TY - JOUR AU - Sather-Wagstaff, Sean PY - 2012/03/01 Y2 - 2024/03/28 TI - Lower bounds for the number of semidualizing complexes over a local ring JF - MATHEMATICA SCANDINAVICA JA - Math. Scand. VL - 110 IS - 1 SE - Articles DO - 10.7146/math.scand.a-15192 UR - https://www.mscand.dk/article/view/15192 SP - 5-17 AB - We investigate the set $\mathfrak(R)$ of shift-isomorphism classes of semi-dualizing $R$-complexes, ordered via the reflexivity relation, where $R$ is a commutative noetherian local ring. Specifically, we study the question of whether $\mathfrak(R)$ has cardinality $2^n$ for some $n$. We show that, if there is a chain of length $n$ in $\mathfrak(R)$ and if the reflexivity ordering on $\mathfrak (R)$ is transitive, then $\mathfrak(R)$ has cardinality at least $2^n$, and we explicitly describe some of its order-structure. We also show that, given a local ring homomorphism $\varphi\colon R\to S$ of finite flat dimension, if $R$ and $S$ admit dualizing complexes and if $\varphi$ is not Gorenstein, then the cardinality of $\mathfrak (S)$ is at least twice the cardinality of $\mathfrak (R)$. ER -