Let $\mathfrak{a}$ be an ideal of a commutative noetherian ring $R$ and $M$ $R$-module with Cosupport in $\mathrm{V}(\mathfrak{a})$. We show that is $\mathfrak{a}$-coartinian if only $\mathrm{Ext}_{R}^{i}(R/\mathfrak{a},M)$ artinian for all $0\leq i\leq \mathrm{cd}(\mathfrak{a},M)$, which provides computable finitely many steps to examine $\mathfrak{a}$-coartinianness. also consider the duality...

In this paper we study the vanishing and non-vanishing of generalized local cohomology and generalized local homology. In particular for a Noetherian local ring (R,m) and two non-zero finitely generated R-modules M and N , it is shown that H m (M,N) 6= 0.

We study the category of discrete modules over the ring of degree zero stable operations in p-local complex K-theory. We show that the K(p)-homology of any space or spectrum is such a module, and that this category is isomorphic to a category defined by Bousfield and used in his work on the K(p)-local stable homotopy category [2]. We also provide an alternative characterisation of discrete modu...

Considering modules of finite complete intersection dimension over commutative Noetherian local rings, we prove (co)homology vanishing results in which we assume the vanishing of nonconsecutive (co)homology groups. In fact, the (co)homology groups assumed to vanish may be arbitrarily far apart

For a commutative Noetherian local ring we define and study the class of modules having reducible complexity, a class containing all modules of finite complete intersection dimension. Various properties of this class of modules are given, together with results on the vanishing of homology and cohomology.

In this paper, we develop the rudiments of a tropical homology theory. We use language “triples” and “systems” to simultaneously treat structures from various approaches mathematics, including semirings, hyperfields, super algebra. enrich algebraic with negation map where it does not exist naturally. obtain an analogue Schanuel’s lemma which allows us talk about projective dimension modules in ...