On the exactness of flat resolvents

Let R be a left coherent ring, FP — idRR the FP — injective dimension of RR and wD(R) the weak global dimension of R. It is shown that 1) FP -idRR < n ( n > 0) if and only if every flat resolvent 0 → M → F° → F¹... of a finitely presented right R—module M is exact at F'(i > n?1) if and only if every nth F -cosyzygy of a finitely presented right R — module has a flat preenvelope which is a monomorphism; 2) wD(R) < n (n > 1) if and only if every (n?l)th F-cosyzygy of a finitely presented right R—module has a flat preenvelope which is an epimorphism; 3) wD(R) 0) if and only if every nth F — cosyzygy of a finitely presented right R — module is flat. In particular, left FC rings and left semihereditary rings are characterized     

n-C-Star Modules and n-C-Tilting Modules

Let C be a faithfully balanced selforthogonal module over an Artin algebra R. We introduce the notion of n-C-star modules, which is a common generalization of n-star modules and n-C-tilting modules. We extend some characterizations of n-star modules to this context and prove that n-C-tilting modules are precisely n-C-star modules n-C-presenting all the injectives.     

Dualità per alcune classi di moduliE-compatti

Summary Starting from a topological module E over a commutative discrete ring A, the category C(E) of E-compact modules is defined as the class of all A-modules topologically isomorphic to closed submodules of direct product of copies of E. Under suitable assumptions it is shown that C(E) is dual of the category of abstract A-modules M for whichHom _A(M, E) separates points of M. The duality theory so obtained contains as particular cases Pontryagin's duality between discrete and compact abelian groups and Macdonald's duality between lineary discrete and linearly compact modules over a complete local ring. There are also some applications to the theory of linearly compact modules over noetherian rings.

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Entrata in Redazione il 10 aprile 1976.

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Lavoro eseguito nell'ambito dell'attività dei gruppi di ricerca matematici del C.N.R.     

An embedding theorem for rings

Kaplansky [2] proved that if P is a projective module, then every f.g. submodule of P is contained in a finitely generated direct summand iff P is the direct sum of f.g. projectives. We show that in order that all injectives have the dual property to the above statement,, for ach pair of simples (S₁, S₂), Hom(?₁,?₂) must be an Artinian and Noet. i.ian End(?₁) module where _i. is the injective hull of ?_i. This leads to a study of universally cotorsionless modules.     

On (I*.n)-regular graphs

ABSTRACT

A walk P in G is said to be a route if the length of P is at least one and no edge of G appears as two consecutive edge-terms of P. For n ≥ 2, a graph G is said to be (I*,n)-regular of degree k if each vertex of G is an end vertex of exactly k routes of length at most n-1. A complete characterization of (I*.n)-regular graphs is obtained.     

On eigenvalues of Seidel matrices and Haemers’ conjecture

For a graph G, let S(G) be the Seidel matrix of G and \({\theta }_1(G),\ldots ,{\theta }_n(G)\) be the eigenvalues of S(G). The Seidel energy of G is defined as \(|{\theta }_1(G)|+\cdots +|{\theta }_n(G)|\). Willem Haemers conjectured that the Seidel energy of any graph with n vertices is at least \(2n-2\), the Seidel energy of the complete graph with n vertices. Motivated by this conjecture, we prove that for any \(\alpha \) with \(0<\alpha <2,|{\theta }_1(G)|^\alpha +\cdots +|{\theta }_n(G)|^\alpha \geqslant (n-1)^\alpha +n-1\) if and only if \(|\hbox {det}\,S(G)|\geqslant n-1\). This, in particular, implies the Haemers’ conjecture for all graphs G with \(|\hbox {det}\,S(G)|\geqslant n-1\). A computation on the fraction of graphs with \(|\hbox {det}\,S(G)|<n-1\) is reported. Motivated by that, we conjecture that almost all graphs G of order n satisfy \(|\hbox {det}\,S(G)|\geqslant n-1\). In connection with this conjecture, we note that almost all graphs of order n have a Seidel energy of order \(\Theta (n^{3/2})\). Finally, we prove that self-complementary graphs G of order \(n\equiv 1\pmod 4\) have \(\det S(G)=0\).     

On the Cohomology of Relative Hopf Modules

Let H be a Hopf algebra over a field k, and A an H-comodule algebra. The categories of comodules and relative Hopf modules are then Grothendieck categories with enough injectives. We study the derived functors of the associated Hom functors, and of the coinvariants functor, and discuss spectral sequences that connect them. We also discuss when the coinvariants functor preserves injectives.     

A SURJECTIVE HOMOMORPHISM OF LOCAL COHOMOLOGY MODULES

Let R be a local ring and M a finitely generated generalized Cohen-Macaulay R-module such that dim _R M = dim _R M/αM + height_Mα a for all ideals α of R. Suppose that H_I ^j(M) ≠ 0 for an ideal I of R and an integer j > height_M I. We show that there exists an ideal J ? such that a. height_M J = j;

b. the natural homomorphism H_I ^j(M) → H_I ^j(M) is an isomorphism, for all i > j; and,

c. the natural homomorphism H_I ^j(M) → H_I ^j(M) is surjective.

By using this theorem, we obtain some results about Betti numbers, coassociated primes, and support of local cohomology modules.     

Köthe’s upper nil radical for modules

Let M be a left R-module. In this paper a generalization of the notion of an s-system of rings to modules is given. Let N be a submodule of M. Define $\mathcal{S}(N):=\{ {m\in M}:\, \mbox{every } s\mbox{-system containing } m \mbox{ meets}~N \}$ . It is shown that $\mathcal{S}(N)$ is equal to the intersection of all s-prime submodules of M containing N. We define $\mathcal{N}({}_{R}M) = \mathcal{S}(0)$ . This is called (Köthe’s) upper nil radical of M. We show that if R is a commutative ring, then $\mathcal{N}({}_{R}M) = {\mathop{\mathrm{rad}}\nolimits}_{R}(M)$ where ${\mathop{\mathrm{rad}}\nolimits}_{R}(M)$ denotes the prime radical of M. We also show that if R is a left Artinian ring, then ${\mathop{\mathrm{rad}}\nolimits}_{R}(M)=\mathcal{N}({}_{R}M)= {\mathop{\mathrm{Rad}}\nolimits}\, (M)= {\mathop{\mathrm{Jac}}\nolimits}\, (R)M$ where ${\mathop{\mathrm{Rad}}\nolimits}\, (M)$ denotes the Jacobson radical of M and ${\mathop{\mathrm{Jac}}\nolimits}\, (R)$ the Jacobson radical of the ring R. Furthermore, we show that the class of all s-prime modules forms a special class of modules.     

广义FP—内射模、广义平坦模与某些环

左（右）R－模A称为GFP－内射模，如果ExtR(M,A)=0对任－2－表现R－模M成立；左（右）R－模称为G－平坦的，如果Tor1^R(M,A)=0(Tor1^R(AM)=0)对于任一2－表现右（左）R－模M成立；环R称左（右）R－半遗传环，如果投射左（右）R－模的有限表现子模是投射的，环R称为左（右）G－正而环，如果自由左（右）R－模的有限表现子模为其直和项，研究了GFP－内射模和G－平坦模的一些性质，给出了它们的一些等价刻划，并利用它们刻划了凝聚环，G－半遗传环和G－正则环。     

Hodge theory on cyclotomic threefolds (with a view towards the higher dimensional case)

Sunto Nel presente lavoro si considerano alcune famiglie di ipersuperfici liscie dello spazio proiettivo complesso quadridimensionale, invarianti per opportune azioni del gruppo delle radici p-esime dell'unità, dove p è un divisore del grado. Si verifica la congettura di Grothendieck-Hodge sull'elemento generale delle famiglie considerate generalizzando alcuni metodi di G. E. Welters o alternativamente esibendo una relazione tra la congettura di unirigatezza di Miyaoka-Mori e la suddettta congettura sulla prima filtrazione di Hodge della coomologia intermedia. La stessa costruzione viene poi applicata ad analoghe famiglie di ipersuperfici dello spazio proiettivo complesso (n+1)-dimensionale ottenendo risultati simili. In particolare si esibisce un insieme numerabile di famiglie di varietà quadridimensionali che verificano la (2, 2)-congettura classica di Hodge.

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Partially supported by the italian M.U.R.S.T. and the european scientific project «Algebraic Geometry in Europe» (A.G.E.).     

有限級有序加羣及有序環的表現

在另一文中,我们讨论了由全体2维實向量所成的有序环,在该文最後並说當维数n>2时(n为有限)也可类似地作初步讨论.为了显示这种向量环的用途,我们考虑用向量环来表现一般有序环的问题.在本文中我们证明:任一“n级的”(见以下定义)有序环都能与一个由若干n维實向量所组成的有序环同构.(主要在於证出关於n级有序加羣的类似结果.)我们希望有较好的结果,即:任一n级有序环都能与由全体n维實向量所成的一个有序环的一个子环同构,但未能证明或否定.     

Minimizers of Higher Order Gauge Invariant Functionals

We introduce higher order variants of the Yang–Mills functional that involve \((n-2)\)-th order derivatives of the curvature. We prove coercivity and smoothness of critical points in Uhlenbeck gauge in dimensions \(\mathrm {dim}M\le 2n\). These results are then used to establish the existence of smooth minimizers on a given principal bundle \(P\rightarrow M\) for subcritical dimensions \(\mathrm {dim}M<2n\). In the case of critical dimension \(\mathrm {dim}M=2n\) we construct a minimizer on a bundle which might differ from the prescribed one, but has the same Chern classes \(c_1,\ldots ,c_{n-1}\). A key result is a removable singularity theorem for bundles carrying a \(W^{n-1,2}\)-connection. This generalizes a recent result by Petrache and Rivière.     

A note on the Schur multiplier of groups of prime power order

By a well-known result of Green (Proc R Soc A 237:574?C581, 1956) and the formal definition of Ellis and Wiegold (Bull Austral Math Soc 60:191?C196, 1999), there is an integer t, say corank(G), such that ${|\mathcal{M}(G)| = p^{\frac{1}{2}n(n-1)-t}}$ . In Niroomand (J Algebra 322:4479?C4482, 2009), the author showed for a non-abelian group G, corank(G)????log _p (|G|)?2 and classified the structure of all non-abelian p-groups of corank log _p (|G|)?2. In the present paper, we are interesting to characterize the structure of all p-groups of corank log _p (|G|)?1.     

Star operations on modules

We present a theory of (semi)star operations for torsion-free modules. This extends the analogous theory of star operations on domains as in [R. Gilmer, Multiplicative Ideal Theory, M. Dekker, New York, 1972] and its generalization to semistar operations studied in [A. Okabe, R. Matsuda, Semistar operations on integral domains, Math. J. Toyama Univ. 17 (1994) 1–21], and recovers some closure operations defined on modules. We investigate some properties of (semi)star operations on a given module over a domain D and their relation with the properties of some classes of semistar operations induced on D.Among other things, this leads to a connection between semistar operations on the D-module and localizing systems on the domain D.     

The Number of Triangulations of the Cyclic Polytope C (n , n -4)

We show that the exact number of triangulations of the standard cyclic polytope C(n,n-4) is (n+4)2 ^(n-4)/2 -n if n is even and \left((3n+11)/2\right)2 ^(n-5)/2 -n if n is odd. These formulas were previously conjectured by the second author. Our techniques are based on Gale duality and the concept of virtual chamber. They further provide formulas for the number of triangulations which use a specific simplex. We also compute the maximum number of regular triangulations among all the realizations of the oriented matroid of C(n,n-4) . Received October 24, 2000, and in revised form July 8, 2001. Online publication November 7, 2001.     

Copolyform Modules

Abstract

In this note all rings R are associative with identity, all modules are unitary right modules and we denote the category of all such R-modules by Mod-R. Let M ? Mod-R and A ? M. Corational submodules of A in M are defined and studied. Examples of modules M for which every submodule has a smallest corational submodule in M and an example of a module M with a submodule A which has no minimal corational submodule in M are given. In the second half of the paper copolyform modules are defined and we characterize when a finite direct sum of weakly supplemented copolyform modules is copolyform.     

Sulla regolarità dei processi termodinamici per alcuni materiali con memoria

Summary We study the thermodynamic restrictions for a class of materials which have memory and emphasize that the different regularity of the temperature gradient brings to a different definition of material state and to different properties for thermodynamic potentials.

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Lavoro eseguito nell'ambito del G.N.F.M. del C.N.R.     

Uniserial modules: sums and isomorphisms of subquotients

Let R be an associative ring with 1. A left R module M is uniserial i f the lattice L(M) of its submodules is totally ordered under inclusion. We give an example of a uniserial module M with the property of having two submodules 0 < H < K < M such that M is isomorphic to K/H (we call a module M with this property shrinkable). Then we give an example of a uniserial module M isomorphic to all its nonzero quotients M/N, N<M, and with L(M) isomorphic to ω²+1; this solves a problem of Hirano and Mogami [7]. Finally we show that for uniserial modules the property of being shrinkable is connected to the problem of deciding whether a module, which is both a homomorphic image of a finite direct sum of uniserial modules and a submodule of a finite direct sum of uniserial modules, is a finite direct sum of uniserial modules     

Characterization of finite $$\varvec{}$$-grous by their Schur multilier

Let G be a finite p-group of order \(p^n\) and M(G) be its Schur multiplier. It is a well known result by Green that \(|M(G)|= p^{\frac{1}{2}n(n-1)-t(G)}\) for some \(t(G) \ge 0\). In this article, we classify non-abelian p-groups G of order \(p^n\) for \(t(G)=\log _p(|G|)+1\).