Let R = k[x1, . . . , xn] be a polynomial ring over a field k. Let J = {j1, . . . , jt} be a subset of {1, . . . , n}, and let mJ ⊂ R denote the ideal (xj1 , . . . , xjt). Given subsets J1, . . . , Js of {1, . . . , n} and positive integers a1, . . . , as, we study ideals of the form I = m1 J1 ∩ · · · ∩ m as Js . These ideals arise naturally, for example, in the study of fat points, tetrahedral...

A complete determination of the prime ideals invariant under winding automorphisms in the generic 3 × 3 quantum matrix algebra Oq(M3(k)) is obtained. Explicit generating sets consisting of quantum minors are given for all of these primes, thus verifying a general conjecture in the 3 × 3 case. The result relies heavily on certain tensor product decompositions for winding-invariant prime ideals, ...

1. Preliminaries 3 2. Gröbner Bases 3 2.1. Motivating Problems 3 2.2. Ordering of Monomials 3 2.3. The Division Algorithm in S = k[x1, . . . , xn] 5 2.4. Dickson’s Lemma 7 2.5. Gröbner Bases and the Hilbert Basis Theorem 10 2.6. Some Further Applications of Gröbner bases 18 3. Hilbert Functions 21 3.1. Macaulay’s Theorem 27 3.2. Hilbert Functions of Reduced Standard Graded k-algebras 37 3.3. Hi...

We define the reduced horseshoe resolution and the notion of conjoined pairs of ideals in order to study the minimal graded free resolution of a class of p-Borel ideals and recover Pardue’s regularity formula for them. It will follow from our technique that the graded betti numbers of these ideals do not depend on the characteristic of the base field k.

Introduction. It is notoriously difficult to deduce anything about the structure of an ideal or scheme by directly examining its defining polynomials. A notable exception is that of monomial ideals. Combined with techniques for making fiat degenerations of arbitrary ideals into monomial ideals (typically, using Gr6bner bases), the theory of monomial ideals becomes a useful tool for studying gen...

Classically, Gröbner bases are computed by first prescribing a fixed monomial order. Moss Sweedler suggested an alternative in the mid 1980s and developed a framework to perform such computations by using valuation rings in place of monomial orders. We build on these ideas by providing a class of valuations on K(x, y) that are suitable for this framework. We then perform such computations for i...

For investigation of the properties of smooth muscle contraction hypertonically added 40 mm K medium has been used as a routine in this laboratory. Others have used isotonic high K/Na deficient medium for the same purpose. The distinct difference xisting between the mechanical response of guinea pig taenia coli in hyper-40 K medium and that in iso 152 K medium has been reported earlier. The res...

In [2] J. Gubeladze and the author have studied the divisorial ideals of an algebra R = K[M ] where K is a field and M a normal affine monoid. The divisorial ideals which have a monomial K-basis cut out from the group gp(M) by a translate of the cone R+M have been called conic. Up to isomorphism the conic ideals are exactly the direct summands of the extension R 1/n of R where R 1/n is the K-al...

Introduction. Fix a base field k. The quantized coordinate ring of n×n matrices over k, denoted by q(Mn(k)), is a deformation of the classical coordinate ring of n×n matrices, (Mn(k)). As such, it is a k-algebra generated by n2 indeterminates Xij , for 1 ≤ i,j ≤ n, subject to relations which we state in (1.1). Here, q is a nonzero element of the field k. When q = 1, we recover (Mn(k)), which is...