2018/04/12 by Helmut Zöschinger, Zöschinger, Helmut
Mathematics · #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #Rings, Modules, and Algebras #math.AC #math.RA #msc:13C05 #msc:13C11 #msc:16L60 #msc:16S90
paper · pdf · doi:10.48550/arxiv.1804.04551
in German
arxiv created 2018/04/12 · arxiv updated 2018/04/13
Let (R, \mathfrak m) be a commutative noetherian local ring and I an ideal of R. For every R-module M, γI(M) = ∑\ Bi f | f ∈ HomR(I,M)\ is called the trace of I in M. It is easy to see that ExtR1(R/I,M) = 0 always implies IM = γI(M). If the second condition holds for all ideals I of R, we say that M is excellent. In part 1, we show a number of conditions for these modules, which are well-known for injective modules. In the second part, we examine the special case M = R. In particular, we show that for every prime ideal \mathfrakp the equality \mathfrakp = γ_\mathfrakp(R) holds iff R_\mathfrakp is not a discrete valuation ring. From the results by Matlis (1973) about 1-dimensional local CM-rings and with the help of the first neighborhood ring Λ, it follows immediately that γ_\mathfrakmn (R) = Λ-1 for almost all n ≥ 1. In the third part, we examine the dual construction κI(M) = \bigcap \ Ke f | f∈ HomR(M,I^∘) \ and reduce the main results about Tor1R(M, R/I) = 0 and κI(M) = M[I] to part 1 by considering the Matlis dual M^∘ = HomR(M, E) and the equalities γI(M^∘) = AnnM^∘(κI(M)), κI(M^∘) = AnnM^∘(γI(M)).