It is shown that a free topological group on a k.space is a k.-space. Using this it is proved that if X is a k.-group then it is a quotient of a free topological group by a free topological group. A corollary to this is that the projective dimension of any k.-group, relative to the class of all continuous epimorphisms admitting sections, is either zero or one. In particular the projective dimen...

We prove the existence of an algorithm that, from a finite set of polynomials defining an algebraic projective variety, computes the Chow form of its equidimensional component of the greatest dimension. Applying this algorithm, a finite set of polynomials defining the equidimensional component of the greatest dimension of an algebraic (projective or affine) variety can be computed. The complexi...

Based on work of Rouquier, some bounds for Aulander’s representation dimension are discussed. More specifically, if X is a reduced projective scheme of dimension n over some field, and T is a tilting complex of coherent OX-modules, then the representation dimension of the endomorphism algebra EndOX (T ) is at least n.

We show that for the edge ideals of a certain class of forests, the arithmetical rank equals the projective dimension.

We prove a new upper bound on the essential p-dimension of the projective linear group PGLn.

We give a proof of Dyson’s Lemma for a product of smooth projective varieties of arbitrary dimension.

Based on work of Rouquier, some bounds for Aulander’s representation dimension are discussed. More specifically, if X is a reduced projective scheme of dimension n over some field, and T is a tilting complex of coherent OX-modules, then the representation dimension of the endomorphism algebra EndOX (T ) is at least n.

In this paper, we prove that the global Gorenstein projective dimension of a ring R is equal to the global Gorenstein injective dimension of R, and that the global Gorenstein flat dimension of R is smaller than the common value of the terms of this equality.

Let X be a smooth projective variety, let L be a very ample invertible sheaf on X and assume N+1 = dim(H(X,L)), the dimension of the space of global sections of L. Let P1, . . . , Pt be general points on X and consider the blowing-up π : Y → X of X at those points. Let Ei = π−1(Pi) be the exceptional divisors of this blowing-up. Consider the invertible sheaf M := π∗(L) ⊗ OY (−E1 − . . . − Et) o...