Satya Mandal

Projective Modules and Complete Intersections

Preliminaires.- Patching modules and other preliminaries.- Extended modules over polynomial rings.- Modules over commutative rings.- The theory of matrices.- Complete intersections.- The techniques of lindel.

On a question of nori: the local case

In this paper we consider an algebraic problem which was motivated by a topological problem posed by Nori, about the homotopy of sections of projective modules. We give an affirmative answer in the case of some local rings, namely when the ring is a powerseries ring k[[X1,… , Xn]] over a field k or when the ring is a regular k‐spot over an infinite perfect field k.

Complete intersection K-theory and Chern classes

The purpose of this paper is to investigate the theory of complete intersection in noetherian commutative rings from the K-Theory point of view. (By complete intersection theory, we mean questions like when/whether an ideal is the image of a projective module of appropriate rank.) The paper has two parts. In part one (Section 1-5), we deal with the relationship between complete intersection and K-theory. The Part two (Section 6-8) is, essentially, devoted to construction projective modules with certain cycles as the total Chern class. Here Chern classes will take values in the Associated graded ring of the Grothedieck γ − filtration and as well in the Chow group in the smooth case. In this paper, all our rings are commutative and schemes are noetherian. To avoid unnecessary complications, we shall assume that all our schemes are connected. For a noetherian schemeX, K0(X) will denote the Grothendieck group of locally free sheaves of finite rank over X. Whenever it make sense, for a coherent sheaf M over X, [M ] will denote the class of M in K0(X). We shall mostly be concerned with X = SpecA, where A is a noetherian commutative ring and in this case we shall also use the notation K0(A) for K0(X).

On set-theoretic intersection in affine spaces

In this paper the main result is that if R = A[X] is a polynomial ring over a Cohen-Macaulay ring A and if I is a locally complete intersection ideal in R such that dim R/I≥1 and that I contains a monic polynomial in it, then I is set theoretically generated by dim A elements. We also prove some interesting consequences of this result for ideals in affine algebras.

Some results on generators of ideals

We prove some results on projective generation of ideals. c

On Automorphisms of Modules over Polynomial Rings

Write f = Xfi - bXf . Then f is in XP* and f(q) = 0. Clearly, 1 + qf is in EL( P, X) and it is a lift of 1 +pg. In a similar way we can lift 1 +pg when g is unimodular. This completes the proof of (2.8). The following is a variant of a proposition of Lindel ([L, 2.71; see 4.3 for the statement). Like the proposition of Lindel in his paper, this proposi- tion plays a key roll in the proof of our main theorems ((3.1) and (3.2)). We shall go into detailed discussions on Lindel’s proposition [L, 2.71 in our later sections (Sects. 4, 5, 6). And now we shall state our proposition. (2.9)

Study of vector bundles by restriction

The present note is motivated by the following well known result about vector bundles on projective space

Quadratic spaces over discrete hodge algebras

Ton Vorst [9] proved the analogue of Serres conjecture for discrete Hodge algebras over fields (R is called a discrete Hodge algebra over A if R = A[X0, . . . ,Xn]/I where/is an ideal generated by monomials). In [1] some more results about modules over polynomial rings were extended to discrete Hodge algebras. Here we shall extend similar results about quadratic spaces to discrete Hodge algebras. We shall prove two theorems (Theorem 2.2 and 2.4), both are extensions of Parimalas theorems (see [3,4]). In Theorem 2.2 we prove that all quadratic spaces over discrete Hodge algebras over dedekind domains, which contain an unimodular isotropy element, are extendable from the base ring. Theorem 2.4 states that quadratic spaces over discrete Hodge algebras with sufficient Witt index are cancellative.