---
abstract: 'We give a $K$-theoretic criterion for a quasi-projective variety to be smooth. If ${\mathbb L}$ is a line bundle corresponding to an ample invertible sheaf on $X$, it suffices that $K_q(X)\cong K_q({\mathbb L})$ for all $q\le\dim(X)+1$.'
address:
- 'School of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia'
- 'Dept. of Mathematics, Rutgers University, New Brunswick, NJ 08901, USA'
author:
- 'C. Haesemeyer'
- 'C. Weibel'
title: '$K$-theory of line bundles and smooth varieties'
---

[^1]

[^2]

 {#section .unnumbered}

Let $X$ be a quasi-projective variety over a field $k$ of characteristic 0. The main result of this paper gives a $K$-theoretic criterion for $X$ to be smooth. For affine $X$, such a criterion was given in [@chw-v]: it suffices that $X$ be $K_{d+1}$-regular for $d=\dim(X)$, i.e., that $K_{d+1}(X)\cong K_{d+1}(X\times{\mathbb{A}}^m)$ for all $m$. If $X$ is affine, we also showed that $K_{d+1}$-regularity of $X$ is equivalent to the condition that $K_i(X)\cong K_i(X\times{\mathbb{A}}^1)$ for all $i\le d+1$.

We also showed that $K_{d+1}$-regularity is insufficient for quasi-projective $X$; see [@chw-v Thm.0.2]. In this paper we prove:

\[main\] Let $X$ be quasi-projective over a field $k$ of characteristic 0, of dimension $d$, and let ${\mathbb L}=\Spec(\Sym\,{\mathcal L})$ be the line bundle corresponding to an ample invertible sheaf ${\mathcal L}$ on $X$.

If $K_i({\mathbb L})\cong K_{i}(X)$ for all $i\le n$ then $X$ is regular in codimension $<n$.

If $K_i({\mathbb L})\cong K_{i}(X)$ for all $i\le d+1$, then $X$ is regular.

For example, if $K_i({\mathbb L})\cong K_{i}(X)$ for all $i\le d$, then $X$ has at most isolated singularities.

In the affine case, of course, every line bundle is ample, and when ${\mathbb L}={\mathbb{A}}^1_R$ we recover our previous result, proven in [@chw-v 0.1]:

\[Vorst\] If $R$ is essentially of finite type over a field of characteristic 0, and $K_i(R)\cong K_i(R[t])$ for all $i\le n$ then $R$ is regular in codimension $<n$.

The affine assumption in this corollary is critical. In [@chw-v], we gave an example of a curve $Y$ which is $K_n$-regular for all $n$, but which is not regular; no affine open $U$ is even reduced. However, $K_0(X)\ne K_0({\mathbb L})$ for the line bundle associated to an ample ${\mathcal L}$; see Example \[ex:DD\] below. In Theorem \[thm:cusp-bundle\] we give a surface $X$ which is $K_n$-regular for all $n$, but which is not regular and such that $K_0(X)\ne K_0({\mathbb L})$ for the line bundle associated to an ample ${\mathcal L}$; it is a cusp bundle over an elliptic curve.

As in our previous papers [@chsw; @chw-v; @chww-fibrant], our technique is to compare $K$-theory to cyclic homology using $cdh$-descent and cyclic homology. The parts of $cdh$ descent we need are developed in Section 1, and applied to give a formula for the cyclic homology of line bundles in Section 2. The main theorem is proven in Section 3, and the two examples are given in Section \[sec:4\].

If $E$ is a presheaf of spectra, we write $\pi_nE$ for the presheaf of abelian groups $X\mapsto \pi_nE(X)$; we say that a spectrum $E$ is [*$n$-connected*]{} if $\pi_qE=0$ for all $q\le n$. For example, $K_n(X)$ is the homotopy group $\pi_nK(X)$ of the spectrum $K(X)$.

Similarly, if $E$ is a cochain complex of presheaves, we may regard it as a presheaf of spectra via Dold-Kan [@WH ch.10]. Thus $\pi_iE(X)$ is another notation for $H^{-i}E(X)$. We will use the cochain shift convention $E[i]^n=E^{i+n}$, so that the spectrum corresponding to $E[1]$ is the suspension of the spectrum of $E$, and $\pi_nE[1]=\pi_{n-1}E$. Thus if $E$ is $n$-connected then $E[1]$ is $(n+1)$-connected.

Zariski and $cdh$ descent
=========================

In this paper, we fix a field of characteristic 0, and work with the category $\Sch$ of schemes $X$ of finite type over the field. We will be interested in the Zariski and $cdh$ topologies on $\Sch$.

If $\tau$ is a Grothendieck topology on $\Sch$, there is an “injective $\tau$-local” model structure on the category $\Psh({\mathbf{Ch}}({\mathbf{Ab}}))$ of presheaves of cochain complexes of abelian groups on $\Sch$. In this model structure, a map $A\to B$ is a cofibration if $A(X)\to B(X)$ is an injection for all $X$, and it is a weak equivalence if $H^n A\to H^n B$ induces an isomorphism on the associated $\tau$-sheaves. The fibrant replacement of $A$ in this model structure is written as $A\to{\mathbb H}_\tau(-,A)$. We say that $A$ [*satisfies $\tau$-descent*]{} if the canonical map $A(X)\to{\mathbb H}_\tau(X,A)$ is a quasi-isomorphism for all $X$. There is a parallel notion of $\tau$-descent for presheaves of spectra.

If $A$ is a sheaf then $A\to{\mathbb H}_\tau(-,A)$ is an injective resolution; it follows that ${\mathbb H}^n_\tau(X,A) = H^n{\mathbb H}_\tau(X,A)$ for all $n$. For a complex $A$, the hypercohomology group ${\mathbb H}^n_\tau(X,A)$ equals $H^{n}{\mathbb H}_\tau(X,A).$ See [@chsw 3.3] for these facts.

The inclusion of complexes of sheaves (for a topology $\tau$) into complexes of presheaves induces an injective $\tau$-local model structure on complexes of sheaves, and the inclusion is a Quillen equivalence; see [@jardine-LHT 5.9].

For the Zariski, Nisnevich and $cdh$ topologies, there is a parallel “injective $\tau$-local” model structure on the category $\Psh({\mathbf{Ch}}(\cO_\tau))$ of presheaves of complexes of $\cO_\tau$-modules, and the functor forgetting the module structure is a Quillen adjunction. In particular, if $A$ is a presheaf of complexes of $\cO_\tau$-modules, the forgetful functor sends its fibrant $\cO_\tau$-module replacement to a presheaf that is objectwise weak equivalent to ${\mathbb H}_\tau(-,A)$.

\[HH descent\] The Hochschild complex $HH/k$ satisfies Zariski descent by [@WG 0.4]. By definition, the cochain complex $HH(X/k)$ is concentrated in negative cohomological degrees and has the (quasi-coherent) Zariski sheaf $\cO_X^{\oo_k n+1}$ in cohomological degree $-n$. When $k$ is understood, we drop the ’$/k$’ from the notation. We sometimes regard $HH$ as a sheaf of spectra, using Dold-Kan, and use the notation $HH_{q}(X)=\pi_{q} HH(X)$ for ${\mathbb H}_\zar^{-q}(X,HH)$. Recall from [@WG 4.6] that if $X$ is noetherian then $HH_q(X)=0$ for $q<-\dim(X)$.

If $E$ is a complex of Zariski sheaves of $\cO$-modules on $\Sch/X$, we may assume that ${\mathbb H}_\zar(-,E)$ is a complex of Zariski sheaves of $\cO$-modules, and similarly for ${\mathbb H}_\cdh(-,E)$. (See [@jardine-LHT 8.6].) Thus it makes sense to form the sheaf tensor product ${\mathbb H}_\tau(-,E)\oozar {\mathcal L}$ with a Zariski sheaf ${\mathcal L}$ of $\cO_X$-modules.

If $E$ is a Zariski sheaf of $\cO_X$-modules on $X$, then there is a Zariski sheaf $E^\prime$ of $\cO$-modules on $\Sch/X$, unique up to unique isomorphism, such that for every $f: Y \to X$ in $\Sch/X$ the restriction of $E^\prime$ to the small Zariski site of $Y$ is naturally isomorphic to the sheaf $f^*E$. In this paper we will always work with this sheaf on the big site; so for example “an invertible sheaf ${\mathcal L}$ on $X$” will indicate the sheaf on the big site associated in this way to an invertible sheaf on $X$.

\[Hzar-autoeq\] If ${\mathcal L}$ is an invertible sheaf on $X$, $\oozar{\mathcal L}$ is an auto-equivalence of the category $\Sh({\mathbf{Ch}}(\cO_\zar))/X$ of sheaves of complexes of $\cO_\zar$-modules on $\Sch/X$ which preserves cofibrations, fibrations and weak equivalences.

The functor $\oozar{\mathcal L}^{-1}$ is a quasi-inverse to $\oozar{\mathcal L}$. Since ${\mathcal L}$ is flat, $\oozar{\mathcal L}$ preserves injections. Since ${\mathcal L}$ is locally trivial on $X$ (and hence on any $X$-scheme), and $A\oozar\cO_X\cong A$, $\oozar{\mathcal L}$ preserves weak equivalences. Now suppose that $C\to D$ is a Zariski-local fibration; we want to see that $C\oozar{\mathcal L}\to D\oozar{\mathcal L}$ is a Zariski-local fibration. By invertibility, it suffices to observe that if $A\to B$ is a trivial cofibration of $\cO_\zar$ modules, then so is $A\oozar{\mathcal L}^{-1}\to B\oozar{\mathcal L}^{-1}$, a fact we have just verified.

\[Hzar-L\] If ${\mathcal L}$ is an invertible sheaf on $X$, and $A$ is a complex of Zariski sheaves of $\cO$-modules, then there is a quasi-isomorphism on $\Sch/X$: $${\mathbb H}_\zar(-,A)\oozar {\mathcal L}\map{\simeq} {\mathbb H}_\zar(-,A\oozar {\mathcal L}).$$

This follows immediately from Lemma \[Hzar-autoeq\].

We write $(a^*,a_*)$ for the usual adjunction between Zariski and $cdh$ sheaves associated to the change-of-topology morphism $a: (Sch/k)_\cdh\to (Sch/k)_\zar$. Thus if $\cF$ is a sheaf of $\cO_\cdh$-modules on $(\Sch/X)_{\cdh}$, $a_*\cF$ is the underlying sheaf of $\cO_\zar$-modules, and for any Zariski sheaf $E$ of $\cO_X$-modules on $X$, we may form the Zariski sheaf $a_*\cF\oo_{\cO_X}E$ on $\Sch/X$. Recall from [@EGA $0_I$(5.4.1)] that a Zariski sheaf $E$ of $\cO_X$-modules is [*locally free*]{} if each point of $X$ has an open neighborhood $U$ such that $E|_U$ is a free $\cO_U$-module, possibly of infinite rank.

\[EoF cdh\] If $E$ is a locally free sheaf on $X$, and $\cF$ is a $cdh$ sheaf of $\cO_\cdh$-modules, then $a_*\cF\oo_{\cO_X}E$ is a $cdh$ sheaf on $(\Sch/X)$.

Since the question is local on $X$, we may replace $X$ by an open subscheme to assume that $E$ is free. Because the $cdh$-topology on $\Sch /X$ is quasi-compact, and therefore arbitrary direct sums of sheaves are sheaves, we are reduced to the trivial case $E=\cO_X$, when $a_*\cF\oo_{\cO_X}E=a_*\cF$.

If $\cF$ is a $cdh$ sheaf of $\cO_\cdh$-modules, we will write $\cF\oozar E$ for the $cdh$ sheaf $a_*\cF\oo_{\cO_X}E$.

Note that ${\mathbb H}_\zar^*(X,\cF\oozar E) \ne {\mathbb H}_\zar^*(X,\cF)\oo E(X)$. For example, $E(X)=0$ does not imply that $(\cF\oozar E)(X)=0$.

If $E$ is locally free on $X$ then $\oozar E$ preserves weak equivalences and cofibrations for complexes of $cdh$ sheaves of $\cO_\cdh$-modules on $\Sch/X$.

As in the proof of Lemma \[EoF cdh\], we may replace $X$ by an open subscheme and assume that $E$ is a sheaf of free modules. Since $A\oozar E$ is a sum of copies of $A$, it follows that $A\mapsto A\oozar E$ preserves weak equivalences and cofibrations.

\[F\_E(X)\] Given a cochain complex $A$ of presheaves of abelian groups on $\Sch$, we write $F_A(X)$ for the homotopy fiber (the shifted mapping cone) of the canonical map $A(X)\to {\mathbb H}_\cdh(X,A)$, so for each $X$ there is a long exact sequence $$\cdots{\mathbb H}_\cdh^{n-1}(X,A)\to H^{n} F_A(X)\to H^{n} A(X)\to
 {\mathbb H}_\cdh^{n}(X,A) \to H^{n+1} F_A(X) \cdots.$$ If $A$ is a complex of sheaves (in some topology) of $\cO$-modules, then ${\mathbb H}_\cdh(X,A)$ can be represented by a complex of sheaves of $\cO$-modules as well (see [@jardine-LHT 8.1]), and hence so can $F_A$. We also write $F_K(X)$ for the homotopy fiber of $K(X)\to KH(X)$.

It is well known that $HH$, $HC$ and $K$-theory satisfy Zariski descent; it follows that $F_{HH}$, $F_{HC}$ and $F_K$ also satisfy Zariski descent.

\[commuteH/L\] If ${\mathcal L}$ is an invertible sheaf on $X$ and $A$ is a complex of Zariski sheaves of $\cO$-modules on $\Sch/X$, then: $${\mathbb H}_\cdh(-,A)\oozar{\mathcal L}\map{\simeq} {\mathbb H}_\cdh(-,A\oozar{\mathcal L}).$$ Consequently, $F_A\oozar{\mathcal L}\map{\simeq} F_{A\oo{\mathcal L}}$.

Arguing as in the proof of Lemma \[Hzar-autoeq\], Lemma \[EoF cdh\] shows that $\oozar {\mathcal L}$ preserves $cdh$-local fibrations (in addition to cofibrations and weak equivalences). The first statement follows immediately from this. Because $\oozar {\mathcal L}$ is exact, the second statement follows from the triangles $$F_A\to A\to {\mathbb H}_\cdh(-,A)\to \quad\text{and}\quad
F_{A\oo {\mathcal L}} \to A\oo {\mathcal L}\to {\mathbb H}_\cdh(-,A\oo {\mathcal L}) \to.
\qedhere$$

\[lem:directsumreplace\] let $A_i$ be cochain complexes of presheaves on $\Sch/X$. Then for every $X$-scheme $Y$, the canonical maps $$\bigoplus\nolimits_i {\mathbb H}_\zar(Y,A_i)\to {\mathbb H}_\zar(Y, \bigoplus\nolimits_i A_i)$$ and $$\bigoplus\nolimits_i {\mathbb H}_\cdh(Y,A_i)\to {\mathbb H}_\cdh(Y, \bigoplus\nolimits_i A_i)$$ are quasi-isomorphisms.

These sites are quasi-compact, and thus cohomology in them commutes with direct limits.

Homology of line bundles {#sec:HH(R[L])}
========================

Suppose that $R$ is a (commutative) noetherian algebra over a field $k$ of characteristic 0. In [@chww-fibrant 3.2, 4.1], we showed that $NK(R)=K(R[t])/K(R)$ is isomorphic to $NF_{HC/{\mathbb{Q}}}(R)[1]$ as well as $\F_{HH/{\mathbb{Q}}}(R)[1]\oo_R tR[t]$. In this section, we replace $R[t]$ by the symmetric algebra $R[L]=\Sym_R(L)$ of a rank 1 projective $R$-module, and the ideal $tR[t]$ by $LR[L]$. More generally, if ${\mathcal L}$ is an invertible sheaf on a scheme $X$, we replace $X\times{\mathbb{A}}^1$ by the line bundle ${\mathbb L}=\Spec(\Sym_X {\mathcal L})$.

\[aff:Kunneth\] Let $L$ be a rank 1 projective $R$-module. Then the symmetric algebra $R[L]=\Sym_R(L)$ satisfies: $$\begin{aligned}
HH(R[L]) \simeq&\; HH(R)\oo_RR[L]  \;\oplus\; HH(R)[1] \oo_R LR[L] \\
HC(R[L]) \simeq&\; HC(R) \;\oplus\; HH(R)\oo_R LR[L]. $$ Sinilarly, if $X$ is a scheme over $R$ and $X[L]$ denotes $X\times_R\Spec(R[L])$, then $$\begin{aligned}
HH(X[L]) \simeq&\; HH(X)\oo_RR[L]  \;\oplus\; HH(X)[1]\oo_R LR[L]) \\
HC(X[L]) \simeq&\; HC(X) \;\oplus\; HH(X)\oo_R LR[L].  $$

Note that, as an $R$-module, $LR[L] = R[L]\oo_R L$ is just $\oplus_{j=1}^\infty L^{\oo j}$.

The cochain complex $HH(R[L])$ ends: $\to R[L]\oo R[L] \smap{0} R[L]\to0$. Therefore there are natural maps from $R[L]$ and $R[L]\oo L[1]$ to $HH(R[L])$. Using the shuffle product, we get a natural map $\mu(R)$ from the direct sum of $HH(R)\oo_R R[L]$ and $HH(R)\oo_R(R[L]\oo L)[1]$ to $HH(R[L])$. For each prime ideal $\wp$ of $R$, we have $R_\wp[L]\cong R_\wp[t]$ and $\mu(R_\wp)$ is a quasi-isomorphism by the Künneth formula [@WH 9.4.1]. It follows that $\mu(R)$ is a quasi-isomorphism. The formula for $HC(R[L])$ follows by induction on the SBI sequence, just as it does for $HC(R[t])$.

If $X$ is a scheme over $R$, the same argument applies to $\pi_* HH(\cO_X [L])$, the direct image along $X[L]\smap{\pi} X$ of the cochain complex $HH(\cO_X [L])$ on $X[L]$ of quasi-coherent sheaves described in Example \[HH descent\]. Because $\pi$ is affine, we have a quasi-isomorphism $${\mathbb H}_\zar(X[L],HH(\cO_X [L]))\cong {\mathbb H}_\zar(X,\pi_* HH(\cO_X [L]).$$ Now the assertions about $X[L]$ follow from Corollary \[Hzar-L\] and Lemma \[lem:directsumreplace\].

\[cor:Kunneth\] $F_{HC}(R[L]) \cong F_{HC}(R) \;\oplus\;
\bigoplus\nolimits_{j=1}^\infty \left(F_{HH}\oo_R L^{\oo j}\right)(R).$

This follows from Lemma \[aff:Kunneth\], Proposition \[commuteH/L\], and Lemma \[lem:directsumreplace\].

Now suppose that $X$ is a scheme of finite type over a field of characteristic 0, containing $k$, and write $HH$, $HC$, etc for $HH/k$, $HC/k$, etc.

\[descent\] Let ${\mathbb L}$ be a line bundle over $X$, and write $\cF_{HH}$ for the cochain complex of Zariski sheaves on $X$ associated to the complex of presheaves $U\mapsto F_{HH}({\mathbb L}|_U)$. Then $F_{HH}({\mathbb L}) \smap{\simeq} {\mathbb H}_\zar(X,\cF_{HH})$.

As observed after \[F\_E(X)\], the presheaf of complexes $F_{HH}$ satisfies Zariski descent: $F_{HH}({\mathbb L})\!\simeq\!{\mathbb H}_\zar({\mathbb L},F_{HH})$. By [@AKTEC 1.56], ${\mathbb H}_\zar({\mathbb L},F_{HH})\smap{\simeq}
{\mathbb H}_\zar(X,\cF_{HH})$.

In what follows, we write $\oo$ for the tensor product of $\cO_X$-modules.

\[Kunneth\] Let ${\mathbb L}$ be the line bundle $\Spec(\Sym\,{\mathcal L})$ on $X$ associated to an invertible sheaf ${\mathcal L}$, and $p:{\mathbb L}\to X$ the projection. Then we have quasi-isomorphisms: $$\begin{aligned}
HC({\mathbb L}) \simeq\ & HC(X) \oplus {\mathbb H}_\zar(X,HH\oo\Sym({\mathcal L})\oo{\mathcal L});
\\
{\mathbb H}_\cdh(X,p_*HC) \simeq\ &
{\mathbb H}_\cdh(X,HC)\oplus {\mathbb H}_\cdh(X,HH\oo\Sym({\mathcal L})\oo{\mathcal L});
\\
F_{HC}({\mathbb L}) \simeq\ &
F_{HC}(X)  \;\oplus\;
\bigoplus\nolimits_{j=1}^\infty\left(F_{HH}\oo{\mathcal L}^{\oo j}\right)(X);
\\
K({\mathbb L},X) \simeq\ & F_{HC}({\mathbb L},X)[1].\end{aligned}$$

Using Zariski descent, we may assume that $X=\Spec(R)$ for some $R$. The first two quasi-isomorphisms are immediate from Lemma \[aff:Kunneth\], while the third is immediate from Corollary \[cor:Kunneth\]. By Theorem 1.6 of [@chw-v], $$K({\mathbb L})/K(X)\cong F_K({\mathbb L})/F_K(X) \simeq
F_{HC/{\mathbb{Q}}}({\mathbb L})[1]/F_{HC/{\mathbb{Q}}}(X)[1].$$ Now use the formula for $F_{HC}({\mathbb L})$ to get the final quasi-isomorphism.

Now suppose that $R$ is a commutative ${\mathbb{Q}}$-algebra. Then $K_n(R[L],R)$ is a ${\mathbb{Q}}$-module [@W-mod], and the Adams operations give an $R$-module decomposition $$K_n(R[L],R)\cong\oplus_{i=0}^\infty K_n^{(i)}(R[L],R)$$ with $K_n^{(0)}(R[L],R)=0$ for all $n$. The relative terms $F_K(R)\cong F_{HC}(R)[1]$ have a similar decomposition, and $F_K^{(i)}(R[L],R) \simeq F_{HC}^{(i-1)}(R[L],R)[1]$.

As in [@chww-fibrant 5.1], we define the [*typical piece*]{} $TK_n(R)$ to be $H^{1-n}(F_{HH}(R))$, and set $TK_n^{(i)}(R)=H^{1-n}(F_{HH}^{(i-1)}(R))$. Since these groups were detemined in [@chww-fibrant], we may rephrase the last part of Proposition \[Kunneth\] as follows:

\[TK(RL)\] If $R$ is a commutative ${\mathbb{Q}}$-algebra, $K_n(R[L],R)\cong TK_n(R)\oo_R LR[L]$ and $$K_n^{(i)}(R[L]) \cong K_n^{(i)}(R) \oplus TK_n^{(i)}(R)\oo_R LR[L].$$ Moreover, $$TK_n^{(i)}(R) \cong\begin{cases}
HH_{n-1}^{(i-1)}(R), &\text{if }~ i<n,\\
H_\cdh^{i-n-1}(R,\Omega^{i-1}),& \text{if }~ i\ge n+2.
\end{cases}$$

(The formulas for $TK_n^{(n)}$ and $TK_n^{(n+1)}$ are more complicated; see [*loc.cit*]{}.) The following special case $n=0$ of \[TK(RL)\], which is an analogue of [@chww-fibrant (0.5)], shows that we cannot twist out the example in [@chw-v Theorem 0.2]

Let $L$ be a rank 1 projective $R$-module, where $R$ is a $d$-dimensional commutative ${\mathbb{Q}}$-algebra, with seminormalization $R^+$, and $R[L]$ the twisted polynomial ring. Then $$K_0(R[L],R)\cong \biggl((R^+/R)\oplus
\bigoplus_{p=1}^{d-1}{\mathbb H}_\cdh^{p}(R,\Omega^{p}) \biggr)\oo_R LR[L].$$ In particular, $K_n(R)=K_n(R[t])$ if and only if $K_n(R)=K_n(R[L])$.

This follows from the fact that ${\mathbb H}_\cdh(X,HH^{(i)})\cong Ra_*a^*\Omega^i[-i]$, so that when $i>1$ we have $K_0^{(i)}(R[L],R)\cong{\mathbb H}_\cdh^{i-1}(R,\Omega^{i-1})\oo_R LR[L]$; see [@chw-v 2.2].

\[S-module\] Corollary \[TK(RL)\] shows that $K_*(R[L],R)$ is a graded $R[L]$-module. As in [@chww-fibrant], this reflects the fact that locally $R[L]$ is a polynomial ring, and $K_*(R[t],R)$ has a continuous module structure over the ring of big Witt vectors $W(R)$, compatible with the operations $V_n$ and $F_n$; when ${\mathbb{Q}}\subset R$, such modules are graded $R[t]$-modules. Since $H^0(\Spec R,\widetilde W)=W(R)$, patching the structures via Zariski descent proves that $K_*(R[L],R)$ is a graded $R[L]$-module.

When $X$ is no longer affine, this Zariski descent argument shows that $$K_n({\mathbb L},X) = \oplus H^{1-n}(X,F_{HH}\oozar {\mathcal L}^{\oo i})$$ is a graded module over $S=\oplus H^0(X,{\mathcal L}^{\oo i})$. This is clear from Proposition \[Kunneth\]. Previously, using [@W-mod], it was only known that the $K_n({\mathbb L},X)$ are continuous modules over $H^0(X,\widetilde{W})=W(k)=\prod_1^\infty k$.

Proof of Theorem \[main\]
=========================

In order to use Proposition \[Kunneth\], we need to analyze ${\mathbb H}_\zar^n\left(X,F_{HH/{\mathbb{Q}}}\oo{\mathcal L}^{j}\right)$. For this, we use the hypercohomology spectral sequence. (See [@WH 5.7.10].) $$\label{eq:SS}
E_2^{p,q} = H^p_\zar(X,H^qE) \Rightarrow {\mathbb H}_\zar^{p+q}(X,E).$$ Here $E$ is a cochain complex which need not be bounded below and (by abuse of notation) the $E_2^{}$ term denotes cohomology with coefficients in the Zariski sheaf associated to $H^qE$; the spectral sequence converges if $X$ is noetherian and finite dimensional. When $E=F_{HH}\oo{\mathcal L}^j$, we have $H^qE=H^q(F_{HH})\oo{\mathcal L}^j$, because ${\mathcal L}^j$ is flat.

\[l:hyper\] If $X$ is noetherian and finite dimensional, and $E$ is a complex of Zariski sheaves such that $H_\zar^p(X,H^qE)=0$ for $1\le p\le\dim(X)$ and $p+q=s,s+1$ then ${\mathbb H}_\zar^s(X,E)\cong H_\zar^0(X,H^sE)$.

This is immediate from the hypercohomology spectral sequence .

In the remainder of this section, we will write $H^p(X,-)$ for $H^p_\zar(X,-)$. By a “quasi-coherent” (or “coherent”) sheaf on $\Sch/k$ we mean a Zariski sheaf whose restriction to every small Zariski site is quasi-coherent (or coherent). When discussing Hochschild homology (or cyclic homology, or differentials, etc.) relative to ${\mathbb{Q}}$, we will suppress the base from the notation. For example, if $X$ is a $k$-scheme then $HH_n(X)$ and $\Omega^n_X$ will mean $HH_n(X/{\mathbb{Q}})$ and $\Omega^n_{X/{\mathbb{Q}}}$.

Recall that when ${\mathbb{Q}}\subseteq k$, the Hochschild homology complex relative to $k$ decomposes into a direct sum of weight pieces $HH^{(j)}(-/k)$; this induces decompositions on ${\mathbb H}_\cdh (-,
HH(/k))$, the fiber $F_{HH(/k)}$, and on their cohomology sheaves and hypercohomology groups as well. As in [@chw-v], we use versions of a spectral sequence introduced by Kassel and Sletsjœ in [@kasle] to obtain information about $F_{HH(/k)}$ from information about $F_{HH}$.

\[lem:KS\] (Kassel-Sletsjœ) Let ${\mathbb{Q}}\subseteq k$ and $p\ge1$ be fixed, and $X$ a scheme over $k$. Then there are bounded cohomological spectral sequences of quasi-coherent sheaves on $\Sch/k$ ($p>s\ge0$): $$E_1^{s,t} = \Omega^s_k \oo_k H^{2s+t-p}HH^{(p-s)}(-/k)
\Rightarrow H^{s+t-p}HH^{(p)}(-/{\mathbb{Q}})$$ (for $s+t\le0$) and $$E_1^{s,t} = \Omega^s_k \oo_k H^{s+t}(Ra_*\Omega_{(-/k),\cdh}^{(p-s)})
\Rightarrow H^{s+t}(Ra_*\Omega_\cdh^p)$$ and a morphism of spectral sequences between them. If $k$ has finite transcendence degree, then both spectral sequences are spectral sequences of coherent sheaves.

We remark that the second spectral sequence is just the sheafification of the spectral sequence in [@chw-v 4.2].

If $X=\Spec(R)$, the homological spectral sequence in [@kasle 4.3a] is $${}_p E^1_{-i,i+j}=\Omega^{i}_{k}\oo_k HH^{(p-i)}_{p-i+j}(R/k)
\Rightarrow HH_{p+j}^{(p)}(R)$$ ($0\le i<p$, $j\ge0$); see [@chw-v 4.1].

We claim that this is a spectral sequence of $R$-modules, compatible with localization of $R$. Indeed, following the construction in [@kasle Theorem 3.2], the exact couple underlying the spectral sequence is constructed by choosing ${\mathbb{Q}}$-cofibrant simplicial resolutions $P_{{\mathbf{\scriptscriptstyle\bullet}}}\to k$ and $Q_{{\mathbf{\scriptscriptstyle\bullet}}}\to R$ and then filtering the differential modules $\Omega^p_{Q_{{\mathbf{\scriptscriptstyle\bullet}}}/{\mathbb{Q}}}$ by certain $Q_{{\mathbf{\scriptscriptstyle\bullet}}}$-submodules, leading to a filtration of $\Omega^p_{Q_{{\mathbf{\scriptscriptstyle\bullet}}}/{\mathbb{Q}}}\otimes_{Q_{{\mathbf{\scriptscriptstyle\bullet}}}} B$ by $B$-modules. (Although the filtration steps are defined as certain $P_{{\mathbf{\scriptscriptstyle\bullet}}}$-submodules in [@kasle Section 3], they are in fact $Q_{{\mathbf{\scriptscriptstyle\bullet}}}$-submodules.) The identification of the associated graded via [@kasle Lemma 3.1] is easily checked to be a $B$-module isomorphism. The whole construction commutes with localization because forming differential modules does.

Setting $\ell=i+j$, the spectral sequence is $${}_p E^1_{-i,\ell}=\Omega^{i}_{k}\oo_k HH^{(p-i)}_{p+\ell-2i}(R/k)
\Rightarrow HH_{p+\ell-i}^{(p)}(R),\quad \ell\le i.$$ As this spectral sequence is a spectral sequence of $R$-modules, compatible with localization and natural in $R$, we may sheafify it for the Zariski topology to obtain a spectral sequence of quasi-coherent sheaves. Reindexing cohomologically, with $s=i$ and $t=-\ell$, we have $${}^p E_1^{s,t}=\Omega^{s}_{k}\otimes_k H^{2s+t-p}(HH^{(p-s)})(-/k)
\Rightarrow H^{s+t-p}(HH^{(p)}).$$ This yields the first spectral sequence. If we sheafify it for the $cdh$ topology, and use the isomorphism $HH^{(p)}_\cdh \cong \Omega^p_\cdh[p]$, we get the second spectral sequence. That it is still a spectral sequence of quasi-coherent sheaves follows from [@chw-v lemma 2.8]. The morphism between the spectral sequences is just the change-of-topology map.

Finally, if $k$ has finite transcendence degree, then the $E_1$-terms of both spectral sequences are coherent (apply [@chw-v lemma 2.8] again for the second one) and hence so are the abutments.

\[cor:KS\] There is a bounded spectral sequence of quasi-coherent sheaves $$E_1^{s,t} = \Omega^s_k \oo_k H^{2s+t-p}(F_{HH/k}^{(p-s)})
\Rightarrow H^{s+t-p}(F_{HH}^{(p)}).$$ If $k$ has finite transcendence degree, this is a spectral sequence of coherent sheaves.

The morphism of spectral sequences in Lemma \[lem:KS\] comes from a morphism $HH^{(p)}\to HH^{(p)}_\cdh$ of filtered complexes of quasi-coherent sheaves on $\Sch/k$. By a lemma of Eilenberg–Moore [@WH Ex.5.4.4], there is a filtration on the \[shifted\] mapping cone $F_{HH}^{(p)}$ of $HH^{(p)}\to HH_\cdh^{(p)}$, yielding a spectral sequence converging to $H^*(F_{HH})$. This is the displayed spectral sequence.

\[twist\] Assume that $k$ has finite transcendence degree. If ${\mathcal L}$ is an ample line bundle on $X$, then for every $n$ and $p\geq 0$ there is an $N_0 = N_0(n,p)$ such that for all $N>N_0$ the Zariski sheaf $H^nF_{HH}^{(p)}\oo{\mathcal L}^{\oo N}$ is generated by its global sections, and $H^q(X,H^nF_{HH}^{(p)}\oo{\mathcal L}^{\oo N})=0$ for all $q>0$.

The complex $F_{HH}^{(0)}$ is quasi-isomorphic to the cone of the map from the structure sheaf $\cO$ to $Ra_* a^* \cO$ and thus has coherent cohomology by [@chsw Lemma 6.5]. If $p > 0$, then by Corollary \[cor:KS\] the cohomology sheaves in question are coherent as well. Now apply Serre’s Theorem B.

Let ${\mathcal L}$ be an ample sheaf on $X$ and ${\mathbb L}$ the line bundle $\Spec(\Sym\,{\mathcal L})$. Recall that for any $Y$, $F_{HC}(Y)$ is $n$-connected if and only if $F_{HH}(Y)$ is $n$-connected; see [@chw-v 1.7]. If ${\mathbb L}$ is a line bundle over $X$, we define $\F_{HH/k}({\mathbb L},X)$ to be the cokernel of the canonical split injection $\F_{HH/k}(X)\to\F_{HH/k}({\mathbb L})$, and similarly for cyclic homology.

\[thm:HH\] If $F_{HC}({\mathbb L},X)$ is $n$-connected for some ample line bundle ${\mathcal L}$ on $X$, then $F_{HH}({\mathbb L},X)$ is $n$-connected and:

1.  The Zariski sheaf $F_{HH}$ is $n$-connected.

2.  $X$ is regular in codimension $\le n$.

3.  If $F_{HC}({\mathbb L},X)$ is $d$-connected for $d=\dim(X)$, then $X$ is regular.

There is a finitely generated subfield $k_0$ of $k$, a $k_0$-scheme $X_0$ and an ample line bundle ${\mathcal L}_0$ such that $X =
X_0\otimes_{k_0} k$ and ${\mathcal L}= {\mathcal L}_0\otimes_{k_0} k$. The Künneth formula for Hochschild homology implies that $F_{HH}({\mathbb L},X)
= F_{HH}({\mathbb L}_0,X_0)\otimes\Omega^\ast_{k/k_0}$, whence $F_{HH}({\mathbb L},X)$ is $n$-connected if and only if $F_{HH}({\mathbb L}_0,X_0)$ is. Thus we may assume that $k$ has finite transcendence degree.

\(1) Recall [@chw-v 2.1] that $F_{HH}({\mathbb L},X)=\prod F_{HH}^{(p)}({\mathbb L},X)$. Thus it suffices to fix $p$ and show that $F_{HH}^{(p)}$ is $n$-connected. Set $\cG_N={\mathcal L}^N\oo F^{(p)}_{HH}$, and note that $H^q\cG_N={\mathcal L}^N\oo H^qF^{(p)}_{HH}$. By Proposition \[twist\] and Lemma \[l:hyper\], $H^s(X,\cG_N)\cong H^0(X,H^s\cG_N)$ for large $N$ and all $s\ge-n$.

By assumption and Lemma \[descent\], the groups $$\pi_sF_{HH}^{(p)}({\mathbb L},X)= {\mathbb H}_\zar^{-s}(X,F^{(p)}_{HH}({\mathbb L},X))
={\mathbb H}_\zar^{-s}(X,F^{(p)}_{HH}({\mathbb L})/F^{(p)}_{HH})$$ vanish for $s\le n$. By Lemma \[Kunneth\], this implies that for all $N>0$: $$H^0(X,H^{-s}\cG_N) \cong H^{-s}(X,\cG_N) =
H^{-s}(X,{\mathcal L}^N\oo F^{(p)}_{HH})=0,
s\le d.$$ Since ${\mathcal L}$ is ample, the sheaves $H^s\cG_N={\mathcal L}^N\oo H^sF^{(p)}_{HH}$ are generated by their global sections $H^0(X,H^s\cG_N)$ for large $N$ and $s\ge-n$. This implies that the sheaves ${\mathcal L}^N\oo H^sF^{(p)}_{HH}$ vanish, and hence that the sheaves $H^sF^{(p)}_{HH}$ vanish for $s\ge-n$. This proves (1).

Given (1), the stalks $F_{HH}(\cO_{X,x})$ are $n$-connected. We proved in [@chw-v 4.8] that this implies that each $F_{HH/k}(\cO_{X,x})$ is $n$-connected. If $\dim(\cO_{X,x})\ge n$, we proved in [@chw-v 3.1] that $\cO_{X,x}$ is smooth over $k$, and hence regular.

\[thm:May16\] Let $X$, ${\mathcal L}$ and ${\mathbb L}$ be as in Proposition \[thm:HH\]. Suppose that $F_{HC/k}({\mathbb L},X)$ is $n$-connected. Then the proof of Theorem \[thm:HH\] goes through to show that:

1.  The sheaf $\cF_{HH/k}$ is $n$-connected.

2.  $X$ is regular in codimension $\le n$.

3.  If $F_{HH/k}({\mathbb L},X)$ is $d$-connected for $d=\dim(X)$, then $X$ is regular.

Suppose that $K_i({\mathbb L})\cong K_i(X)$ for all $i\le n$. By Proposition \[Kunneth\], $F_{HC/{\mathbb{Q}}}({\mathbb L},X)$ is $(n-1)$-connected. By Theorem \[thm:HH\], $F_{HH/{\mathbb{Q}}}({\mathbb L},X)$ is $(n-1)$-connected and $X$ is regular in codimension $<n$

Two examples {#sec:4}
============

We conclude with two quick examples. Let $E$ be an elliptic curve over ${\mathbb{Q}}$ with basepoint $Q$, and $P$ a point such that $P-Q$ does not have finite order in $\Pic(E)$.

\[ex:DD\] Consider the non-reduced scheme $Y=\Spec(\cO_E\oplus J)$, where $J$ is the invertible sheaf $\cO(P-Q)$. We showed in [@chw-v 0.2] that $Y$ is $K_n$-regular for all $n$, because $K_n(Y\times{\mathbb{A}}^1)\cong K_n(Y)\cong K_n(E)$ for all $n$.

Let ${\mathcal L}$ be the sheaf $\cO(Q)$ and set ${\mathbb L}=\Spec_Y(\Sym{\mathcal L})$. Then $$K_0({\mathbb L}) \cong K_0(Y) \oplus {\mathbb{Q}}[x,y].$$

For our second example, recall that if $R$ is a regular ${\mathbb{Q}}$-algebra and $J$ is a rank 1 projective $R$-module and $A$ is the subring $R[J^2,J^3]$ of $R[J]=\Sym_R(J)$ then $\Spec(A)$ is an affine cusp bundle over $\Spec(R)$. For $n\ge2$, set $$V_n(R) =
\begin{cases}
J^{6(i-1)}\oplus (J^{6(i-2)}\oo\Omega_R^2)\oplus\cdots
 \oplus (R\oo\Omega_R^{n-2}), & n=2i\ge2; \\
J^{6(i-1)}\oo\Omega_R^1\oplus(J^{6(i-2)}\oo\Omega_R^3)\oplus\cdots
 \oplus (R\oo\Omega_R^{n-2}), & n=2i+1\ge3.
\end{cases}$$ In particular, $V_2(R)=R$ and $V_3(R)=\Omega_R^1$. Let us write $\widetilde{K}_n(A)$ for $K_n(A)/K_n(R)$.

\[GRW 9.2\] If $A=R[J^2,J^3]$ and $R$ is a regular ${\mathbb{Q}}$-algebra then $$\widetilde{K}_n(A)
\cong
(J^5\oplus J^6) \oo V_n(R) \oplus (J\oo\Omega_R^n).$$ In particular, $\widetilde{K}_0(A)\cong J$, $\widetilde{K}_1(A)\cong J\oo\Omega^1_R$ and $$\widetilde{K}_2(A)
\cong (J^5\oplus J^6) \oplus (J\oo\Omega_R^2).$$

For $J=R$, this is Theorem 9.2 of [@GRW], which holds for any regular ${\mathbb{Q}}$-algebra $R$ (not just for any field). In order to pass to $R[J^2,J^3]$, we need more detail. Using the classical Mayer-Vietoris sequence for $A\subset R[J]$, it is easy to see that $K_0(A)/K_0(R)\cong J$ and $K_1(A)/K_1(R)\cong J\oo\Omega^1_R$.

For $n\ge2$ the factors in $K_n(A)$ come from $HH_{n-1}(A)$ via the maps $HH_*(A)\to HC_*(A)$ and $\widetilde{K}_n(A)\to \widetilde{HC}_{n-1}(A)$. The summand $J\oo\Omega_R^n$ of $K_n(A)$ comes from the $J\oo\Omega_R^1$ in $K_1(A)$ (or $HH_0(A,R[J],J)$) by multiplication by $HH_{n-1}(R)\cong\Omega^{n-1}_R$.

The $V_n$ factors come from the explicit description of the corresponding cyclic homology cycles (coming from cycles in Hochschild homology $HH_{n-1}(A)$) in 4.3, 4.7 and 5.8 of [@GRW]. Locally, $J$ is generated by an element $t$; we set $x=t^2\in J^2$, $y=t^3\in J^3$ so that $y^2=x^3$. The summands $J^5$ and $J^6$ of $K_2(A)$ are locally generated by the cycles $z=2x[y]+3y[x]$ and $tz=2y[y]+3x^2[x]$ in $HH_1(A)$. Multiplication by $\Omega_R^{n-2}$ gives the summands $(J^5\oplus J^6)\oo\Omega_R^{n-2}$ in $K_{n}(A)$.

Now consider the summand $J^6$ in the degree 2 part $A^{\oo3}$ of the Hochschild complex for $A$, locally generated by the element $w=[y|y]-x[x|x]-[x^2|x]$. The product $zw^{i-1}$ is a cycle in $HH_{2i-1}(A)$, and locally generates a summand $J^{5+6(i-1)}$ of $HH_{2i-1}(A)$, corresponding to the factor $J^{5+6(i-1)}$ of the summand $J^5\oo V_{2i}(R)$ of $K_{2i}(A)$. As above, multiplication by $\Omega_R^{*}$ gives the rest of the summands.

\[MVcusp\] In the spirit of Corollary \[TK(RL)\], we note that $NK_n(A) \cong TK_n(A)\oo_R LR[L]$, where $$TK_n(A) = \widetilde{K}_n(A) \oplus \widetilde{K}_n(A).$$

\[thm:cusp-bundle\] Let $J$ be the invertible sheaf $\cO(P-Q)$ on the ellipic curve $E$ and let $X$ denote the affine cusp bundle $\Spec_E(\cO_E[J^2,J^3])$ over $E$. ($X$ has a codimension 1 singular locus.) If $J$ does not have finite order in $\Pic(E)$ then $X$ is $K_n$-regular for all integers $n$: for all $m\ge0$ we have $$K_n(X) \cong K_n(X\times{\mathbb{A}}^m) \cong K_n(E)$$ On the other hand, if ${\mathbb L}=\Sym_E(\cO(Q))$ then $K_{-1}({\mathbb L})\ne K_{-1}(X)$ and $K_0({\mathbb L})\ne K_0(X)$.

Since $\Omega_E\cong\cO_E$, $V_n(\cO_E)$ is a sum of terms $J^i$ for $i>0$; the same is true for the pushforward of the sheaf $V_n(\cO_E[t_1,...,t_m])$ to $E$. Recall that $H^p(E,J^r)=0$ for all $r\ne0$. From the Zariski descent spectral sequence $$E_2^{p,q} \!=\! H^p(E,K_{-q}(\cO_E[J^2,J^3][t_1,...,t_m])/K_{-q}(\cO_E))
\Rightarrow K_{-p-q}(X\times{\mathbb{A}}^m)/K_{-p-q}(E)$$ we see that $K_n(X\times{\mathbb{A}}^m)\cong K_n(E)$ for all $n$.

On the other hand, Proposition \[GRW 9.2\] yields $\widetilde{K}_{-1}({\mathbb L})\cong\oplus_{j\ge1}H^1(E,J\oo{\mathcal L}^j)$ and $$\widetilde{K}_{0}({\mathbb L})\cong \oplus_{j\ge1}H^0(E,J\oo{\mathcal L}^j)
\oplus \widetilde{K}_{-1}({\mathbb L}).$$ These groups are nonzero because ${\mathcal L}$ is ample.

Acknowledgements {#acknowledgements .unnumbered}
----------------

The authors would like to thank the Tata Institute, the University of Melbourne and the Hausdorff Institute for Mathematics for their hospitality during the preparation of this paper.

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[^1]: Haesemeyer was supported by ARC DP-170102328

[^2]: Weibel was supported by NSF grant DMS-146502
