A short proof of a theorem of M. Hashimoto

Advances in Mathematics, 1981

Journal of Algebra, 2001

Bulletin of the American Mathematical Society, 1970

COROLLARY 1. If !#,» has grade gH, n then it is grade unmixed, i.e. the associated primes of I H , n all have grade gH,n* COROLLARY 2. If R is Cohen-Macaulay (locally), and lH, n has grade gH t n> then In, n is rank unmixed, i.e. the associated primes all have rank (== altitude) gH, n ; moreover, R/I is Cohen-Macaulay. COROLLARY 3. The rank of any minimal prime of In,n is at most gst, n (with no conditions on the grade of I).

arXiv: Commutative Algebra, 2018

Let $R=K[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal generated in degree $d$. Bandari and Herzog conjectured that a monomial ideal $I$ is polymatroidal if and only if all its monomial localizations have a linear resolution. In this paper we give an affirmative answer to the conjecture in the following cases: $(i)$ ${\rm height}(I)=n-1$; $(ii)$ $I$ contains at least $n-3$ pure powers of the variables $x_1^d,...,x_{n-3}^d$; $(iii)$ $I$ is a monomial ideal in at most four variables.

Archiv der Mathematik, 1989

Journal of Algebra, 1997

Journal of Pure and Applied Algebra, 1991

Proceedings of the American Mathematical Society, 1999

Abstract. We give a combinatorial proof of the Dedekind–Mertens formula by computing the initial ideal of the content ideal of the product of two generic polynomials. As a side effect we obtain a complete classification of the rank 1 Cohen–Macaulay modules over the determinantal ...

The Michigan Mathematical Journal, 2013

We consider ideals generated by general sets of m-minors of an m×n-matrix of indeterminates. The generators are identified with the facets of an (m−1)-dimensional pure simplicial complex. The ideal generated by the minors corresponding to the facets of such a complex is called a determinantal facet ideal. Given a pure simplicial complex ∆, we discuss the question when the generating minors of its determinantal facet ideal J∆ form a Gröbner basis and when J∆ is a prime ideal.

Bulletin of the American Mathematical Society, 1990

Determinantal varieties were considered for the first time in the nineteenth century in connection with the first and second fundamental theorem of invariant theory. Let us consider a vector space F of dimension r over a field k. Let Z denote the space of (ra + n)-tuples