We show that any quasi-Gorenstein deformation of a 3-dimensional Buchsbaum local ring with I-invariant 1 admits maximal Cohen-Macaulay module, provided it is quotient Gorenstein ring. Such class rings includes two instances unique factorization domains constructed by Marcel-Schenzel and Imtiaz-Schenzel, respectively. Apart from this result, motivated the small conjecture in prime characteristic...

We study the ring Γ of all functions N → K, endowed with the usual convolution product. Γ, which we call the ring of number-theoretic functions, is an inverse limit of the “truncations” Γn = { f ∈ Γ ∀m > n : f(m) = 0 } . Each Γn is a zero-dimensional, finitely generated K-algebra, which may be expressed as the quotient of a finitely generated polynomial ring with a stable (after reversing the o...

Introduction. Several authors have established that many classical codes are ideals in certain ring constructions. Berman [3], in the case of characteristic two, and Charpin [5], in the general case, proved that all generalized Reed-Muller codes coincide with powers of the radical of the quotient ring A = Fq[x1, . . . , xn]/(x q1 1 − 1, . . . , xn n − 1), where Fq is a finite field, p = charFq ...

The idea of associating a free resolution to a finitely generated module was introduced in two famous papers by Hilbert in 1890 [Hi1] and 1893 [Hi2]. He proved Hilbert’s Syzygy Theorem 4.9, which says that the minimal free resolution of every finitely generated graded module over a polynomial ring is finite. Since then, there has been a lot of progress on the structure and properties of finite ...

An eta-quotient of levelN is a modular form of the shape f(z) = ∏ δ|N η(δz) rδ . We study the problem of determining levels N for which the graded ring of holomorphic modular forms for Γ0(N) is generated by (holomorphic, respectively weakly holomorphic) eta-quotients of level N . In addition, we prove that if f(z) is a holomorphic modular form that is nonvanishing on the upper half plane and ha...

Let X be a finite set and let k be a commutative ring. We consider the k-algebra of the monoid of all relations on X, modulo the ideal generated by the relations factorizing through a set of cardinality strictly smaller than Card(X), called inessential relations. This quotient is called the essential algebra associated to X. We then define a suitable nilpotent ideal of the essential algebra and...

A natural problem in algebraic geometry is the formation of quotients. This is particularly important in the theory of moduli, where many moduli spaces are naturally constructed as quotients of parameter spaces by linear algebraic groups. Examples of quotient moduli spaces include moduli spaces of curves, stable maps and stable vector bundles (with fixed determinant). Unfortunately, the quotien...

In 2004, J.-C. Aval, F. Bergeron and N. Bergeron studied the algebra of diagonally quasi-symmetric functions DQSym in the ring Q[x,y] with two sets of variables. They made conjectures on the structure of the quotient Q[x,y]/〈DQSym〉, which is a quasi-symmetric analogue of the diagonal harmonic polynomials. In this paper, we construct a Hilbert basis for this quotient when there are infinitely ma...