In this article we propose a theory of Chow-Witt groups, in the paradigm of Quillen's arrow based approach to K-theory. We refer to this style of arguments as arrowtic paradigm. Other than establishing the machinery needed to work in this paradigm, we establish the homotopy invariance property of the Arrowtic Chow-Witt groups. We also compute the arrowtic Chow-Witt groups of the projective spaces, over fields.

Let X be a quasi projective scheme over a noetherian affine scheme Spec(A), U subset of X be an open subset, and Z = X-U. Assume that Z has a complete intersection subscheme structure, with k = co dim Z. Consider the map q : K (1/ (X))-* K (1/ (U)) of the K-theory spectra. We give a description of the homotopy fiber of q. Let CMZ (X) denote the full subcategory of perfect modules is an element of Coh(X) such that (1) |U = 0, (2) grade( ) = dimV(X) = k. It turns out that the homotopy fiber of q is the K-theory spectra K (CMZ (X)). Likewise, we compute the homotopy fiber of the pullback map g : GW (1/ (X))-* GW (1/ (U)) of Karoubi Grothendieck-Witt bispectra. Consequently, we obtain long exact sequences of K-groups and of GW-groups. These results settle some of the long standing open problems. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

Let A denote an affine algebra over an algebraically closed field k, with A=d≥ 3. In the light of availability of cancellation theorems for stably free modules P with rank(P)=d-1 (corank one), we try to implement the methods of complete intersections theory in corank zero, to the corank one case. Our conclusion is that cancellation theorems need to clean up some of the lack of minor generalities, for such an approach to work. However, we hypothesize and derive some of the consequences to complete intersections, of such hypotheses.

Throughout $A$ will denote commutative noetherian ring, with $\dim A=d\geq 2$, and $P$ denote a projective $A$-module with $rank(P)=n$. In \cite{MM1} we considered the Homotopy obstruction sets $\pi_0\left({\mathcal LO}(P)\right)$, which has a structure of an abelian monoid, under suitable regularity and other conditions. In this article, we provide some perspective on these sets $\pi_0\left({\mathcal LO}(P)\right)$. Under similar regularity and other conditions, we prove if $P, Q$ are two projective $A$-modules, with $rank(P)=rank(Q)=d$ and $\det(P) \cong \det Q$, then $\pi_0\left({\mathcal LO}(Q)\right)\cong \pi_0\left({\mathcal LO}(P)\right)$. Further, for any projective $A$-module $P$ with $rank(P)=n$, we define a natural set theoretic map $\pi_0\left({\mathcal LO}(P)\right)\rightarrow CH^n(A)$, where $CH^n(A)$ Chow groups of codimension $n$ cycles.

Let $X=Spec{A}$ denote a regular affine scheme, over a field $k$, with $1/2\in k$ and $\dim X=d$. Let $P$ denote a projective $A$-module of rank $n\geq 2$. Let $\pi_0\left({\mathcal LO}(P)\right)$ denote the (Nori) Homotopy Obstruction set, and $\widetilde{CH}^n\left(X, \Lambda^nP\right)$ denote the Chow Witt group. In this article, we define a natural (set theoretic) map} $$ \Theta_P: \pi_0\left({\mathcal LO}(P)\right) \longrightarrow \widetilde{CH}^n\left(X, \Lambda^nP\right) $$

Let A be a regular ring over a field k, with 1/ 2 is an element of k, and dim A = d. We discuss the Homotopy Obstruction Program, in the complete intersection case. Fix an integer n = 2. A local orientation is a pair (I, omega), where I is an ideal and omega : A(n) >> I/I-2 is a surjective map. The goal is to define and detect homotopy obstructions, for omega to lift to a surjective map A(n) ->-> I. Denote the set of all local orientations by LO(A, n). A homotopy relations on LO(A, n) is induced by the maps LO(A, n) <-(T=0) LO(A[T], n) ->(T=1) LO(A, n). The homotopy obstruction set pi(0)(LO(A, n)) is defined to be the set of all equivalence classes. Assume 2n >= d+2. We prove that pi(0)(LO(A, n)) is an abelian group. We also establish a surjective map rho : E-n(A) ->-> pi(0)(LO(A, n)), where E-n(A) denotes the Euler class group. When 2n >= d + 3, and A is essentially smooth, we prove rho is an isomorphism. This settles a conjecture of Morel.

Let A be a commutative noetherian ring, containing a field k, with 1/2∈k, dim⁡A=d, and let P be a projective A-module, with rank(P)=n. Let LO(P) denote the set of all pairs (I,ω), where I is an ideal of A and ω:P↠I/I2 is a surjective map. The homotopy relations on LO(P), induced by LO(P[T]), leads to a set π0(LO(P)) of equivalence classes in LO(P). There are two distinguished elements e0,e1∈π0(LO(P)), respectively, the images of (0,0) and (A,0). Define the obstruction classε(P)=e0∈π0(LO(P)), to be called the (Nori) homotopy class of P. The following results are under suitable smoothness or regularity hypotheses. We prove, if 2n≥d+2, then π0(LO(P)) has a natural structure of a monoid, which is a group if P≅Q⊕A. When 2n≥d+3, we proveP≅Q⊕A⟺ε(P)=e1(“the additive zero”).Further, we give a definition of a Euler class group E(P). Under suitable smoothness hypotheses, we prove, if P≅Q⊕A and 2n≥d+3, then there is natural isomorphism E(P)⟶∼π0(LO(P)) of groups.

This is an erratum to [4] and, as well, to [3]. This article provides examples to indicate inconsistencies, in [3], [4].

In this article, we prove some results on Witt, Grothendieck–Witt (GW) and K-theory of noetherian quasi-projective schemes X, over affine schemes Spec(A). For integers k≥0, let CMk(X) denote the category of coherent OX-modules F, with locally free dimension dimV(X)⁡(F)=k=grade(F). We prove that there is an equivalence Db(CMk(X))→Dk(V(X)) of the derived categories. It follows that there is a sequence of zig-zag maps K(CMk+1(X))⟶K(CMk(X))⟶∐x∈X(k)K(CMk(Xx)) of the K-theory spectra that is a homotopy fibration. In fact, this is analogous to the homotopy fiber sequence of the G-theory spaces of Quillen (see proof of [16, Theorem 5.4]). We also establish similar homotopy fibrations of GW-spectra and GW-bispectra, by application of the same equivalence theorem.

This is essentially an erratum, with some example to indicate inconsistencies. Suppose $A=k[X_1, X_2, \ldots, X_n]$ is a polynomial ring over a field $k$. The Complete Intersection conjecture states that, for any ideal $I$ in $A$, $\mu(I)=\mu(I/I^2)$, where $\mu$ denotes the minimal number of generators. When $k$ is an infinite field, with $1/2\in k$, a proof of this conjecture was claimed recently, which was a consequence of a stronger claim. A counter example of this stronger claim surfaced recently. This note discusses such examples and attempts to provide some clarity to the inconsistencies in the literature.

Suppose A=k[X1,X2,…,Xn] is a polynomial ring over a field k and I is an ideal in A. M.P. Murthy conjectured that μ(I)=μ(I/I2), where μ denotes the minimal number of generators. Recently, Fasel [3] settled this conjecture, affirmatively, when k is an infinite perfect field, with 1/2∈k (always). We are able to do the same, when k is an infinite field. In fact, we prove similar results for ideals I in a polynomial ring A=R[X], that contains a monic polynomial and R is essentially smooth algebra over an infinite field k, or R is a regular ring over a perfect field k.

In this paper we extend and apply the work of Paul Balmer and others on derived and triangular Witt Groups. We obtain a generalized form of dévissage for derived Witt Groups over Cohen-Macaulay rings.

We define isomorphic Binary Structures. Main point is, if two binary structures are isomorphic, then propertes of one translate over to properties of the other, via the isomorphism. So, if we know one we know the other. We do not have to study two of them seperately. Definition 3.1. By a binary structure 〈S, ∗〉 , we mean a set S with a binary operation ∗ on it. Definition 3.2. Let 〈S, ∗〉 and 〈T, ∗′〉 be two binary structures. 1. A map φ : S −→ T is called (a map of) or a homomorphism of binary structures if φ(x ∗ y) = φ(x) ∗ φ(y) ∀ x, y ∈ S. 2. A map φ : S −→ T is called an isomorphism of binary structures if φ(x ∗ y) = φ(x) ∗ φ(y) ∀ x, y ∈ S. and if φ is a bijection. (Emphasis in this section is on isomorphic structures; not on homomorphisms) Example 3.3. Let U = {z ∈ C : |z| = 1} be the unit circle. Then, with usual multiplication, 〈U, ·〉 is a binary structure. On the interval [0, 2π) the addition "modulo 2π provides a binary structure ([0, 2π),+). The map φ : [0, 2π) −→ U defined by φ(t) = e is an isomorphism of binary structures.

In this article we show multiple ways of constructing sublagrangians of symmetric forms phi : epsilon(center dot) (->) over tilde epsilon(#)(center dot) in the bounded derived category D-b(V(X)) of complexes of locally free sheaves over quasi-projective regular (sometimes without regularity) schemes X, over noetherian affine schemes Spec(A). By application of the sublagrangian theorem of Balmer, for Witt theory of triangulated categories, we prove some results regarding structure of forms in the Witt groups W-n(D-k(X)) of the filtered subcategories

For quasi-projective schemes X over affine schemes Spec(A), resolving subcategories A of Coh(X) were considered. The equivalences of derived categories were established, where Mgk(A)={F∈Coh(X):dimA⁡(F)<∞,grade(F)≥k} and Dk denote the corresponding filtration of the derived category.

We consider bounded complexes P• of finitely generated projective A-modules whose homologies have finite projective dimension and are locally Cohen–Macaulay. We give a necessary and sufficient condition so that its dual P•⁎ also has the same property.

In this article we establish some formalism of derived Witt theory for resolving subcategories of abelian categories. Results directly apply to noetherian schemes.