For finitely generated module M over a local ring R, the conventional notions of complete intersection dimension CI-dimRM and CohenMacaulay dimension CM-dimRM do not extend to cover the case of infinitely generated modules. In this paper we introduce similar invariants for not necessarily finitely generated modules, (namely, complete intersection flat and Cohen-Macaulay flat dimensions) which f...

Basic Definition: Let R be a commutative ring with 1. A (unital) R-module is an abelian group M together with a operation R ×M → M , usually just written as rv when r ∈ R and v ∈ M . This operation is called scaling . The scaling operation satisfies the following conditions. 1. 1v = v for all v ∈M . 2. (rs)v = r(sv) for all r, s ∈ R and all v ∈M . 3. (r + s)v = rv + sv for all r, s ∈ R and all ...

The purpose of this note is to prove the following “boundedness” stated in [Ol]. Let X and Y be separated Deligne–Mumford stacks of finite presentation over an algebraic space S and define HomS(X ,Y) as in [Ol, 1.1]. Assume that X is flat and proper over S, and that locally in the fppf topology on S, there exists a finite flat surjection Z → X from an algebraic space Z. Let Y → W be a quasi-fin...

of zeros of d F ( e j W ) / d w [2]. Generally speaking, the maximally In this paper, we derive some properties of maximally flat R-regular Mth-band FIR filters. We show that the R-regularity implies maximally flat frequency response at w = 0. The R-regular constraints are a set of linear equations with complex coefficients. We can convert these complex-value equations to equivalent ones with o...

It is well known that Verma module homomorphisms correspond to invariant operators on homogeneous spaces, which in certain situations can be regarded as the flat models of specific differential geometries. This can be generalised to curved space by introducing semiholonomic Verma modules, whose homomorphisms give rise to invariant operators on curved space. In this article we investigate from a...

background: the adverse effects of noise are well known and noise problems due to industrialization of communities are increasing over the time. oil industries due to the process and nature of production; contain many noise sources such as compressors, turbines, and pumps, which cause excessive noise exposure. the objective of this study was to evaluate the noise characteristics of compressors ...

Given a homomorphism of commutative noetherian rings R → S and an S–module N , it is proved that the Gorenstein flat dimension of N over R, when finite, may be computed locally over S. When, in addition, the homomorphism is local and N is finitely generated over S, the Gorenstein flat dimension equals sup {m ∈ Z | Torm(E,N) 6= 0}, where E is the injective hull of the residue field of R. This re...

Given a homomorphism of commutative noetherian rings R → S and an S–module N , it is proved that the Gorenstein flat dimension of N over R, when finite, may be computed locally over S. When, in addition, the homomorphism is local and N is finitely generated over S, the Gorenstein flat dimension equals sup {m ∈ Z | TorRm(E,N) 6= 0}, where E is the injective hull of the residue field of R. This r...

This a first step to develop a theory of smooth, étale and unramified morphisms between noetherian formal schemes. Our main tool is the complete module of differentials, that is a coherent sheaf whenever the map of formal schemes is of pseudo finite type. Among our results we show that these infinitesimal properties of a map of usual schemes carry over into the completion with respect to suitab...

We investigate the structure of pure-syzygy modules in a pure-projective resolution of any right R-module over an associative ring R with an identity element. We show that a right R-module M is pure-projective if and only if there exists an integer n ≥ 0 and a pure-exact sequence 0 → M → Pn → · · · → P0 → M → 0 with pure-projective modules Pn, . . . , P0. As a consequence we get the following v...