We develop a theory of G-dimension for modules over local homomorphisms which encompasses the classical theory of G-dimension for finite modules over local rings. As an application, we prove that a local ring R of characteristic p is Gorenstein if and only if it pos- sesses a nonzero finite module of finite projective dimension that has finite G-dimension when considered as an R-module via some power of the Frobenius endomorphism of R. We also prove results that track the behavior of Gorenstein properties of local homomorphisms under (de)composition.

/pdf/g-dimension-over-local-homomorphisms-applications-to-the-3o893lu3b3.pdf

G-dimension over local homomorphisms. Applications to the Frobenius endomorphism

New homotopy invariant finiteness conditions on modules over commutative rings are introduced, and their properties are studied systematically. A number of finiteness results for classical homological invariants like flat dimension, injective dimension, and Gorenstein dimension, are established. It is proved that these specialize to give results concerning modules over complete intersection local rings. A noteworthy feature is the use of techniques based on thick subcategories of derived categories.

https://ems.press/content/serial-article-files/1123

Finiteness in derived categories of local rings

The notions of Betti numbers and of Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S , for some local homomorphism φ: R → S . Various techniques are developed to study the new invariants and to establish their basic properties. In some cases they are computed in closed form. Applications go in several directions. One is to identify new classes of finite R -modules whose classical Betti numbers or Bass numbers have extremal growth. Another is to transfer ring theoretical properties between R and S in situations where S may have infinite flat dimension over R . A third is to obtain criteria for a ring equipped with a "contracting" endomorphism—such as the Frobenius endomorphism—to be regular or complete intersection; these results represent broad generalizations of Kunz's characterization of regularity in prime characteristic.

/pdf/homology-over-local-homomorphisms-qhwzi0zhwk.pdf

Homology over local homomorphisms

The notions of Betti numbers and of Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S, for some local homomorphism f: R --> S. Various techniques are developed to study the new invariants and to establish their basic properties. In several cases they are computed in closed form. Applications go in several directions. One is to identify new classes of finite R-modules whose classical Betti numbers or Bass numbers have extremal growth. Another is to transfer ring theoretical properties between R and S in situations where S may have infinite flat dimension over R. A third is to obtain criteria for a ring equipped with a `contracting' endomorphism -- such as the Frobenius endomorphism -- to be regular or complete intersection; these results represent broad generalizations of Kunz's characterization of regularity in prime characteristic.

Assume that ϕ(R, m, k) → (S, n, l) is a local homomorphism between commutative noetherian local rings R and S. We say that an S-module M is almost finite over R if it is finitely generated over S (the R-structure on M is induced by ϕ). We investigate the homological behaviour of such modules, as well as various properties of the rings R and S in the presence of an almost finite module of finite flat dimension over R.

Almost finite modules

For a complex X of modules over a commutative ring R the weak annihilator is defined by AnnRX  T i2ZAnnRHiX, the intersection of the annihilators of the homology modules, homotopy annihilator hannRX as the kernel of the map R! H0RHomRX ;X, and when X is homologically bounded, say HiX  0 for jij > n, the small annihilator is annRX  AnnRHÿnX AnnRHnX, the product of the annihilators of the homology modules. Various properties of annihilators are investigated; in particular it is proved that for suitably bounded complexes X and Y the homotopy annihilator hannRX is contained in hannRRHomRX ;Y and hannRX R Y.

/pdf/annihilating-complexes-of-modules-3grgdn5yzw.pdf

Annihilating complexes of modules

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Almost finite modules

Annihilating complexes of modules