4.3. (i) Since HomA(A (I),M) ∼= M (I) for a set I and ⊕ is exact, it is clear that HomA(A (I),−) is exact, i.e. A(I) is projective. (ii) “Only if” part: Let A(P ) = ⊕ p∈P Ap be the free A-module indexed by P and ψ : A (P ) → p be the A-homomorphism such that ψ(1p) = p. Then ψ is surjective. If P is projective, then ψ has a section, i.e. an A-homomorphism φ : P → A(P ) such that ψ ◦ φ = idP . Th...

We show, in full generality, that Lusztig’s a-function describes the projective dimension of both indecomposable tilting modules and indecomposable injective modules in the regular block of the BGG category O, proving a conjecture from the first paper. On the way we show that the images of simple modules under projective functors can be represented in the derived category by linear complexes of...

Let C be a binary self-dual code with an automorphism g of order 2p, where p is an odd prime, such that g is a xed point free involution. If C is extremal of length a multiple of 24 all the involutions are xed point free, except the Golay Code and eventually putative codes of length 120. Connecting module theoretical properties of a self-dual code C with coding theoretical ones of the subcode C...

4.3. (i) Since HomA(A (I),M) ∼= M (I) for a set I and ⊕ is exact, it is clear that HomA(A (I),−) is exact, i.e. A(I) is projective. (ii) “Only if” part: Let A(P ) = ⊕ p∈P Ap be the free A-module indexed by P and ψ : A (P ) → p be the A-homomorphism such that ψ(1p) = p. Then ψ is surjective. If P is projective, then ψ has a section, i.e. an A-homomorphism φ : P → A(P ) such that ψ ◦ φ = idP . Th...

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...

Let R be a ring. A right R-module is said to C-flat if the kernel of any epimorphism B → C-pure in B, i.e. induced map Hom(C,B) Hom(C,A) surjective for cyclic C. Projective modules are and weakly-flat neat-flat. In this article, it discussed connections between C-flat, weakly-flat, neat-flat singly flat modules. It shown that coincide with singly-projective over arbitrary rings. Next, several c...

Let M be a finitely generated module over an Artin algebra. By considering the lengths of each module in the minimal projective resolution of M , we obtain its Betti sequence. This sequence must be bounded if M is eventually periodic, but the converse fails to hold in general. We give conditions under which this holds, using techniques from Hochschild cohomology. We also provide a result which ...

Let $mathcal{X}$ be a class of $R$-modules‎. ‎In this paper‎, ‎we investigate ;$mathcal{X}$-injective (projective) and DG-$mathcal{X}$-injective (projective) complexes which are generalizations of injective (projective) and DG-injecti‎‎ve (projective) complexes‎. ‎We prove that some known results can be extended to the class of ;$mathcal{X}$-injective (projective) and DG-$mathcal{X}$-injective ...

Projective modules are a link between geometry and algebra as established by the theorem of Serre-Swan. In this paper, we define super analog projective explore in case some particular geometric objects. We consider tangent bundle over supersphere show that module vector field is ring supersmooth functions. Also, discuss class can be constructed from projection map on defined functions supersph...