arXiv subjects
Publications and source records attributed to Satya Mandal.
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 ${\mathbb F}_0$ be an algebraically closed field, with $char({\mathbb F}_0)=0$. In this article, for prime numbers $p\geq 2$, we construct smooth affine algebras $B$ over ${\mathbb F}_0$, with $\dim B=p+2$. Further, we construct projective $B$-modules $Q$ with $rank(Q)=p$, such that $x=[Q] -[B^p]\neq 0$ in $K_0(B)$ and the total Chern class $C(Q)=1+\sum_{i=1}^{p}C^k(Q) =1$ is trivial. We use the splitting theorem in \cite{ABH} that for projective $B$-modules $P$ with $rank(P)=r=\dim B-1$, vanishing $C^r(P)=0 \Longrightarrow P\cong Q\oplus B$.
Let $A$ denote an affine algebra over an algebraically closed field ${\mathbb F}$, with $\dim A=d\geq 3$. Due to the recent exciting activities regarding projective $A$-modules $P$, with $rank{P}=d-1$ (corank one), we explore the corank zero methods of N. Mohan Kumar and M. P. Murthy, in the complete intersections theory. We hypothesize regarding cancellation properties projective modules of corank one, and derive some of analogues of the results in corank zero case. We obtain some affirmative results, when $A$ is an affine algebra over $\overline{{\mathbb F}}_p$.
We prove Dévissage theorems for Hermitian $K$ Theory (or $GW$ theory), analogous to Quillen's Dévissage theorem for $K$-theory. For abelian categories ${\mathscr A}:=({\mathscr A}, ^{\vee}, \varpi)$ with duality, and appropriate abelian subcategories ${\mathscr B}\subseteq {\mathscr A}$, we prove Dévissage theorems for ${\bf GW}$ spaces, $G{\mathcal W}$-spectra and ${\mathbb G}W$ bispectra. As a consequence, for regular local rings $R$ with $1/2\in R$, we compute the ${\mathbb G}W$ groups ${\mathbb G}W^{[n]}_k(Spec{R})$ forall $k, n\in {\mathbb Z}$, where $n$ represent the translation.
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 $π_0\left({\mathcal LO}(P)\right)$ denote the (Nori) Homotopy Obstruction set, and $\widetilde{CH}^n\left(X, Λ^nP\right)$ denote the Chow Witt group. In this article, we define a natural (set theoretic) map} $$ Θ_P: π_0\left({\mathcal LO}(P)\right) \longrightarrow \widetilde{CH}^n\left(X, Λ^nP\right) $$ The main Results are included in my recently published book on Algebraic $K$-Theory.
Let $X$ be a quasi projective scheme over a noetherian affine scheme $Spec(A)$, $U\subseteq X$ be an open subset, and $Z=X-U$.Assume that $Z$ is complete intersection, with $k=codim Z$. Consider the map $$ q:{\mathbb K}\left({\mathscr V}(X)\right) \rightarrow {\mathbb K}\left({\mathscr V}(U)\right) $$ of the ${\mathbb K}$-theory spectra. We give a description of the homotopy fiber of $q$. Let $C{\mathbb M}^Z\left(X\right)$ denote the full subcategory of perfect modules ${\mathscr F} \in Coh(X)$ such that(1) ${\mathscr F} _{|U}=0$, (2) $grade({\mathscr F} )=\dim_{{\mathscr V}(X)}{\mathscr F}=k $. It turns out that the homotopy fiber of $q$ is the ${\mathbb K}$-theory spectra ${\mathbb K}\left(C{\mathbb M}^Z\left(X\right)\right)$. Likewise, we compute the homotopy fiber of the pullback map $$ g: {\mathbb G}W\left({\mathscr V}(X)\right) \rightarrow {\mathbb G}W\left({\mathscr V}(U)\right) $$ of Karoubi Grothendieck-Witt bispectra. Consequently, we obtain long exact sequences of ${\mathbb K}$-groups and of ${\mathbb G}W$-groups. These results settle some of the long standing open problems. We also inserted a conjecture.
Let $A$ be a commutative noetherian ring, containing a field $k$, with $1/2\in k$, $\dim A=d$, and let $P$ be a projective $A$-module or $rank(P)=n$. In continuation of \cite{MM}, we study Homotopy obstructions for $P$ to split off a free direct summand. Let ${\mathcal LO}(P)$ be the set of all pairs $(I, ω)$, where $I$ is an ideal of $A$ and $ω: P\rightarrow I/I^2$ is a surjective map. The homotopy relations on ${\mathcal LO}(P)$, induced by ${\mathcal LO}(P[T])$, leads to a set $π_0\left({\mathcal LO}(P)\right)$ of equivalence classes in ${\mathcal LO}(P)$. There are two distinguished elements ${\bf e}_0, {\bf e}_1\in π_0\left({\mathcal LO}(P)\right)$, respectively, the images of $(0, 0)$ and $(A, 0)$. Define the obstruction class $e(P)={\bf e}_0\in π_0\left({\mathcal LO}(P)\right)$. The following results are under suitable smoothness or regularity hypotheses. When $2n\geq d+3$, we prove $e(P)={\bf e}_1 \Leftrightarrow P\cong Q\oplus A$. We prove, if $2n\geq d+2$, then $π_0\left({\mathcal LO}(P)\right)$ has a natural structure of a monoid, which is a group if $P\cong Q\oplus A$. Further, we give a definition of a Euler class group $E(P)$. Under suitable smoothness hypotheses, we prove, if $P\cong Q\oplus A$ and $2n\geq d+3$, then there is natural isomorphism $E(P) \rightarrow π_0\left({\mathcal LO}(P)\right)$ of groups.
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 $π_0\left({\mathcal LO}(P)\right)$, which has a structure of an abelian monoid, under suitable regularity and other conditions. In this article, we provide some perspective on these sets $π_0\left({\mathcal LO}(P)\right)$. 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 $π_0\left({\mathcal LO}(Q)\right)\cong π_0\left({\mathcal LO}(P)\right)$. Further, for any projective $A$-module $P$ with $rank(P)=n$, we define a natural set theoretic map $π_0\left({\mathcal LO}(P)\right)\rightarrow CH^n(A)$, where $CH^n(A)$ Chow groups of codimension $n$ cycles.
Let $A$ be a regular ring over a field $k$, with $1/2\in k$ and dimension $d$. We discuss the Homotopy Conjecture of Madhav V. Nori, in the complete intersection case (meaning when the projective module in question if free, of rank at least 2). Recently, an obstruction set (sheaf) $π_0(Q_{2n})(A)$ was introduced [F] to detect when a surjective map $A^n\to I/I^2$ lifts to a surjective map $A^n\to I$. We establish that $π_0(Q_{2n})(A)$ coincides with the obstruction set of equivalence classes, originally suggested by Nori. We also establish that $π_0(Q_{2n})(A)$ has a natural groups structure, when $2n\geq d+2$. Further, we establish that, when $2n\geq d+2$, there is a surjective homomorphism $E^n(A) \to π_0(Q_{2n})(A) $, where $E^n(A)$ denotes the Euler class group defined by Bhatwadekar and Sridharan [BS2]. This homomorphism is an isomorphism, whenever triviality, in $π_0(Q_{2n})(A)$, of an orientation $(I, ω_I), guarantees that $omega_I$ lifts to a surjective map $A^n\to I$. We also give a Quadratic version of Lindel's Theorem, on extendibility of projective modules.
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 $A$, $μ(I)=μ(I/I^2)$, where $μ$ 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 $A=k[X_1, X_2, \ldots, X_n]$ is a polynomial ring over a field $k$ and $I$ is an ideal in $A$. Then M. P. Murthy conjectured that $μ(I)=μ(I/I^2)$, where $μ$ denotes the minimal number of generators. Recently, Fasel \cite{F} settled this conjecture, affirmatively, when $k$ is an infinite perfect field, with $1/2\in k$ {\rm (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 finite type smooth algebra over an infinite field $k$, or $R$ is a regular ring over a perfect field $k$.
In this article we continue our investigation of the Derived Equivalences over noetherian quasi-projective schemes $X$, over affine schemes $\spec{A}$. For integers $k\geq 0$, let $C{\mathbb M}^k(X)$ denote the category of coherent ${\CO}_X$-modules ${\mathcal F}$, with locally free dimension $proj\dim(\CF)=k=grade({\mathcal F})$. We prove that there is a zig-zag equivalence ${\mathcal D}}^b\left(C{\mathbb M}^k(X)\right) \to {\mathcal D}^k\left({\mathcal V}(X)\right)$ of the derived categories. It follows that there is a sequence of zig-zag maps ${\mathbb K}\left(C{\mathbb M}^{k+1}(X)\right) \to {\mathbb K}\left(C{\mathbb M}^{k}(X)\right) \to \coprod_{x\in X^{(k)}} {\mathbb K}\left(C{\mathbb M}^{k}(X_x)\right) \\ $of the $\K$-theory spectra that is a homotopy fibration. In fact, this is analogous to the fibrations of the $G$-theory spaces of Quillen (see proof of \cite[Theorem 5.4]{Q}). We also establish similar homotopy fibrations of ${\bf GW}$-spectra and ${\mathbb G}W$-bispectra.
In this article we establish some formalism of Derived Witt-Dévissage theory for resolving subcategories of abelian categories. Results directly apply to noetherian schemes.
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 will give another definition of Euler class group of a Noetherian ring.
Let $k$ be a field of characteristic $\ne 2$ and let $Q_{n,m}(x_1, ...,x_n,y_1, ...,y_m)=x_1^2+ ... +x_n^2-(y_1^2+ ... +y_m^2)$ be a quadratic form over $k$. Let $R(Q_{n,m})=R_{n,m}=k[x_1, ...,x_n,y_1, ...,y_m]/(Q_{n,m}-1)$. In this note we will calculate $\wt K_0(R_{n,m})$ for every $n,m \geq 0$.
Throughout this abstruct $A$ will denote a noetherian commutative ring of dimension $n$. The paper has two parts. Among the interesting results in Part-1 are the following: 1) {\it suppose that $f_1, f_2, ..., f_r$ (with $r \leq n$) is a regular sequence in $A$ and suppose $Q$ is a projective $A$-module of rank $r$ that maps onto the ideal $(f_1, f_2, ..., f_{r-1},f_r^{(r-1)!})$. Then $[Q]=[Q_0 \oplus A]$ in $K_0(A)$ for some projective $A-module~Q_0$ of rank $r-1$.} 2) The set $$F_0K_0(A) = \{[A/I] \in K_0(A): I~ is~ a~ locally~ complete ~intersection~ ideal~ in~ A~ of~ height~n \}$$ is a {\it subgroup} of $K_0(A)$. We also show that if $A$ is a reduced affine algebra over a field $k$ then $F_0K_0(A)$ {\it is indeed the Zero Cycle Subgroup of} $K_0(A)$ {\it that is generated by smooth maximal ideals} $\Cal M$ {\it of height} $n$. 3){\it let $A$ be such that whenever $I$ is a locally complete intersection ideal of height $n$ with $[A/I]=0$ then $I$ is the image of a projective $A-module$ of rank $n$. Then for any locally complete intersection ideal $J$ of height $n$ with $[A/J]$ divisible by $(n-1)!$ in $F_0K_0(A)$, there is a projective $A-module$ of rank $n$ that maps onto $J$}. The main result in Part-2 is the following construction: 1) {\it let $X=Spec A$ be a Cohen-Macaulay scheme of dimension $n$ and let $r_0,~r$ be two integers with $n/2 \leq r_0 \leq r \leq n$. Let i) $Q_0$ be a projective $A-module$ of rank $r_0-1$ such that the restriction $Q_0|Y$ is trivial for all locally complete intersection subvarieties $Y$ of codimension at least $r_0$. Also ii) for $k= r_0$ to $r$, let $I_k$ be locally complete intersection ideals of height $k$ so that $I_k/I_k^2$ has a generators of the type $f_1, ..., f_{k-1}, f_k^{(k-1)!}$. Then there is a projective $A-module~Q$ of rank $r$ such that