Some results on the annihilators and attached primes of local cohomology modules

Let ((R, mathfrak {m})) be a local ring and M a finitely generated R-module. It is shown that if M is relative Cohen–Macaulay with respect to an ideal (mathfrak {a}) of R, then ({text {Ann}}_R(H_{mathfrak {a}}^{{text {cd}}(mathfrak {a}, M)}(M))={text {Ann}}_RM/L={text {Ann}}_RM) and ({text {Ass}}_R (R/{text {Ann}}_RM)subseteq {mathfrak {p}in {text {Ass}}_R M|,mathrm{cd}(mathfrak {a}, R/mathfrak {p})={text {cd}}(mathfrak {a}, M)},) where L is the largest submodule of M such that (mathrm{cd}(mathfrak {a}, L)< mathrm{cd}(mathfrak {a}, M)). We also show that if (H^{dim M}_{mathfrak {a}}(M)=0), then ({text {Att}}_R(H^{dim M-1}_{mathfrak {a}}(M))= {mathfrak {p}in {text {Supp}}(M)|mathrm{cd}(mathfrak {a}, R/mathfrak {p})=dim M-1},) and so the attached primes of (H^{dim M-1}_{mathfrak {a}}(M)) depend only on ({text {Supp}}(M)). Finally, we prove that if M is an arbitrary module (not necessarily finitely generated) over a Noetherian ring R with (mathrm{cd}(mathfrak {a}, M)=mathrm{cd}(mathfrak {a}, R/{text {Ann}}_RM)), then ({text {Att}}_R(H^{mathrm{cd}(mathfrak {a}, M)}_{mathfrak {a}}(M))subseteq {mathfrak {p}in {text {V}}({text {Ann}}_RM)|,mathrm{cd}(mathfrak {a}, R/mathfrak {p})=mathrm{cd}(mathfrak {a}, M)}.) As a consequence of this, it is shown that if (dim M=dim R), then ({text {Att}}_R(H^{dim M}_{mathfrak {a}}(M))subseteq {mathfrak {p}in {text {Ass}}_R M|mathrm{cd}(mathfrak {a}, R/mathfrak {p})=dim M}).