Suppose that $k$ is an algebraically closed field and $R$ is a finitely generated $k$-algebra such that if $X$ denotes Spec$R$, then the only closed, singular point of $X$ is the origin. Let $\tilde{X}$ denote the blow-up of $X$ along the origin and $E$ denote the exceptional divisor. I only know that the functorial property of $G$-theory for proper morphisms of Noetherian schemes implies that the induced diagram of $G$-theory spaces for the blow-up diagram of $X$ is commutative. It seems that this diagram of $G$-theory spaces is also a Cartesian square. Is this true?If so, how do we derive this? Also, assuming the existence of the Cartesian square in this context, does it give some exact sequence or formula for the $G$-theory of the blow-up $\tilde{X}$ in terms of the $G$-theory of the schemes $X$, the origin of $X$ and the exceptional divisor $E$? For a specific example I am thinking about, let $R$ be the ring $k[a,b,c,d]/(ac-b^2,bd-c^2,ad-bc)$. Thank you very much for your kind help in advance.
5
-
$\begingroup$ This follows directly from the localization sequence $G(Z)\to G(X)\to G(U)$ for a closed subscheme $Z\subset X$ with open complement $U$. See Theorem 3.12 in preschema.com/papers/kstack.pdf. But this cartesian square of spectra does not give a direct formula for $G_*(\tilde X)$, only a long exact sequence. $\endgroup$– Marc HoyoisCommented Aug 29, 2022 at 14:16
-
$\begingroup$ Thank you so much for your kind help and providing a reference. $\endgroup$– BorisCommented Aug 29, 2022 at 17:42
-
$\begingroup$ I noticed that theorem 3.16 in the paper you mentioned gives a formula for the G-theory of the blow-up. But I am not sure whether this result applies to the specific example I mentioned at the end of my initial post. Could you provide some guidance on determining whether the closed point of X corresponding to the origin is a local complete intersection? Thank you very much. $\endgroup$– BorisCommented Aug 29, 2022 at 18:44
-
$\begingroup$ What you want is 3.12, not 3.16: 3.12 applies to arbitrary blowups. $\endgroup$– Marc HoyoisCommented Aug 29, 2022 at 20:11
-
$\begingroup$ Thank you very much for your kind help. Maybe I did not phrase my question clearly.What I was asking just now was the special case that X equals Spec $k[a,b,c,d]/(ac-b^2, bd-c^2,ad-bc)$ and the center of the blow-up is the origin.I was asking whether in this case, the center of the blow-up is a local complete intersection, so that I could apply theorem 3.16. Thank you very much. $\endgroup$– BorisCommented Aug 29, 2022 at 20:32
Add a comment
|