In terms of the duality property of injective preenvelopes and flat precovers, we get an equivalent characterization of left Noetherian rings. For a left and right Noetherian ring R, we prove that the flat dimension of the injective envelope of any (Gorenstein) flat left R-module is at most the flat dimension of the injective envelope of RR. Then we get that the injective envelope of RR is (Gor...

Let (R,m, k) be a noetherian local ring. It is well-known that R is regular if and only if the injective dimension of k is finite. In this paper it is shown that R is Gorenstein if and only if the Gorenstein injective dimension of k is finite. On the other hand a generalized version of the so-called Bass formula is proved for finitely generated modules of finite Gorenstein injective dimension. ...

Gillespie posed two questions in [Front. Math. China 12 (2017) 97-115], one of which states that “for what rings R do we have K(AC)=K(R-Inj)?”. We give an answer to such a question. As applications, obtain new homological approach unifies some well-known conditions Krause’s recollement holds, and example show there exists Gorenstein injective module is not AC-injective. also improve Neeman’s an...

An immediate generalization of the classical McKay correspondence for Gorenstein quotient spaces C/G in dimensions r ≥ 4 would primarily demand the existence of projective, crepant, full desingularizations. Since this is not always possible, it is natural to ask about special classes of such quotient spaces which would satisfy the above property. In this paper we give explicit necessary and suf...

We consider the question of simplicity a ring $R$ under action its differential operators $D_R$. give examples to show that even when is Gorenstein and has rational singularities need not be simple $D_R$-module; for example, this case homogeneous coordinate smooth cubic surface. Our are rings Fano varieties, our proof proceeds by showing tangent bundle such variety big. also partial converse pr...

In [Ru2] we have classified the smooth projective symmetric G-varieties with Picard number one (and G semisimple). In this work we give a geometrical description of such varieties. In particular, we determine their group of automorphisms. When this group, Aut(X), acts non-transitively on X, we describe a G-equivariant embedding of the variety X in a homogeneous variety (with respect to a larger...

Abstract Let ( X , L ) be a polarized smooth projective variety. For any basepoint-free linear system $$\mathcal {L}_{V}$$ L V with $$V\subset {{\,\textrm{H}\,}}^{0}(X,\mathcal {O}_{X}(L))$$ ⊂ H</mml:mtext...

We generalize the theory developed by K. Takeuchi in [T1] and restrict the birational type of a Q-factorial Q-Fano 3-fold X with the following properties: (1) the Picard number of X is 1; (2) the Gorenstein index of X is 2; (3) the Fano index of X is 1 2 ; (4) h(−KX) ≥ 4; (5) there exists an index 2 point P such that (X, P ) ≃ ({xy + f(z, u) = 0}/Z2(1, 1, 1, 0), o) with ordf(Z, U) = 1. This giv...

Abstract The purpose of this paper is to lay the foundations for study problem when in Banach spaces. We provide a number examples couples $X$?> , $Y$?> such that $\operatorname{Ext}^n(X,Y)$?> (or not) $0$?> . show $\operatorname{Ext}^n(\mathcal K, \mathcal K)\neq 0$?> all $n\in \mathbb{N}$?> $\mathcal K$?> Kadec space. In particular, both projective and...

‎Let $mathcal{F}$ be an field of zero characteristic and $N_{infty‎}(‎mathcal{F})$ be the algebra of infinite strictly upper triangular‎ ‎matrices with entries in $mathcal{F}$‎, ‎and $f:N_{infty}(mathcal{F}‎)rightarrow N_{infty}(mathcal{F})$ be a non-additive Lie centralizer of $‎N_{infty }(mathcal{F})$; that is‎, ‎a map satisfying that $f([X,Y])=[f(X),Y]$‎ ‎for all $X,Yin N_{infty}(mathcal{F})...