Classifying resolving subcategories over a Cohen–Macaulay local ring

Let R be a Cohen–Macaulay local ring. Denote by mod R the category of finitely generated R-modules. In this paper, we consider the classification problem of resolving subcategories of mod R in terms of specialization-closed subsets of Spec R. We give a classification of the resolving subcategories closed under tensor products and transposes. Under restrictive hypotheses, we also give better classification results.     

Codismantlability and projective dimension of the Stanley–Reisner ring of special hypergraphs

In this paper, we generalize the concept of codismantlable graphs to hypergraphs and show that some special vertex decomposable hypergraphs are codismantlable. Then we generalize the concept of bouquet in graphs to hypergraphs to extend some combinatorial invariants of graphs about disjointness of a set of bouquets. We use these invariants to characterize the projective dimension of Stanley–Reisner ring of special hypergraphs in some sense.     

A-hypergeometric series and the Hasse–Witt matrix of a hypersurface

We give a short combinatorial proof of the generic invertibility of the Hasse–Witt matrix of a projective hypersurface. We also examine the relationship between the Hasse–Witt matrix and certain A-hypergeometric series, which is what motivated the proof.     

∑–pure—injective modules over a commutative prüfer ring

We give the complete classification of ∑-pure-injective modules over a commutative Prüfer ring.     

Every order is the endomorphism ring of a projective module over a quasi–hereditary order

We give in this note some new bounds on p _c(G) in terms of σ _c for several classes of finite groups, in particular, we prove that ρ _c ≤4σ _c (G) for any finite solvable group G, which improves some known results. We also pose some related open problems.     

The Witt Group of Symmetric Bilinear Forms over a Brauer–Severi Variety with Values in a Line Bundle

For a Brauer–Severi variety X over a field of characteristic not two, the Witt groups of symmetric bilinear forms over X with values in a line bundle which generates X are calculated.     

Finiteness theorems for the shifted Witt and higher Grothendieck–Witt groups of arithmetic schemes

For smooth varieties over finite fields, we prove that the shifted (aka derived) Witt groups of surfaces are finite and the higher Grothendieck–Witt groups (aka Hermitian K-theory) of curves are finitely generated. For more general arithmetic schemes, we give conditional results, for example, finite generation of the motivic cohomology groups implies finite generation of the Grothendieck–Witt groups.     

The Mahler measure of a Calabi–Yau threefold and special $$$$-values

The aim of this paper is to prove a Mahler measure formula of a four-variable Laurent polynomial whose zero locus defines a Calabi–Yau threefold. We show that its Mahler measure is a rational linear combination of a special \(L\)-value of the normalized newform in \(S_4(\Gamma _0(8))\) and a Riemann zeta value. This is equivalent to a new formula for a \(_6F_5\)-hypergeometric series evaluated at 1.     

Normalizer of a group of “diagonal” automorphisms in algebras over a commutative ring

Let S be a faithful algebra over commutative ring R. It is assumed that S is additively generated by its invertible elements. It is shown that the nomalizer of subgroup Aut(S_s) of group Aut(S_R) coincides with the semidirect product Aut(S_S) Aut(S/R),where the second factor is the group of all ring automorphisms of ring S identical on R.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 191, pp. 5–8, 1991.     

Consistency of a λ-theory withn-tuples and easy term

We give here a model-theoretical solution to the problem, raised by J.L: Krivine, of the consistency of +U(G)+=t, wheret is an arbitrary -term,G an arbitrary finite group of order, sayn, andU(G) the theory which expresses the existence of a surjectiven-tuple notion, such that each element ofG behaves simultaneously as a permutation of the components of then-tuple and as an automorphism of the model. This provides in particular a semantic proof of the -easiness of the -term .     

Vaught’s conjecture for modules over a commutative Prüfer ring

We prove that Vaught’s conjecture is true for modules over a commutative Prüfer ring. It is shown that a positive solution to Vaught’s conjecture for modules over 1-dimensional Noetherian domains would imply the same for modules over finitely presented algebras. This article was written during the visit of the second author to the University of Manchester supported by EPSRC grant GR/L68827. She would like to thank the University for hospitality. Translated fromAlgebra i Logika, Vol. 38, No. 4, pp. 419–435, July–August, 1999.     

Lines on cubic surfaces,Witt invariants and Stiefel–Whitney classes

The aim of this note is to give a formula expressing the trace form associated with the 27 lines of a cubic surface.