Symplectic K-theory

Question

For a ring $R$ consider symplectic K-theory defined as follows: let $\operatorname{Sp}(R) = \lim_n \operatorname{Sp}_{2n}(R)$, let $\operatorname{ESp}(R)$ be the subgroup generated by elementary matrices, let $\operatorname{BSp}(R)$ be a classifying space, let $\operatorname{BSp}(R)^+$ be the result of applying Quillen's $+$-construction (with respect to $\operatorname{ESp}(R)$) and define $\operatorname{KSp}_*(R) := \pi_*(\operatorname{BSp}(R)^+)$. [Is this the right definition?]

1. Is the following statement true (and is there a reference)? Let $\tilde{\operatorname{Sp}}_{2n}(R)$ be the Steinberg group of $\operatorname{Sp}_{2n}(R)$. There is an exact sequence $$ 1 \to \operatorname{KSp}_2(R) \to \tilde{\operatorname{Sp}}_{2n}(R) \to \operatorname{Sp}_{2n}(R) \to 1 $$ for $n$ large enough. [Is the kernel always $\pi_2(\operatorname{BSp}_{2n}(R)^+)$?]

2. What is the right analogue of the ``fundamental theorem of K-theory'', i.e. how can $\operatorname{KSp}_*(R[t])$ and $\operatorname{KSp}_*(R[t,t^{-1}])$ be expressed in terms of $\operatorname{KSp}_*(R)$?

So essentially I am wondering what changes if one replaces $\operatorname{GL}(R)$ by $\operatorname{Sp}(R)$ in the definition of algebraic K-theory. I am mostly interested in the case where $R$ is a field. In that case $\operatorname{KSp}_1(R)$, $\operatorname{KSp}_1(R[t])$, and $\operatorname{KSp}_1(R[t,t^{-1}])$ all vanish at least if the characteristic is not $2$.

Answer 1

Concerning the definition of symplectic K-theory: there are various possible definitions, the homotopy groups of the plus-construction of the classifying space of the infinite symplectic group is one such definition. Other definitions can be given using categories of forms (similar to Q-construction or $S^{-1}S$-construction for algebraic K-theory) or a Karoubi-Villamayor style definition (as homotopy of a polynomial resolution $Sp_\infty(R[\Delta^\bullet])$ of the symplectic group). In the end, as for algebraic K-theory, all these definitions agree for regular rings of characteristic $\neq 2$. Various definitions and results are easier to find by using alternative keywords such as "hermitian K-theory" or "higher Grothendieck-Witt groups".

Concerning the first question: the statement is not quite true because the map $\tilde{Sp}_{2n}(R)\to Sp_{2n}(R)$ is not surjective. The morphism surjects onto the elementary symplectic subgroup $ESp_{2n}(R)$. This is normal for $n\geq 2$, and the quotient is $KSp_1(R)$. If you are interested in fields, this is trivial, but not generally.

Otherwise, the statement is true. For $n\geq 3$, the Steinberg group is a central extension, cf. this paper of Lavrenov and the historical references in there. In the paper, you also find explained that for $n\geq 4$, the Steinberg group is the universal central extension, which implies that there is an exact sequence $$ 0\to KSp_{2,n}(R)\to \tilde{Sp}_{2n}(R)\to Sp_{2n}(R)\to KSp_{1,n}(R)\to 0 $$ and $KSp_{2,n}(R)\cong H_2(Sp_{2n}(R))\cong \pi_2(BSp_{2n}(R)^+)$. There are also stabilization results for homology of symplectic groups (e.g. Mirzaii-van der Kallen) which imply that $KSp_{2,n}(R)$ stabilizes to $KSp_2(R)$ for $n\to \infty$, depending on the Krull dimension of $R$.

Concerning the fundamental theorems: For the symplectic K-theory of polynomial rings, there is the homotopy invariance theorem of Karoubi which implies that for regular rings of characteristic $\neq 2$ there is an isomorphism $KSp_\ast(R[T])\cong KSp_\ast(R)$.

For the Laurent-polynomials, the right keyword here is the fundamental theorem of hermitian K-theory. It was proved by Karoubi for rings, but see also the recent generalization by Schlichting. The short answer is, there is a version of the fundamental theorem, but the notation or results may not be transparent on first reading. I try to give some rough idea.

The fundamental theorem of algebraic/hermitian K-theory is closely linked to Bott periodicity. (Ok, I am restricting to regular rings here, characteristic not $2$...) This necessitates viewing algebraic/hermitian K-theory as a cohomology theory on schemes, as it is done in motivic homotopy theory. The fundamental theorem tells us what the cohomology of $X\times\mathbb{G}_m$ is, where $\mathbb{G}_m=\operatorname{Spec}\mathbb{Z}[t,t^{-1}]$ is the multiplicative group. In the stable $\mathbb{A}^1$-homotopy, $X\times\mathbb{G}_m$ splits as $X\vee \Sigma^{1,1}X$, hence its $E$-cohomology splits as direct sum of two things - one is $E$-cohomology of $X$ and the other is $\Omega_{\mathbb{P}^1}$-cohomology of $X$ shifted by one (because $S^{2,1}\cong\mathbb{P}^1$). Now algebraic K-theory is represented by a spectrum $KGL$ which by Bott periodicity satisfies $\Omega_{\mathbb{P}^1}KGL\cong KGL$. This gives us the fundamental theorem of algebraic K-theory.

From this point of view, it should now be clear what changes if we consider symplectic K-theory: Bott periodicity for hermitian K-theory is more complicated and there are four spectra related by $\mathbb{P}^1$-loop spaces - symplectic K-theory, orthogonal K-theory and inbetween Karoubi's U- and V-theory. The fundamental theorem for symplectic K-theory expresses symplectic K-theory of $R[t,t^{-1}]$ as a direct sum of symplectic K-theory of $R$ and the $\Omega_{\mathbb{P}^1}$-loop spectrum of symplectic K-theory. I can never tell if this is U or V. Anyway, as a motivic spectrum, I think it is the spectrum associated to $Sp/GL$. See the paper of Schlichting and Tripathi on $\mathbb{A}^1$-representability of higher Grothendieck-Witt groups.

I hope that this is helpful. In a way, the fundamental theorem for hermitian K-theory is not so easy to grasp if one only considers constructions for rings. It is much more natural to approach it from a motivic homotopy point of view.

Answer 2

1. Matthias has already pointed out that the map $\widetilde{\mathrm{ESp}}_{2n}(R)\rightarrow Sp_{2n}(R)$ need not be surjective. In fact, the same phenomenon arises already for $\mathrm{SL}_n$ (the image of $\widetilde{\mathrm{E}}(n, R) = \mathrm{St}(n, R)$ is exactly the elementary subgroup $\mathrm{E}(n, R)$ which is typically strictly smaller than $\mathrm{SL}(n, R)$).

2. I would also like to add that from Lavrenov's result it follows that an unstable version of Gersten's formula holds (i.e. $\mathrm{H}_3(\widetilde{\mathrm{ESp}}_{2n}(R), \mathbb{Z}) = \pi_3(B\mathrm{Sp}_{2n}(R)^+)$ for $n\geq 4$). The proof is the same as in the stable case (it is only relevant that $\widetilde{\mathrm{ESp}_{2n}}$ is a universal central extension of $\mathrm{ESp}_{2n}$).

3. There is a theorem by Grunewald, Mennicke and Vaserstein which states the following:

   • If $R$ is a "locally principal" ring (i.e. localization $R_\mathfrak{M}$ at every maximal ideal $\mathfrak{M}\triangleleft R$ is a principal ideal domain) then $\mathrm{K_1Sp}_{2n}(R[x_1,\ldots x_n]) = \mathrm{K_1Sp_{2n}}(R)$, $n\geq 2$.
   • If $R$ is a p.i.d. then $\mathrm{K_1Sp}_{2n}(R[x, x^{-1}])=\mathrm{K_1Sp}_{2n}(R)$, $n\geq 2$.

   Both statements apply to a rather narrow class of rings (either assumption implies that Krull dimension of $R$ is $\leq 1$). There exist also theorems asserting triviality of $\mathrm{K_1Sp}_{2n}(R[t_1,\ldots, t_n])$ for certain classes of rings (e.g in this paper of Kopeiko such theorem is proved for rings of geometric type).