Dorin Popescu

General Néron desingularization and approximation

Let A be a noetherian ring (all the rings are supposed here to be commutative with identity), a ⊂ A a proper ideal and  the completion of A in the α -adic topology. We consider the following conditions (WAP) Every finite system of polynomial equations over A has a solution in A iff it has one in  .

STANLEY CONJECTURE IN SMALL EMBEDDING DIMENSION

We show that Stanleys conjecture holds for a polynomial ring over a field in four variables. In the case of polynomial ring in five variables, we prove that the monomial ideals with all associated primes of height two, are Stanley ideals.

Maximal Cohen-Macaulay Modules over the Cone of an Elliptic Curve

Let R=k[Y1,Y2,Y3]/(f), f=Y13+Y23+Y33, where k is an algebraically closed field with chark≠3. Using Atiyah bundle classification over elliptic curves we describe the matrix factorizations of the graded, indecomposable reflexive R-modules, equivalently we describe explicitly the indecomposable bundles over the projective curve V(f)⊂Pk2. Using the fact that over the completion R of R every reflexive module is gradable, we obtain a description of the maximal Cohen–Macaulay modules over R=k〚Y1,Y2,Y3〛/(f).

Stanley depth and size of a monomial ideal

Lyubeznik introduced the concept of size of a monomial ideal and showed that the size of a monomial ideal increased by

The Stanley Conjecture on Intersections of Four Monomial Prime Ideals

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Hilbert Functions and Generic Forms

is a lower bound for its depth. We show that the size is also a lower bound for its Stanley depth. Applying Alexander duality we obtain upper bounds for the regularity and Stanley regularity of squarefree monomial ideals.

Deformations of maximal Cohen-Macaulay modules

We show that the Stanleys Conjecture holds for an intersection of four monomial prime ideals of a polynomial algebra S over a field and for an arbitrary intersection of monomial prime ideals (P i ) i∈[s] of S such that each P i is not contained in the sum of the other (P j ) j≠i .

Depth of factors of square free monomial ideals

Let A be a homogeneous K-algebra where K is a field of characteristic 0, and h ∈ A a generic form. We bound the Hilbert function H(A/(h),-) in terms of H(A,-) which extends the bound given by M.Green for generic linear forms. We apply this to some conjectures from Higher Castelnuovo Theory and Cayley-Bacharach Theory.

Hilbert Series and Lefschetz Properties of Dimension One Almost Complete Intersections

Let (R,m) be a complete Cohen-Macaulay isolated singularity over a field K which is either perfect or [K : K p] 0. According to Dieterich [Di] and [Yo] Ch. 6 (see also [Pol], [PR] or, more generally, [CHP]) there exists a system of parameters x = (xl . . . . . xr) of R such that the base change functor R/(x)®Rdefines an injection v preserving the indecomposability from the set of isomorphism classes of maximal Cohen-Macaulay R-modules to the set of isomorphism classes of R/(x)-modules (in this case the ideal generated by x is called (after [Di]) a reduction ideal. Trying to describe the image of v we noticed in [Po2] that a finitely generated R/(x)module is in Imv if and only if it has the form Xl.. .x~P for a finitely generated R2 := R/(x 2 . . . . . x~ )-module satisfying

Indecomposable Cohen-Macaulay modules and irreducible maps

Let I be an ideal of a polynomial algebra over a eld, generated by r-square free monomials of degree d. If r is bigger (or equal) than the number of square free monomials of I of degree d + 1 then depthS I = d. Let J I, J 6 0 be generated by square free monomials of degree d + 1. If r is bigger than the number of square free monomials of InJ of degree d + 1 then depthS I=J = d. In particular Stanleys Conjecture holds in both cases.