G-dimension over local homomorphisms. Applications to the Frobenius endomorphism
نویسندگان
چکیده
منابع مشابه
G-dimension over Local Homomorphisms. Applications to the Frobenius Endomorphism
We develop a theory of G-dimension over local homomorphisms which encompasses the classical theory of G-dimension for finitely generated modules over local rings. As an application, we prove that a local ring R of characteristic p is Gorenstein if and only if it possesses a nonzero finitely generated module of finite projective dimension that has finite G-dimension when considered as an R-modul...
متن کاملLocal Cohomology and Gorenstein Injective Dimension over Local Homomorphisms
Let φ : (R, m)→ (S, n) be a local homomorphism of commutative noetherian local rings. Suppose that M is a finitely generated S-module. A generalization of Grothendieck’s non-vanishing theorem is proved for M (i.e. the Krull dimension of M over R is the greatest integer i for which the ith local cohomology module of M with respect to m, Hi m(M), is non-zero). It is also proved that the Gorenstei...
متن کاملAlgebraic Dimension over Frobenius Fields
We prove that each perfect Frobenius field is algebraically bounded and hence has a dimension function in the sense of v.d. Dries on the collection of all definable sets. Given a definable set S over Q (resp. Fp) we can effectively determine for each k ∈ {−∞, 0, 1, . . .} whether there exists a perfect Frobenius fieldM of characteristic 0 (resp., of characteristic p) such that the dimension of ...
متن کاملGorenstein Dimension of Modules over Homomorphisms
Given a homomorphism of commutative noetherian rings R → S and an S–module N , it is proved that the Gorenstein flat dimension of N over R, when finite, may be computed locally over S. When, in addition, the homomorphism is local and N is finitely generated over S, the Gorenstein flat dimension equals sup {m ∈ Z | Torm(E,N) 6= 0}, where E is the injective hull of the residue field of R. This re...
متن کاملThe Frobenius endomorphism and homological dimensions
In 1969 Kunz [Ku] proved a fundamental result, connecting the regularity of a local ring of positive characteristic with the flatness of its Frobenius endomorphism φ. This was a first indication of the important role that φ would play in homological commutative algebra, especially in reflecting basic homological properties of the ring. Some results in Peskine and Szpiro’s groundbreaking thesis,...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Illinois Journal of Mathematics
سال: 2004
ISSN: 0019-2082
DOI: 10.1215/ijm/1258136183