Narrow Results

The notion of κ-tameness of a pseudovariety was introduced by Almeida and Steinberg and is a strong property which implies decidability of pseudovarieties. In this paper we prove that the pseudovariety LSl, of local semilattices, is κ-tame.

We combine two notions in AECs, tameness and good λ-frames, and show that they together give a very well-behaved nonforking notion in all cardinalities. This helps to fill a longstanding gap in classification theory of tame AECs and increases the applicability of frames. Along the way, we prove a complete stability transfer theorem and uniqueness of limit models in these AECs.

We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness (a locality property for orbital types), and are stable (in terms of the number of orbital types) in some cardinal. Assuming the singular cardinal hypothesis (SCH), we prove a full characterization of the (high-enough) stability cardinals, and connect the stability spectrum with the behavior of saturated models.

We deduce (in ZFC) that if a class is stable on a tail of cardinals, then it has no long splitting chains (the converse is known). This indicates that there is a clear notion of superstability in this framework.

We also present an application to homogeneous model theory: for $D$ a homogeneous diagram in a first-order theory $T$, if $D$ is both stable in $\left|T\right|$ and categorical in $\left|T\right|$ then $D$ is stable in all $\lambda \ge \left|T\right|$.

We provide comprehensive, level-by-level characterizations of large cardinals, in the range from weakly compact to strongly compact, by closure properties of powerful images of accessible functors. In the process, we show that these properties are also equivalent to various forms of tameness for abstract elementary classes. This systematizes and extends results of [W. Boney and S. Unger, Large cardinal axioms from tameness in AECs, Proc. Amer. Math. Soc.145(10) (2017) 4517–4532; A. Brooke-Taylor and J. Rosický, Accessible images revisited, Proc. AMS145(3) (2016) 1317–1327; M. Lieberman, A category-theoretic characterization of almost measurable cardinals (Submitted, 2018), http://arxiv.org/abs/1809.06963; M. Lieberman and J. Rosický, Classification theory for accessible categories. J. Symbolic Logic81(1) (2016) 1647–1648].

Let R = ⨁_j≥0 R_j be a homogeneous Noetherian ring with irrelevant ideal R₊ = ⨁_j≥1 R_j. Let M, N be two finitely generated graded R-modules. Several results on the Artinianness, tameness and asymptotic property of the graded R-modules will be investigated.

Let R=⊕_{n ≥ 0} R_n be a standard graded ring, 𝔞 ⊇ ⊕_{n > 0} R_n an ideal of R, and M, N two finitely generated graded R-modules. This paper studies the homogeneous components of graded generalized local cohomology modules. We show that for any i ≥ 0, the n-th graded component of the i-th generalized local cohomology module of M and N with respect to 𝔞 vanishes for all n ≫ 0. Some sufficient conditions are proposed to satisfy the equality . Also, some sufficient conditions are proposed for the tameness of such that or i=cd_𝔞(M,N), where and cd_𝔞(M,N) denote the R₊-finiteness dimension and the cohomological dimension of M and N with respect to 𝔞, respectively. Finally, we consider the Artinian property of some submodules and quotient modules of , where j is the first or last non-minimax level of .

The first decade of the 2000's has seen remarkable progress in the theory of hyperbolic 3-manifolds. We report on some of these developments.