Combinatorial bases of modules for affine Lie algebra B_2^(1)

In this paper we construct bases of standard (i.e. integrable highest weight) modules $L(\Lambda)$ for affine Lie algebra of type B_2\sp{(1)} consisting of semi-infinite monomials. The main technical ingredient is a construction of monomial bases for Feigin-Stoyanovsky type subspaces $W(\Lambda)$ of $L(\Lambda)$ by using simple currents and intertwining operators in vertex operator algebra theory. By coincidence $W(k\Lambda_0)$ for B_2\sp{(1)} and the integrable highest weight module $L(k\Lambda_0)$ for A_1\sp{(1)} have the same parametrization of combinatorial bases and the same presentation $\mathcal P/\mathcal I$\,.Comment: AMS-LaTeX, 29 pages, to appear in Central European Journal of Mathematics. v2: The claim that presentation for B_2^(1) implies linear independence for A_1^(1) is incorrect and omitted; other minor changes; a new section on vertex operator formula is adde

Regularity and linearity defect of modules over local rings

Given a finitely generated module $M$ over a commutative local ring (or a standard graded $k$-algebra) (R,\m,k) we detect its complexity in terms of numerical invariants coming from suitable \m-stable filtrations $\mathbb{M}$ on $M$. We study the Castelnuovo-Mumford regularity of $gr_{\mathbb{M}}(M)$ and the linearity defect of $M,$ denoted \ld_R(M), through a deep investigation based on the theory of standard bases. If $M$ is a graded $R$-module, then \reg_R(gr_{\mathbb{M}}(M)) <\infty implies \reg_R(M)<\infty and the converse holds provided $M$ is of homogenous type. An analogous result can be proved in the local case in terms of the linearity defect. Motivated by a positive answer in the graded case, we present for local rings a partial answer to a question raised by Herzog and Iyengar of whether \ld_R(k)<\infty implies $R$ is Koszul.Comment: 15 pages, to appear in Journal of Commutative Algebr

Triangular bases of integral closures

In this work, we consider the problem of computing triangular bases of integral closures of one-dimensional local rings. Let $(K, v)$ be a discrete valued field with valuation ring $\mathcal{O}$ and let $\mathfrak{m}$ be the maximal ideal. We take $f \in \mathcal{O}[x]$, a monic irreducible polynomial of degree $n$ and consider the extension $L = K[x]/(f(x))$ as well as $\mathcal{O}_{L}$ the integral closure of $\mathcal{O}$ in $L$, which we suppose to be finitely generated as an $\mathcal{O}$-module. The algorithm $\operatorname{MaxMin}$, presented in this paper, computes triangular bases of fractional ideals of $\mathcal{O}_{L}$. The theoretical complexity is equivalent to current state of the art methods and in practice is almost always faster. It is also considerably faster than the routines found in standard computer algebra systems, excepting some cases involving very small field extensions