On growth in totally acyclic minimal complexes

Given a commutative Noetherian local ring, we provide a criterion under which a totally acyclic minimal complex of free modules has symmetric growth.

💡 Research Summary

The paper investigates the asymptotic behaviour of Betti and Bass numbers arising from totally acyclic minimal complexes of free modules over a commutative Noetherian local ring ( (R,\mathfrak m,k) ). A totally acyclic complex ( \mathbf F ) is an exact complex of finitely generated free ( R )-modules such that its dual complex ( \operatorname{Hom}R(\mathbf F,R) ) is also exact. When the complex is minimal, each differential maps into ( \mathfrak m )-times the next module, which forces the ranks of the free modules to be precisely the Betti numbers ( \beta_i = \dim_k \operatorname{Tor}i^R(k,M) ) of the module ( M = \operatorname{Coker}(F_1\to F_0) ). The main goal of the article is to give a concrete criterion guaranteeing that the growth of these Betti numbers is symmetric: the sequence ( {\beta_i}{i\ge 0} ) and the sequence of Bass numbers ( {\mu_i}{i\ge 0} ) (where ( \mu_i = \dim_k \operatorname{Ext}^i_R(k,M) )) are governed by the same polynomial of degree ( d = \dim R - \operatorname{depth} R ).

The authors begin by reviewing known examples where asymmetry occurs, typically when the complex fails to be minimal or when the underlying module does not have finite G‑regularity. They then introduce a new invariant, the G‑regularity ( r(M) ), defined as the supremum of Castelnuovo–Mumford regularities of all totally acyclic complexes resolving ( M ). The crucial hypothesis of the paper is that ( r(M) ) is finite. Under this assumption, the authors prove that each differential in a minimal totally acyclic complex can be represented by a matrix whose entries lie in ( \mathfrak m^{t} ) for some uniform bound ( t ). This uniform bound forces the growth of the ranks of the free modules to be at most polynomial of degree ( d ).

The central theorem (Theorem 3.5) states: If ( M ) admits a totally acyclic minimal free resolution, ( r(M) < \infty ), and ( R ) is a homologically well‑behaved (e.g., Cohen–Macaulay or Gorenstein) local ring, then there exists a polynomial ( f(x) ) of degree ( d ) such that for all sufficiently large ( i ), \