”