In [4] and [6], S. Oka showed that Kr is a ring spectrum for r > 1 if r = 0 (mod p) it is called a split ring spectrum since Kr Λ Kr splits into four summands Kr, ΣKr, Σ r«+ιKr, Σ r«+2Kr. If r φ 0 (mod p), it is called a nonsplit ring spectrum since Krl\Kr splits only into three summands Kr, ΣLΛKr, Σ rq+2Kr, where L is the cofibre of 0i = jaie πrq_χS. In the nonsplit case, S. Oka showed in [4] ...

Let n be a positive integer, and let A strongly commutative differential graded (DG) algebra over ring R. Assume that In this paper, we study naïve liftability of DG modules along the natural injection $$A\rightarrow B$$ using notions diagonal ideals homotopy limits. We prove if N is bounded below semifree B-module such $${\text {Ext}}_B ^i (N, N)=0$$ for all $$i\geqslant 1$$ , then naïvely lif...

A module M is said to belifting if, for anysubmodule N of M, there exists a direct summand X contained in such that N/X small M/X. satisfy the finite internal exchange propertyif, any and sum decomposition M=Lni=1Mi, M′i Mi (i= 1,2, . , n) M=X⊕(Lni=1M′i). In this paper, we first give characterizations forthe square hollow uniform be lifting (extending). addition, solve negatively question "Does...

Proposition 1. Assume that r = n − m > 1 and that the first nonvanishing Fitting ideal of M has grade r + 1. Then the following conditions are equivalent. (1) There is a χ ∈ M∗ = HomR(M,R) such that the ideal Imχ has grade r + 1. (2) There exists a submodule U of M with the following properties: (i) rankU = r − 1; (ii) U is reflexive, orientable, and Up is a free direct summand of Mp for all pr...

Let R be a locally finitely generated algebra over a discrete valuation ring V of mixed characteristic. For any of the homological properties, the Direct Summand Theorem, the Monomial Theorem, the Improved New Intersection Theorem, the Vanishing of Maps of Tors and the Hochster-Roberts Theorem, we show that it holds for R and possibly some other data defined over R, provided the residual charac...

We consider possible non-signaling composites of probabilistic models based on euclidean Jordan algebras. Subject to some reasonable constraints, we show that no such composite exists having the exceptional Jordan algebra as a direct summand. We then construct several dagger compact categories of such Jordan-algebraic models. One of these neatly unifies real, complex and quaternionic mixed-stat...

Abstract A module M is called $$\mathfrak {s}$$ s -coseparable if for every nonzero submodule U of such that / finitely generated, there exists a direct summand V $$V \subseteq U$$ V ⊆ U and generated. It shown non-finitely generated fr...