let r be an associative ring with identity, c(r) be the category of com-plexes of r-modules and flat(c(r)) be the class of all at complexes of r-modules. we show that the at cotorsion theory (flat(c(r)); flat(c(r))−)have enough injectives in c(r). as an application, we prove that for each atcomplex f and each complex y of r-modules, exti (f,x)= 0, whenever ris n-perfect and i > n.

An artin algebra is called CM-free provided that all its finitely generated Gorenstein projective modules are projective. We show that a connected artin algebra with radical square zero is either self-injective or CM-free. As a consequence, we prove that a connected artin algebra with radical square zero is Gorenstein if and only if its valued quiver is either an oriented cycle with the trivial...

The Gorenstein projective modules are proved to form a precovering class in the module category of a ring which has a dualizing complex. 0. Introduction This paper proves over a wide class of rings that the Gorenstein projective modules form a precovering class in the module category. Let me explain this statement. There are two terms of mystery, “Gorenstein projective modules” and “precovering...

Using new number-theoretic bounds on the denominators of unit fractions summing up to one, we show that in any dimension d ≥ 4 there is only one d-dimensional reflexive simplex having maximal volume. Moreover, only these reflexive simplices can admit an edge that has the maximal number of lattice points possible for an edge of a reflexive simplex. In general, these simplices are also expected t...

If V is an equidimensional codimension c subscheme of an n-dimensional projective space, and V is linked to V ′ by a complete intersection X, then we say that V is minimally linked to V ′ if X is a codimension c complete intersection of smallest degree containing V . Gaeta showed that if V is any arithmetically Cohen-Macaulay (ACM) subscheme of codimension two then there is a finite sequence of...

In this paper we use "ring changed'' Gorenstein homologicaldimensions to define Cohen-Macaulay injective, projective and flatdimensions. For doing this we use the amalgamated duplication of thebase ring with semi-dualizing ideals. Among other results, we prove that finiteness of these new dimensions characterizes Cohen-Macaulay rings with dualizing ideals.

Max Noether’s Theorem asserts that if ω is the dualizing sheaf of a nonsingular nonhyperelliptic projective curve then the natural morphisms SymH(ω) → H(ω) are surjective for all n ≥ 1. This is true for Gorenstein nonhyperelliptic curves as well. We prove this remains true for nearly Gorenstein curves and for all integral nonhyperelliptic curves whose non-Gorenstein points are unibranch. The re...

A central problem in liaison theory is to decide whether every arithmetically Cohen-Macaulay subscheme of projective n-space can be linked by a finite number of arithmetically Gorenstein schemes to a complete intersection. We show that this can be indeed achieved if the given scheme is also generically Gorenstein and we allow the links to take place in an (n + 1)-dimensional projective space. F...

Following the well-established terminology in commutative algebra, any (not necessarily commutative) finite-dimensional local algebra A $A$ with radical J $J$ will be said to short provided 3 = 0 $J^3 0$ . As case, we show: If a has an indecomposable non-projective Gorenstein-projective module M $M$ , then either is self-injective (so that all modules are Gorenstein-projective) and then, of cou...