Standard monomial bases and geometric consequences for certain rings of invariants (original) (raw)

On Cohen–Macaulay Rings of Invariants

Journal of Algebra, 2001

We investigate the transfer of the Cohen-Macaulay property from a commutative ring to a subring of invariants under the action of a finite group. Our point of view is ring theoretic and not a priori tailored to a particular type of group action. As an illustration, we briefly discuss the special case of multiplicative actions, that is, actions on group algebras k[Z n ] via an action on Z n .

Standard monomial bases, Moduli spaces of vector bundles, and Invariant theory

Transformation Groups, 2006

Consider the diagonal action of SO n (K) on the affine space X = V ⊕m where V = K n , K an algebraically closed field of characteristic = 2. We construct a "standard monomial" basis for the ring of invariants K[X] SOn(K) . As a consequence, we deduce that K[X] SOn(K) is Cohen-Macaulay. As the first application, we present the first and second fundamental theorems for SO n (K)-actions. As the second application, assuming that the characteristic of K is = 2, 3, we give a characteristicfree proof of the Cohen-Macaulayness of the moduli space M 2 of equivalence classes of semi-stable, rank 2, degree 0 vector bundles on a smooth projective curve of genus > 2. As the third application, we describe a K-basis for the ring of invariants for the adjoint action of SL 2 (K) on m copies of sl 2 (K) in terms of traces.

On the Cohen-Macaulay property of multiplicative invariants

2006

We investigate the Cohen-Macaulay property for rings of invariants under multiplicative actions of a finite group G. By definition, these are G-actions on Laurent polynomial algebras k[x ±1 1 , . . . , x ±1 n ] that stabilize the multiplicative group consisting of all monomials in the variables x i . For the most part, we concentrate on the case where the base ring k is Z. Our main result states that if G acts non-trivially and the invariant ring Z[x ±1 1 , . . . , x ±1 n ] G is Cohen-Macaulay then the abelianized isotropy groups G ab m of all monomials m are generated by the bireflections in G m and at least one G ab m is non-trivial. As an application, we prove the multiplicative version of Kemper's 3-copies conjecture.

Invariants of Cohen–Macaulay rings associated to their canonical ideals

Journal of Algebra

The purpose of this paper is to introduce new invariants of Cohen-Macaulay local rings. Our focus is the class of Cohen-Macaulay local rings that admit a canonical ideal. Attached to each such ring R with a canonical ideal C, there are integers-the type of R, the reduction number of C-that provide valuable metrics to express the deviation of R from being a Gorenstein ring. We enlarge this list with other integers-the roots of R and several canonical degrees. The latter are multiplicity based functions of the Rees algebra of C.

Algebraic invariants of projective monomial curves associated to generalized arithmetic sequences

Journal of Symbolic Computation, 2016

Let K be an infinite field and let m 1 < • • • < m n be a generalized arithmetic sequence of positive integers, i.e., there exist h, d, m 1 ∈ Z + such that m i = hm 1 + (i − 1)d for all i ∈ {2,. .. , n}. We consider the projective monomial curve C ⊂ P n K parametrically defined by x 1 = s m1 t mn−m1 ,. .. , x n−1 = s mn−1 t mn−mn−1 , x n = s mn , x n+1 = t mn. In this work, we characterize the Cohen-Macaulay and Koszul properties of the homogeneous coordinate ring K[C] of C. Whenever K[C] is Cohen-Macaulay we also obtain a formula for its Cohen-Macaulay type. Moreover, when h divides d, we obtain a minimal Gröbner basis G of the vanishing ideal of C with respect to the degree reverse lexicographic order. From G we derive formulas for the Castelnuovo-Mumford regularity, the Hilbert series and the Hilbert function of K[C] in terms of the sequence.

The Hilbert series of Invariants of Sl n (k)

1993

In this paper we consider the problem of computing the Hilbert series of Invariants of Sln(k), in order in a futur step to have a good description of these algebras. Using the Mollien-Weyl theorem, we have to nd the constant term of a series in several variables. Using linear algebra via Vandermond matrices, we give examples of this experimentation for the rst degrees.

Cohen-Macaulayness of monomial ideals and symbolic powers of Stanley-Reisner ideals

2011

We present criteria for the Cohen-Macaulayness of a monomial ideal in terms of its primary decomposition. These criteria allow us to use tools of graph theory and of linear programming to study the Cohen-Macaulayness of monomial ideals which are intersections of prime ideal powers. We can characterize the Cohen-Macaulayness of the second symbolic power or of all symbolic powers of a Stanley-Reisner ideal in terms of the simplicial complex. These characterizations show that the simplicial complex must be very compact if some symbolic power is Cohen-Macaulay. In particular, all symbolic powers are Cohen-Macaulay if and only if the simplicial complex is a matroid complex. We also prove that the Cohen-Macaulayness can pass from a symbolic power to another symbolic powers in different ways.