Speaker: Satyagopal Mandal
Affiliation: University of Kansas
Title: Perfect Modules and K-Theory localization sequences
Date and Time: May 22, 2024, 14:30:00 Hours
Venue: AG-77
Asbtract: Given a map $f:(Y, \star) \to (X, \star)$ of pointed topological spaces, it extends to a homotopy fibration sequence ${\mathscr F}(f), \star)\to (Y, \star) \xrightarrow{f}(X, \star)$. One says that every arrow $f$ fits into a "triangle". This leads to a long exact sequence of homotopy groups: $$\cdots \to \pi_{n-1}(X, \star) \to \pi_n({\mathscr F}(f), \star)\to \pi_n(Y, \star) \xrightarrow{\iota}\to \pi_n(X, \star)\to \cdots .$$ The homotopy fiber $({\mathscr F}(f), \star)$ is determined up to homotopy. In the context of ${\mathbb K}$-theory, we consider a scheme $X$ and an open subscheme $U\subseteq X$. Let ${\mathscr E}:={\mathscr V}(X)$ denote the exact category of locally free sheaves (vector bundles) on $X$. The Quillen ${\mathbb K}$-theory of exact categories ${\mathscr E}$ is defined as a CW-complex ${\mathbb K}({\mathscr E})$ (more generally as a spectrum). The restriction map ${\mathscr V}(X) \to {\mathscr V}(U)$ induces $\iota:{\mathbb K}({{\mathscr V}(X)}) \to {\mathbb K}({{\mathscr V}(U)})$, which induces the homotopy fibration $${\mathscr F}(\iota) \to {\mathbb K}({{\mathscr V}(X)})\xrightarrow{\iota} {\mathbb K} ({{\mathscr V}(U)}).$$ Algebraic description of ${\mathscr F}(\iota)$ is under contention. Assume $X$ is quasi-projective over a noetherian affine scheme ${\rm Spec}(A)$. The point of this talk asserts that the ${\mathbb K}$-Theory of the category $C{\mathbb M}^Z(X)$ of perfect modules is homotopic to ${\mathscr F}(\iota)$, where $Z=X-U$ and a coherent module $M$ on $X$ is said to be in $C{\mathbb M}^Z(X)$, if ${\rm Supp}(M)\subseteq Z$ and $\dim_{{\mathscr V}(X)}M = grade(M) = grade(X, Z)$.
Affiliation: University of Kansas
Title: Perfect Modules and K-Theory localization sequences
Date and Time: May 22, 2024, 14:30:00 Hours
Venue: AG-77
Asbtract: Given a map $f:(Y, \star) \to (X, \star)$ of pointed topological spaces, it extends to a homotopy fibration sequence ${\mathscr F}(f), \star)\to (Y, \star) \xrightarrow{f}(X, \star)$. One says that every arrow $f$ fits into a "triangle". This leads to a long exact sequence of homotopy groups: $$\cdots \to \pi_{n-1}(X, \star) \to \pi_n({\mathscr F}(f), \star)\to \pi_n(Y, \star) \xrightarrow{\iota}\to \pi_n(X, \star)\to \cdots .$$ The homotopy fiber $({\mathscr F}(f), \star)$ is determined up to homotopy. In the context of ${\mathbb K}$-theory, we consider a scheme $X$ and an open subscheme $U\subseteq X$. Let ${\mathscr E}:={\mathscr V}(X)$ denote the exact category of locally free sheaves (vector bundles) on $X$. The Quillen ${\mathbb K}$-theory of exact categories ${\mathscr E}$ is defined as a CW-complex ${\mathbb K}({\mathscr E})$ (more generally as a spectrum). The restriction map ${\mathscr V}(X) \to {\mathscr V}(U)$ induces $\iota:{\mathbb K}({{\mathscr V}(X)}) \to {\mathbb K}({{\mathscr V}(U)})$, which induces the homotopy fibration $${\mathscr F}(\iota) \to {\mathbb K}({{\mathscr V}(X)})\xrightarrow{\iota} {\mathbb K} ({{\mathscr V}(U)}).$$ Algebraic description of ${\mathscr F}(\iota)$ is under contention. Assume $X$ is quasi-projective over a noetherian affine scheme ${\rm Spec}(A)$. The point of this talk asserts that the ${\mathbb K}$-Theory of the category $C{\mathbb M}^Z(X)$ of perfect modules is homotopic to ${\mathscr F}(\iota)$, where $Z=X-U$ and a coherent module $M$ on $X$ is said to be in $C{\mathbb M}^Z(X)$, if ${\rm Supp}(M)\subseteq Z$ and $\dim_{{\mathscr V}(X)}M = grade(M) = grade(X, Z)$.