Tilting modules over almost perfect domains

Abstract

a b s t r a c tWe provide a complete classification of all tilting modules and tilting classes over almost perfect domains, which generalizes the classifications of tilting modules and tilting classes over Dedekind and 1-Gorenstein domains. Assuming the APD is Noetherian, a complete classification of all cotilting modules is obtained (as duals of the tilting ones).

One-tilting classes and modules over commutative rings

“…The classification of tilting classes and modules was done gradually, starting with abelian groups ([GT00]), then small Dedekind domains, first assuming V=L ( [TW02], [TW03]), and then in ZFC ( [BET05]), for Prüfer domains ( [Baz07]), and almost perfect domains ( [AJ11]). Recently, in [AHP ŠT14] the authors classified tilting classes of a commutative noetherian ring in terms of finite sequences of subsets of the Zariski spectrum of R. In particular, they proved that 1-tilting classes correspond bijectively to specialization closed subsets of Spec(R) that do not contain associated primes of R. We generalize this result to arbitrary commutative rings by showing that there is a one-to-one correspondence between 1-tilting classes and Thomason subsets of Spec(R) that avoid primes "associated" to R in certain sense.…”

Tilting for regular rings of Krull dimension two

“…This enables classification of tilting modules and classes over Dedekind domains [6], and is essential in extending this classification in various directions: to Prüfer domains [24], almost perfect domains [1], and Gorenstein rings of Krull dimension one [25]. However, commutative noetherian rings of Krull dimension 2 are known to be finlen-wild [21].…”

Divisibility classes are seldom closed under flat covers

“…Finally, by [1,Proposition 3.2], any R-module of finite projective dimension has projective dimension at most 1. Therefore, pd R/J = 1, and thus J is actually projective, and hence also finitely generated.…”

“…(i) R has finite character, that is, each non-zero element belongs to only finitely many maximal ideals of R, and 1 Here, "T -preenvelope" is a shorthand for "codomain of the T -preenvelope", which we will use throughout the paper. Also, note that "flat T -preenvelope" is not to be confused with (T ∩ F 0 )preenvelope.…”