Some properties of invariants for Cohen-Macaulay local rings

Let (R,m,k) be a Cohen-Macaulay local ring with canonical module ω. In this note, we study how relations between index(R) and e(R) and between index(R) and gll(R) are preserved when factoring out regular sequences and localizing at prime ideals. In particular, we give conditions for when index(R)=e(R) implies index(R)=index(R¯)=e(R¯) and when index(R)=gll(R) implies index(R)=index(R¯)=gll(R¯), where R¯=R/xR and x∈m is a regular sequence. When R is an abstract hypersurface and k is infinite, we also show that the inequality index(Rp)≤index(R) holds for each unramified prime ideal p∈Spec(R). Finally, we prove the inequalities index(Rp)≤gll(Rp)≤index(Rp)+t−1 for height one prime ideals p∈Spec(R) that satisfy pRp⊆trR(ω)Rp and contain an element in their tth symbolic power p(t) which is regular on ⊕i=0∞p(i)/p(i+1).