Let (R,m) be a Noetherian normal local domain with m as the maximal ideal. To this ring R, we assign a simple undirected graph RSG(R) which we call as a regular sequence graph of R. The vertices, V(RSG(R)), of this graph are distinct non-zero elements of m and two vertices x, y are adjacent if and only if {x,y} forms a R-regular sequence. In this paper, we show that the RSG(R) is connected, Hausdorff, and it has an infinite clique. Moreover we show that the regular sequence graph RSG(R) is compact if and only if the class group Cl(R) is torsion.

Let k denote an algebraically closed field of characteristic zero and let B be an affine normal k-domain. Let delta be a locally nilpotent derivation on B and let A be the kernel of delta. It is known that A need not be affine over k. In this paper we show that even if A is not affine, A is almost affine in the sense that there is an affine normal k-domain C and two nonzero elements x, y in C such that A = C[1/x] boolean AND C[1/y].

Let k be a field and let B be an affine normal domain over k. Let $$\phi $$ be a non-trivial exponential map on B and let $$A = B^{\phi }$$ be the ring of $$\phi $$-invariants. Since A is factorially closed in B, $$A = K \cap B$$ where K denotes the field of fractions of A. Hence A is a Krull domain. We investigate here a relation between the class group $$\mathrm{Cl}(A)$$ of A and the class group $$\mathrm{Cl}(B)$$ of B. In this direction, we give a sufficient condition for an injective group homomorphism from $$\mathrm{Cl}(A)$$ to $$\mathrm{Cl}(B)$$. We also give an example to show that $$\mathrm{Cl}(A)$$ may not be realized as a subgroup of $$\mathrm{Cl}(B)$$.

We will briefly describe the basic theory of non-complete algebraic varieties developed by Japanese algebraic geometers and some of the main contributions of Indian mathematicians in this area during the last ten years.

Let k be an algebraically closed field and let A be an affine domain of dimension one over k. Let P be a finitely generated projective A-module of rank d and let R=SymA(P) be the symmetric algebra of P over A. Assume that A is not a polynomial algebra over k. In this article we show that, under these assumptions, the Makar-Limanov invariant ML(R)=A.

Let k be an algebraically closed field of characteristic zero, D a locally nilpotent derivation on the polynomial ring k[X-1, X-2, X-3, X-4] and A the kernel of D. A question of M. Miyanishi asks whether projective modules over A are necessarily free. Implicit is a subquestion: whether the Grothendieck group K-0(A) is trivial.In this paper we shall demonstrate an explicit k[X-1]-linear fixed point free locally nilpotent derivation D of k[X-1, X-2, X-3, X-4] whose kernel A has an isolated singularity and whose Grothendieck group K-0(A) is not finitely generated; in particular, there exists an infinite family of pairwise non-isomorphic projective modules over the kernel A.We shall also show that, although Miyanishi's original question does not have an affirmative answer in general, suitably modified versions of the question do have affirmative answers when D annihilates a variable. For instance, we shall establish that in this case the groups G(0)(A) and G(1)(A) are indeed trivial. Further, we shall see that if the above kernel A is a regular ring, then A is actually a polynomial ring over k; in particular, by the Quillen-Suslin theorem, Miyanishi's question has an affirmative answer.Our construction involves rings defined by the relation umv = F(z, t), where F(Z, T) is an irreducible polynomial in k[Z, T]. We shall show that a necessary and sufficient condition for such a ring to be the kernel of a k[X-1]linear locally nilpotent derivation D of a polynomial ring k[X-1,..., X-4] is that F defines a polynomial curve.

In [On affine-ruled rational surfaces, Math. Ann.255(3) (1981) 287–302], Russell had proved that when k is a perfect field of positive characteristic, the polynomial ring k[X, Y] is cancellative. In this note, we shall show that this cancellation property holds even without the hypothesis that k is perfect.

Let A be an affine domain of dimension n over a field of characteristic zero. Let I. A[T] be a local complete intersection ideal of height n such that mu(I/I-2) = n. This paper examines under what condition I is surjective image of a projective A[T]-module of rank n. More specifically, one is interested in knowing when is an element (I,omega(I)) of the Euler class group E(A[T]) obtained as the Euler class of a projective A[T]-module. It is proved that such a phenomenon occurs if and only if the naturally induced element (I (0),omega(I(0))) of the Euler class group E(A) is obtained as the Euler class of a projective A-module.

. Let S be a principal ideal domain. Re- call that a Laurent polynomial algebra over S is an S algebra of the form S [ T 1 ;:::;T n ;T (cid:0) 1 1 ;:::;T (cid:0) 1 n ]. Gener- alizing this notion, we call an S -algebra of the form S [ T 1 ;:::;T n ;f (cid:0) 1 1 ;:::;f (cid:0) 1 n ] a quasi Laurent polynomial al- gebra in n variables over S if T 1 ;:::;T n are algebraically independent over S and f i = a i T i + b i , where a i 2 S n 0 and b i 2 S are such that ( a i ;b i ) S = S , for each i = 1 ;:::;n . It has been shown recently that a locally Laurent polynomial algebra in n variables over S is itself a Laurent polynomial algebra. Now suppose A is a locally quasi Laurent polynomial algebra in n variables over S . In this note, we investi- gate the question: ‘is A necessarily quasi Laurent polynomial in n variables over S ?’ We (cid:12)rst give a su(cid:14)cient condition for the question to have an a(cid:14)rmative answer. Moreover, when S is semi-local with two maximal ideals and contains the (cid:12)eld of rationals Q , we give examples of S -algebras which are locally quasi Laurent polynomial in two variables but not quasi Laurent polynomial in two variables.

This is my PhD thesis from 2004 under Prof. S.M. Bhatwadekar. Here we answer a question of Nori and prove the following result. Let A be a smooth affine domain of dimension d over an infinite perfect field. Let I be an ideal of A[T] of height n such that 2n ≥ d+3. Given surjections ϕ:(A[T]/I)^n → I/I^2 and ρ :A^n → I(0) such that ϕ(0)=ρ⊗ A/I(0), then there exist a surjection Φ :A[T]^n → I such that Φ(0)=ρ and Φ⊗ A/I =ϕ. This is a joint work with S.M. Bhatwadekar.

Let (R,π) be a D.V.R. with quotient field K and residue field k. We call an R-algebra A to be quasi Laurent polynomial (abbreviated as quasi LP) in n variables over R if A=R[T1,…,Tn,(a1T1+b1)−1,…,(anTn+bn)−1], where T1,…,Tn are algebraically independent over R and ai∈R∖0, bi∈R are such that (ai,bi)R=R, for i=1,…,n. If an R-algebra A is quasi LP in n variables, then (1) A is a finitely generated, faithfully flat R-algebra, (2) the generic fibre A⊗RK is a Laurent polynomial algebra in n variables over K and (3) the closed fibre A/πA≅k[X1,…,Xr,Y1,Y1−1,…,Ys,Ys−1], where r+s=n. Therefore, it is natural to ask: if an R-algebra A satisfies the above three conditions, then is A necessarily quasi LP in n variables? We give examples to show that, in general, this question does not have an affirmative answer if n=2 and r≥1.

We exhibit an example of a smooth affine threefold A over a field of characteristic 0 for which there exist non-trivial 2-torsion elements in the Euler class group E(A) vanishing in the weak Euler class group E0(A). This gives a positive answer to a question of the first author and Raja Sridharan.

Let R R be a Noetherian normal domain. Call an R R -algebra A A “locally A 1 \mathbb {A}^{1} in codimension-one” if R P ⊗ R A R_P \otimes _R A is a polynomial ring in one variable over R P R_P for every height-one prime ideal P P in R R . We shall describe a general structure for any faithfully flat R R -algebra A A which is locally A 1 \mathbb {A}^{1} in codimension-one and deduce results giving sufficient conditions for such an R R -algebra to be a locally polynomial algebra. We also give a recipe for constructing R R -algebras which are locally A 1 \mathbb {A}^{1} in codimension-one. When R R is a normal affine spot (i.e., a normal local domain obtained by a localisation of an affine domain), we give criteria for a faithfully flat R R -algebra A A , which is locally A 1 \mathbb {A}^{1} in codimension-one, to be Krull and a further condition for A A to be Noetherian. The results are used to construct intricate examples of faithfully flat R R -algebras locally A 1 \mathbb {A}^{1} in codimension-one which are Noetherian normal but not finitely generated.

Bass, Connell and Wright have proved that any finitely presented locally polynomial algebra in n variables over an integral domain R is isomorphic to the symmetric algebra of a finitely generated projective R-module of rank n. In this paper we prove a corresponding structure theorem for a ring A which is a locally Laurent polynomial algebra in n variables over an integral domain R, viz., we show that A is isomorphic to an R-algebra of the form (SymR(Q))[I−1], where Q is a direct sum of n finitely generated projective R-modules of rank one and I is a suitable invertible ideal of the symmetric algebra SymR(Q). Further, we show that any faithfully flat algebra over a Noetherian normal domain R, whose generic and codimension-one fibres are Laurent polynomial algebras in n variables, is a locally Laurent polynomial algebra in n variables over R.

Let R be Noetherian normal domain. We shall call an R-algebra A quasi A* if A = R[X (aX + b)(-1)] where X is an element of A is a transcendental element over R, a is an element of R \ 0, b is an element of R and (a, b)R = R. In this paper we shall describe a general structure for any faithfully flat R-algebra A which is locally quasi A* in codimension-one over R. We shall also investigate minimal sufficient conditions for such an algebra to be finitely generated. (C) 2011 Elsevier B.V. All rights reserved.

Let X=Spec(A) be a smooth, affine variety of dimension n⩾2 over the field R of real numbers. Let P be a projective A-module of rankn such that its nth Chern class Cn(P)∈CH0(X) is zero. In this set-up, Bhatwadekar–Das–Mandal showed (amongst many other results) that P≃A⊕Q in the case that either n is odd or the topological space X(R) of real points of X does not have a compact, connected component. In this paper, we prove that similar results hold for smooth, affine varieties over an arbitrary real closed field R. The proof is algebraic and does not make use of Tarski's principle, nor of the earlier result for R.

Let R be a noetherian domain containing the field of rationals. We show that if R is Dedekind then the kernel of any locally nilpotent R-derivation of R[X,Y,Z] is a finitely generated R-algebra. Conversely, we show that if R is neither a field nor a Dedekind domain then there exists a locally nilpotent R-derivation of R[X,Y,Z] whose kernel is not finitely generated over R.

Let R be a discrete valuation ring with quotient field K and residue field k. For a finitely generated integrally closed domain A over R, we give an explicit algebraic structure of the reduced k-algebra (A circle times(R) k)(red) when the generic fibre A circle times(R) K is a polynomial ring or a Laurent polynomial ring in one variable over K.

Let k be a C1-field of characteristic zero. Let A be an affine algebra of dimension d⩾2 over k. In this set up, Suslin proved that the free module Ad is cancellative (in other words, stably free A-modules of rank d are free). In this note we show that, in fact, all finitely generated projective A-modules of rank d are cancellative.