A note on projective modules over real affine algebras

Let A be an affine algebra over the field of real numbers of dimension d. Let f \in A be an element not belonging to any real maximal ideal of A. Let P be a projective A-module of rank \geq d-1. Let (a,p) \in A_f \oplus P_f be a unimodular element. Then the projective A_f module Q=A_f \oplus P_f/(a,p)A_f is extended from A.Comment: 11 page

Stability result for projective modules over blowup rings

Let R be an affine algebra of dimension n \geq 3 over an algebraically closed field k. Suppose char k =0 or char k =p \geq n. Let g,f_1,...,f_r be a R-regular sequence and A=R[f_1/g,...,f_r/g]. Let P be a projective A-module of rank n-1 which is extended from R. Let (a,p) \in (A \op P) be a unimodular element and Q=A\op P/(a,p)A. Then, Q is extended from R. A similar result for affine algebras over reals are also proved.Comment: 15 page

4-Weighted Fractional Fourier Transform based Multiple Image Encryption Approach with PAN

In this manuscript, a new encryption approach for multiple images is proposed based on 4-weighted fractionalfrequency transform (4-WFRFT) domain. First, the low frequency-components of all the images are obtained byapplying Fourier Transform on each image, which positioned at corner position of image, shifted to the centralposition. Low-frequency component of each individual image is then scrambled with help of Arnold cat mapwith its parameters and combined all scrambled image to form a single image with the same size that of originalimage which is now ready for encryption process. Here, parameters of Arnold cat map and transform order of 4-WFRFT treated as secret keys which are converted from Permanent Account Number (PAN) of authorize person.The encrypted image information generated by authorize person can be recovered by applying PAN at receiverside

A Question of Nori: Projective Generation of Ideals

Abstract. Let A be a smooth affine domain of dimension d over an infinite perfect field k and let n be an integer such that 2n . Under these assumptions, it is proved in this paper that I Mathematics Subject Classification

Euler class group of a Laurent polynomial ring: local case

Let A be a commutative Noetherian ring of dimension d. A classical result of Serre [18] asserts that if P is a projective A-module of rank> d, then P has a unimodular element. It is well known that this result is not true in general if rank P = d = dim A. Therefore, it is interesting to know the obstruction for projective A-modules of rank = dimA to hav

A question of Nori: projective generation of ideals

Let A be a smooth affine domain of dimension d over an infinite perfect field k and let n be an integer such that 2n ≥ d + 3. Let I ⊂A[T] be an ideal of height n. Assume that I = (f 1,...,f n ) + (I 2 T). Under these assumptions, it is proved in this paper that I = (g 1,...,g n ) with f i - g i⊂ (I 2 T), thus settling a question of Nori affirmatively