We show that the functor from curved differential graded algebras to differential graded categories, defined by the second author, sends Cartesian diagrams to homotopy Cartesian diagrams, under certain reasonable hypotheses. This is an extension to the arena of dg-categories of a construction of projective modules due to Milnor. As an example, we show that the functor satisfies descent for certain partitions of a complex manifold.

In this note we recall the history of Serre's problem and its beautiful solution given by D. Quillen in 1976

This paper examines the relation between the Euler class group of a Noetherian ring and the Euler class group of its polynomial extension. When the ring is a smooth affine domain, the two groups are canonically isomorphic. This is a consequence of a theorem of Bhatwadekar-Sridharan, which they proved in order to answer a question of Nori on sections of projective modules over such rings. If the smoothness assumption is removed, the result of Bhatwadekar-Sridharan is no longer valid and also the Euler class groups above are not in general isomorphic. In this paper we investigate a variant of Nori's question for arbitrary Noetherian rings and derive several consequences to understand the relation between various groups in the theory of Euler classes.

Let A be an affine algebra of dimension n over an algebraically closed field k with 1/n! ∈ k. Let P be a projective A-module of rank n − 1. Then, it is an open question due to N. Mohan Kumar, whether P is cancellative. We prove the following results:

(i) If A = R[T,T−1], then P is cancellative.

(ii) If A = R[T,1/f] or A = R[T,f1/f,…,fr/f], where f(T) is a monic polynomial and f,f1,…,fr is R[T]-regular sequence, then An−1 is cancellative. Further, if k = p, then P is cancellative.

The purpose of this paper is to extend the class of pairs A, H where H is a Hopf algebra over a field and A a right coideal subalgebra for which H is proved to be either projective or flat as an A-module. The projectivity is obtained under the assumption that H is residually finite dimensional, A has semilocal localizations with respect to a central subring, and there exists a Hopf subalgebra B of H such that the antipode of B is bijective and B is a finitely generated A-module. The flatness of H over A is shown to hold when H is a directed union of residually finite dimensional Hopf subalgebras, and there exists a Hopf subalgebra of H whose center contains A. More general projectivity and flatness results are established for (co)equivariant modules over an H-(co)module algebra under similar assumptions.

We show that over polynomial extensions of normal affine domains of dimension two over perfect fields (char. ≠ 2) of cohomological dimension [les ] 1, all finitely generated projective modules are cancellative, thus answering a question of Weibel affirmatively in the case of polynomial extensions.

For a commutative Noetherian ring A of finite Krull dimension containing the field of rational numbers, an Abelian group called the Euler class group is defined. An element of this group is attached to a projective A-module of rank = dimA and it is shown that the vanishing of this element is necessary and sufficient for P to split off a free summand of rank 1. As one of the applications of this result, it is shown that for any n-dimensional real affine domain, a projective module of rank n (with trivial determinant), all of whose generic sections have n generated vanishing ideals, necessarily splits off a free direct summand of rank 1.