It is proved that EJ is injective if E is an injective module over a valuation ring R, for each prime ideal J 6= Z. Moreover, if E or Z is flat, then EZ is injective too. It follows that localizations of injective modules over h-local Prüfer domains are injective too. If S is a multiplicative subset of a noetherian ring R, it is well known that SE is injective for each injective R-module E. The...

Throughout this lecture, we let k denote an algebraically closed field, ` a prime number which is invertible in k. In the previous, we define the `-adic cohomology H∗(X; Λ), where X is a quasi-projective k-scheme and Λ ∈ {Z`,Q`,Z/`Z}. Our first goal in this lecture is to review the corresponding theory of `-adic homology. Definition 1. Let Λ be a commutative ring, and let ModΛ denote the∞-categ...

In this paper, we extend the constructions of Boava and Exel to present C∗-algebra associated with an injective endomorphism a group finite cokernel as partial algebra consequently crossed product. With representation, another way study such C∗-algebras, only using tools from products.

We obtain some results about Borel maps with meager fibers on completely metrizable separable spaces. The results are related to a recent dichotomy by Sabok and Zapletal, concerning Borel maps and σ-ideals generated by closed sets. In particular, we give a “classical” proof of this dichotomy. We shall also show that for certain natural σ-ideals I generated by closed sets in compact metrizable s...

LetH be a Hilbert space, L(H) the algebra of all bounded linear operators onH and 〈, 〉A : H ×H → C the bounded sesquilinear form induced by a selfadjoint A ∈ L(H), 〈ξ, η〉A = 〈Aξ, η〉 , ξ, η ∈ H. Given T ∈ L(H), T is A-selfadjoint if AT = T A. If S ⊆ H is a closed subspace, we study the set of A-selfadjoint projections onto S, P(A,S) = {Q ∈ L(H) : Q = Q , R(Q) = S , AQ = QA} for different choices...

It is well-known that a countably injective module is Σ-injective. In Proc. Amer. Math. Soc. 316, 10 (2008), 3461-3466, Beidar, Jain and Srivastava extended it and showed that an injective module M is Σ-injective if and only if each essential extension of M(א0) is a direct sum of injective modules. This paper extends and simplifies this result further and shows that an injective module M is Σ-i...

We prove that if a non-singular planar map $\Lambda \in C^2(\mathbb R^2,\mathbb R^2)$ has convex component, then it is injective. do not assume strict convexity.

‎let $m$ be a non-zero finitely generated module over a commutative noetherian local ring $(r,mathfrak{m})$ with $dim_r(m)=t$‎. ‎let $i$ be an ideal of $r$ with $grade(i,m)=c$‎. ‎in this article we will investigate several natural homomorphisms of local cohomology modules‎. ‎the main purpose of this article is to investigate when the natural homomorphisms $gamma‎: ‎tor^{r}_c(k,h^c_i(m))to kotim...