Interplay between homological dimensions of a complex and its right derived section

Author name not available (Why is that?)

Publication date: 16 November 2017

Abstract: Let

(R, m a t h f r a k m)

be a commutative Noetherian local ring,

m a t h f r a k a

be a proper ideal of

R

and

M

be an

R

-complex in

m a t h r m D (R)

. We prove that if

M i n m a t h r m D_{s}^{f} q s u b s e t (R)

(respectively,

M i n m a t h r m D_{s}^{f} q s u p s e t (R)

), then

m a t h r m {i d}_{R} m a t h b f R G a m m a_{m a t h f r a k a} (M) = m a t h r m {i d}_{R} M

(respectively,

m a t h r m {f d}_{R} m a t h b f R G a m m a_{m a t h f r a k a} (M) = m a t h r m {f d}_{R} M

). Next, it is proved that the right derived section functor of a complex

M i n m a t h r m D_{s} q s u b s e t (R)

(

R

is not necessarily local) can be computed via a genuine left-bounded complex

G s i m e q M

of Gorenstein injective modules. We show that if

R

has a dualizing complex and

M

is an

R

-complex in

m a t h r m D_{s}^{f} q u a r e (R)

, then

m a t h r m {G f d}_{R} m a t h b f R G a m m a_{m a t h f r a k a} (M) = m a t h r m {G f d}_{R} M

and

m a t h r m {G i d}_{R} m a t h b f R G a m m a_{m a t h f r a k a} (M) = m a t h r m {G i d}_{R} M

. Also, we show that if

M

is a relative Cohen-Macaulay

R

-module with respect to

m a t h f r a k a

(respectively, Cohen-Macaulay

R

-module of dimension

n

), then

m a t h r m {G f d}_{R} m a t h b f H_{m a t h f r a k a}^{m a t h r m h t_{M} m a t h f r a k a} (M) = m a t h r m {G f d}_{R} M + n

(respectively,

m a t h r m {G i d}_{R} m a t h b f H_{m a t h f r a k m}^{n} (M) = m a t h r m {G i d}_{R} M - n

). The above results generalize some known results and provide characterizations of Gorenstein rings.

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