All rings in this paper are commutative with unity; we will deal mainly with integral domains. Let R be a ring with total quotient ring K. A fractional ideal I of R is invertible if II−1 = R; equivalently, I is a projective module of rank 1 (see, e.g., [Eis95, Section 11.3]). Here, I−1 = (R : I) = {x ∈ K |xI ⊆ R}. Moreover, a projective R-module of rank 1 is isomorphic to an invertible ideal. (...

Let I be a square-free monomial ideal of projective dimension one. Starting with the Taylor complex on generators $$I^r$$ , we use discrete Morse theory to describe CW that supports minimal free resolution . To do so, concretely acyclic matching faces complex.

in this paper we show that if q is a power of a prime p , then the projective special linear group psl(2, q) and the stabilizer of a point of the projective line have maximum sum element orders among all proper subgroups of projective general linear group pgl(2, q) for q odd and even respectively

We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal of morphisms with certain properties, and that if has these properties, then so does each of its powers. We show how a projective class leads to an Adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence. We use this to study v...

We investigate containment statements between symbolic and ordinary powers bounds on the Waldschmidt constant of defining ideals points in projective spaces. establish stable Harbourne conjecture for ideal a general set points. also prove Chudnovsky’s Conjecture version Harbourne–Huneke conjectures sufficiently many

The main result of the present article is a (practically optimal) criterium for the pseudoeffectivity of the twisted relative canonical bundles of surjective projective maps. Our theorem has several applications in algebraic geometry; to start with, we obtain the natural analytic generalization of some semipositivity results due to E. Viehweg and F. Campana. As a byproduct, we give a simple and...

In this paper we will compare the connectivity dimension c(P/I) of an ideal I in a polynomial ring P with that of any initial ideal of I. Generalizing a theorem of Kalkbrener and Sturmfels [18], we prove that c(P/LT≺(I)) ≥ min{c(P/I), dim(P/I)−1} for each monomial order ≺. As a corollary we have that every initial complex of a Cohen-Macaulay ideal is strongly connected. Our approach is based on...

We study different notions of slope of a vector bundle over a smooth projective curve with respect to ampleness and affineness in order to apply this to tight closure problems. This method gives new degree estimates from above and from below for the tight closure of a homogeneous R+-primary ideal in a two-dimensional normal standard-graded algebra R in terms of the minimal and the maximal slope...

We exhibit an example of a line bundle M on a smooth complex projective variety Y s.t. M satisfies Property Np for some p, the p-module of a minimal resolution of the ideal of the embedding of Y by M is nonzero and M does not satisfy Property Np. Let M be a very ample line bundle on a smooth complex projective variety Y and let φM : Y → P(H (Y,M)) be the map associated to M . We recall the defi...

In this paper we will compare the connectivity dimension c(P/I) of an ideal I in a polynomial ring P with that of any initial ideal of I. Generalizing a theorem of Kalkbrener and Sturmfels [18], we prove that c(P/LT≺(I)) ≥ min{c(P/I), dim(P/I)−1} for each monomial order ≺. As a corollary we have that every initial complex of a Cohen-Macaulay ideal is strongly connected. Our approach is based on...