Complete Intersection Hom Injective Dimension
- PDF / 486,845 Bytes
- 19 Pages / 439.642 x 666.49 pts Page_size
- 89 Downloads / 150 Views
Complete Intersection Hom Injective Dimension Sean K. Sather-Wagstaff1 · Jonathan P. Totushek2 Received: 28 February 2019 / Accepted: 19 November 2019 / © Springer Nature B.V. 2020
Abstract We introduce and investigate a new injective version of the complete intersection dimension of Avramov, Gasharov, and Peeva. It is like the complete intersection injective dimension of Sahandi, Sharif, and Yassemi in that it is built using quasi-deformations. Ours is different, however, in that we use a Hom functor in place of a tensor product. We show that (a) this invariant characterizes the complete intersection property for local rings, (b) it fits between the classical injective dimension and the G-injective dimension of Enochs and Jenda, (c) it provides modules with Bass numbers that are bounded by polynomials, and (d) it improves a theorem of Peskine, Szpiro, and Roberts (Bass’ conjecture). Keywords Bass conjecture · Bass numbers · Complete intersection dimensions · Injective dimension Mathematics Subject Classification (2010) Primary 13D05 · Secondary 13C11 · 13D09 · 13D22
1 Introduction Convention In Sections 1–5, let (R, m, k) be a local ring. It is well known that R-modules with finite projective dimension are particularly nice, behaving like modules over regular local rings. Furthermore, modules of finite projective dimension have the ability to detect when the ring is regular according to the famous result of Auslander, Buchsbaum, and Serre [1, 33]. The injective dimension has similar behavior.
Presented by: Peter Littelmann Jonathan P. Totushek
[email protected] http://mcs-web.uwsuper.edu/jtotushe Sean K. Sather-Wagstaff [email protected] https://ssather.people.clemson.edu/ 1
Department of Mathematical Sciences, Clemson University, O-110 Martin Hall, Box 340975, Clemson, SC 29634, USA
2
Mathematics and Computer Science Department, University of Wisconsin-Superior, Swenson Hall 3030, Belknap and Catlin Ave, P.O. Box 2000, Superior, WI 54880, USA
S. K. Sather-Wagstaff, J. P. Totushek
The complete intersection dimension of Avramov, Gasharov and Peeva [5] similarly identifies modules modules that behave like modules over over formal complete intersection is of the rings. (The local ring R is a formal complete intersection if its m-adic completion R ∼ form R Q/a where Q is a regular local ring and a is generated by a Q-regular sequence.) = For instance, the Betti numbers of such modules are bounded above by a polynomial, and these modules have the ability to detect when the ring is a formal complete intersection. The complete intersection dimension is defined in terms of quasi-deformations of R, ϕ τ → R ← − Q such that ϕ is flat, and τ which are diagrams of local ring homomorphisms R − is surjective with kernel generated by a Q-regular sequence. One then defines the complete intersection dimension of a finitely generated R-module M as CI-dimR (M) = inf{pdQ (M ) − pdQ (R ) | R → R ← Q is a quasi-deformation} where M = R ⊗R M. Sahandi, Sharif, and Yassemi [28] define the complete intersection f
Data Loading...
