Localization problems of Quillen
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909339325825024 |
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| author | Mandal, Satya |
| author_facet | Mandal, Satya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_09284 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Localization problems of Quillen Mandal, Satya K-Theory and Homology Commutative Algebra Algebraic Geometry 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. |
| title | Localization problems of Quillen |
| topic | K-Theory and Homology Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2306.09284 |