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Register a new userRadical factorization in finitary ideal systems
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In this paper we investigate the concept of radical factorization with respect to finitary ideal systems of cancellative monoids. We present new characterizations for r-almost Dedekind r-SP-monoids and provide specific descriptions of t-almost Dedekind t-SP-monoids and w-SP-monoids. We show that a monoid is a w-SP-monoid if and only if the radical of every nontrivial principal ideal is t-invertible. We characterize when the monoid ring is a w-SP-domain and describe when the *-Nagata ring is an SP-domain for a star operation * of finite type.
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In this paper, we survey the theory of Cartwright-Sturmfels ideals. These are Z^n-graded ideals, whose multigraded generic initial ideal is radical. Cartwright-Sturmfels ideals have surprising properties, mostly stemming from the fact that their Hilbert scheme only contains one Borel-fixed point. This has consequences, e.g., on their universal Groebner bases and on the family of their initial ideals. In this paper, we discuss several known classes of Cartwright-Sturmfels ideals and we find a new one. Among determinantal ideals of same-size minors of a matrix of variables and Schubert determinantal ideals, we are able to characterize those that are Cartwright-Sturmfels.
We show that the Specht ideal of a two-rowed partition is perfect over an arbitrary field, provided that the characteristic is either zero or bounded below by the size of the second row of the partition, and we show this lower bound is tight. We also establish perfection and other properties of certain variants of Specht ideals, and find a surprising connection to the weak Lefschetz property. Our results in particular give a self-contained proof of Cohen-Macaulayness of certain $h$-equals sets, a result previously obtained by Etingof-Gorsky-Losev over the complex numbers using rational Cherednik algebras.
What kind of reduced monomial schemes can be obtained as a Grobner degeneration of a smooth projective variety? Our conjectured answer is: only Stanley-Reisner schemes associated to acyclic Cohen-Macaulay simplicial complexes. This would imply, in particular, that only curves of genus zero have such a degeneration. We prove this conjecture for degrevlex orders, for elliptic curves over real number fields, for boundaries of cross-polytopes, and for leafless graphs. We discuss consequences for rational and F-rational singularities of algebras with straightening laws.
Let $A={{bf a}_1,...,{bf a}_m} subset mathbb{Z}^n$ be a vector configuration and $I_A subset K[x_1,...,x_m]$ its corresponding toric ideal. The paper consists of two parts. In the first part we completely determine the number of different minimal systems of binomial generators of $I_A$. We also prove that generic toric ideals are generated by indispensable binomials. In the second part we associate to $A$ a simplicial complex $Delta _{ind(A)}$. We show that the vertices of $Delta_{ind(A)}$ correspond to the indispensable monomials of the toric ideal $I_A$, while one dimensional facets of $Delta_{ind(A)}$ with minimal binomial $A$-degree correspond to the indispensable binomials of $I_{A}$.
We study basic properties of the generalized ideal transforms $D_I(M, N)$ and the set of associated primes of the modules $R^iD_I(M,N).$
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