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Simion Stoilow Institute of Mathematics of the Romanian Academy
Faculty of Mathematics and Computer Science, Ovidius University of Constant¸a
The Romanian Mathematical Society – Constant¸a branch
SPECIAL SESSION ON ALGEBRA
In memory of Professor S
¸ erban Basarab
22nd May, 2015
ABSTRACTS
Adeles and models of arithmetic: Several ways in which S¸erban Basarab’s
work is proving relevant right now
Angus Macintyre
Queen Mary University of London, UK
Basarab did fundamental work on the model theory of Henselian fields, by giving invariants closely related to the residue rings. His formalism, nowadays called the RV
formalism, is being used in motivic integration. It turns out that adelic versions of it,
due to Derakhshan and Macintyre, are closely related to the hyperrings used by Connes
and Consani in recent work (generalizing old ideas of Krasner). Very recently D’Aquino
and Macintyre have begun giving a detailed analysis of the model theory of the higher
residue rings, for primes in models of arithmetic, leading to refined results on definability
in arbitrary quotients of nonstandard models by principal ideals. Again, Basarab’s work
is the starting point.
From lattice isomorphic groups to Abelian G-Cogalois field extensions
Toma ALBU
IMAR, Romania
This talk, based on a joint paper with S¸erban Basarab, shows how a 75-year-old paper
of Reinhold Baer on groups having their lattices of subgroups isomorphic is involved in
proving the following result: for an arbitrary Abelian G-Cogalois field extension E/F , the
Kneser group Kne(E/F ) of E/F is isomorphic, but not canonically, with the Character
group Ch(Gal(E/F )) of the (profinite) Galois group Gal(E/F ) of E/F .
A theorem of Ploski’s type
Dorin Popescu
IMAR, Romania
Let C{x}, x = (x1 , . . . , xn ), f = (f1 , . . . , fs ) be some convergent power series from
C{x, Y }, Y = (Y1 , . . . , YN ) and yˆ ∈ C[[x]]N with yˆ(0) = 0 be a solution of f = 0.
Then Ploski proved that the map v : B = C{x, Y }/(f ) → C[[x]] given by Y → y factors
through an A-algebra of type B 0 = C{x, Z} for some variables Z = (Z1 , . . . , Zs ), that is
v is a composite map B → B 0 → C[[x]].
Now, let (A, m) be an excellent Henselian local ring, Aˆ its completion, B a finite type
A-algebra and v : B → Aˆ an A-morphism. Then we show that v factors through an
A-algebra of type A < Z >= A[Z]h for some variables Z = (Z1 , . . . , Zs ), that is A < Z >
is the Henselization of A[Z](m,Z) .
Hessian ideals of a homogeneous polynomial and generalized Tjurina
algebras
Alexandru DIMCA
Univ. Sophia Antipolis, Nice, France
The study of the Jacobian algebra of a homogeneous polynomial is a classical subject
in Algebraic Geometry, especially due to its relations with the Hodge theory. In this
talk I will present the beginnings of a related study, involving the Jacobian ideal and
various Hessian ideals of a homogeneous polynomial. The results, algebraic in nature, are
a joint work with Gabriel Sticlaru. The geometric interpretation of these results is for the
moment an open question.
Equations and syzygies of Cohen-Macaulay multiple structures of degree 4
on a line in the projective space P3
Nicolae MANOLACHE
IMAR, Romania
The structures which we present are the subject of a section of the paper arXiv:1502.05553.
They are used there to study globally generated vector bundles on the projective spaces.
These structures, called ”quasi-primitive” were introduced in a paper by C. B˘anic˘a and
O. Forster in 1982 (manuscript with a limited circulation), published some years later.
The explicit description represents interest in itself and could have also other applications.
The results are part of a joint work with C. Anghel and I. Coand˘a.
About almost Cohen-Macaulay rings
Cristodor IONESCU
IMAR, Romania
The almost Cohen-Macaulay rings originated in a flaw in the first edition of Matsumura’s book Commutative Algebra. They were studied by Y. Han (Acta Math. Sinica
- 1998; unfortunately the paper is written in Chinese), M.C. Kang (Comm. Algebra 2001, 2002) and the present author (J. Commut. Algebra - to appear). We shall review
the most important properties and examples concerning this class of rings.
From bicrossproducts of finite groups to primality conditions
Nicolae BONCIOCAT
IMAR, Romania
We present Takeuchi’s bicrossproduct construction and the corresponding notion of
a matched pair of groups, and provide necessary and sufficient conditions for a pair of
arbitrary groups to be matched by automorphisms. We then show that a pair of finite
cyclic groups can be matched by automorphisms, provided some exponential congruences
are satisfied. Then, by studying the solutions of these congruences in the symmetric case
when the two groups coincide, as well as their mutual group actions, we obtain some
curious primality conditions. Prime numbers appearing in this way are then investigated.
The results are a joint work with M. Cipu, M.-T. Tsai and A. Zaharescu.
L-functions of algebraic number fields
Florin NICOLAE
IMAR, Romania
A Galois number field is determined by any Artin L-function corresponding to a faithful
character of the Galois group.