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Lars Winther Christensen (Texas Tech University)
6月1日（水）16：30-17：30
大阪府立大学なかもずキャンパスA14棟321教室
Injective dimension of modules over Noetherian rings
Let R be a commutative ring. For those who think in terms of cohomology,
Baer’s criterion says that an R-module M has injective dimension at most n ≥ 0
if Ext^{n+1}R(R/a, M) vanishes for every ideal a in R. When R is also noetherian, it
suffices to test against prime ideals; that is, inj.dim_R M ≤ n holds if and only if
Ext^{n+1}_R 1(R/p, M) = 0 for every prime ideal p in R . Further, by the existence of minimal injective resolutions, one can even test locally;
i.e. inj.dim_R M ≤ n holds if and only if one has Ext^{n+1}{Rp}(k(p), Mp) = 0 for every prime ideal p in R .Here k(p) denotes the field Rp/pRp. The result I will report on says that injectivity of an R-module can be detected by vanishing of Ext globally—over R—against these fields. Among other consequences, this result provides a shortcut to cosupport theory in the derived category. The talk is based on joint work with Srikanth B.Iyengar.