Some examples of algebraic groups

In this post I want to give a few examples of the known “pathological” behavior of algebraic groups defined over general bases. In particular, this post contains examples of the following.

  • A smooth group scheme over a DVR with generic fiber \(\mathbf{G}_m\) and special fiber \(\mathbf{G}_a\),
  • An affine smooth group scheme \(G\) over a regular dimension \(2\) base such that the (open) relative identity component \(G^0\) is not affine.

In the first bullet point, we note that the reverse degeneration cannot happen, i.e., there is no smooth group scheme over a DVR with generic fiber \(\mathbf{G}_a\) and special fiber \(\mathbf{G}_m\). One way to see this is to note that if \(\ell\) is a prime number invertible on the base, then multiplication by \(\ell\) is flat (by the fibral flatness criterion), so the rank of its kernel cannot jump after specialization. In the second bullet point, \(2\) is the minimal dimension in which examples can occur: in dimension \(0\) (i.e., over a field), this follows from the elementary fact that \(G^0\) is an open and closed subscheme of \(G\). In dimension \(1\), \(G^0\) may not be closed in \(G\), but nonetheless it is always affine by a theorem of Raynaud (see Prasad-Yu, On quasi-reductive group scheme, Prop. 3.1).

These are both well-known phenomena, but examples appear somewhat out of the blue in published accounts, in which the group schemes are defined by rather complicated equations. (For the second example, see Raynaud, Faisceaux amples sur les schémas en groupes et les espaces homogènes, VII, 3(iii).) Our goal is to realize examples of the above two phenomena as centralizers of certain sections of \(\mathrm{SL}_n\). For this we will need to set up a few preliminary notions.

If \(k\) is a field and \(g \in \mathrm{GL}_n(k)\), then we say that \(g\) is regular if its schematic centralizer \(Z_{\mathrm{GL}_n}(g)\) has dimension \(n\). Using the Jordan decomposition, it is easy to check that \(n\) is the minimal dimension of a schematic centralizer in \(\mathrm{GL}_n\). If \(g\) is semisimple, this is equivalent to the statement that \(g\) has \(n\) distinct eigenvalues. If \(g\) is unipotent, it is equivalent to the statement that the Jordan normal form of \(g\) has a single Jordan block (so there is a unique regular unipotent element up to conjugacy). In the general case, the statement is that the restriction of \(g\) to each of its generalized eigenspaces has a single Jordan block in its Jordan normal form. It is easy to check that in any of these cases, \(Z_{\mathrm{GL}_n}(g)\) is smooth and connected. If instead \(g \in \mathrm{SL}_n(k)\), then we say that \(g\) is regular if it is regular as an element of \(\mathrm{GL}_n(k)\). Equivalently, the schematic centralizer \(Z_{\mathrm{SL}_n}(g)\) has dimension \(n-1\).

The following lemma will not be used in any serious way later; I only include it to indicate the lay of the land.

Lemma. The locus \(V\) of unipotent elements of \(\mathrm{GL}_n\) is closed and irreducible of dimension \(n^2 - n\); the unique conjugacy class of regular unipotent elements is open and dense in \(V\).

Proof. Let \(B\) denote the upper-triangular Borel subgroup of \(G = \mathrm{GL}_n\), and let \(U\) be the unipotent radical of \(B\) (so \(U\) consists of the strictly upper-triangular matrices in \(\mathrm{GL}_n\)). Let \(W \subset G \times G\) be the closed subscheme defined functorially as \(\{(g, x): g^{-1}xg \in U\}\). This is the preimage of \(U\) under the conjugation map \(G \times G \to G\), \((g, x) \mapsto g^{-1}xg\), so \(W\) is closed. Moreover, there is an isomorphism \(G \times U \to W\) given by \((g, u) \mapsto (g, gug^{-1})\), so that \(W\) is irreducible of dimension \(n^2 + \frac{n^2 - n}{2}\). Since \(B\) normalizes \(U\), the image \(\overline{W}\) of \(W\) in \(G/B \times G\) is closed and irreducible of dimension \(n^2 - n\). Since every unipotent element of \(G\) is conjugate to an element of \(U\), the second projection \(\pi: \overline{W} \to V\) is surjective. Thus \(V\) is closed and irreducible (since \(G/B\) is proper), and \(V\) has dimension at most \(n^2 - n\). Since any regular unipotent element has centralizer of dimension \(n\), its conjugacy class \(C\) has dimension \(n^2 - n\), and so \(V\) has dimension exactly \(n^2 - n\). Moreover, \(C\) is dense in \(V\) and open by the closed orbit lemma. This completes the proof.

In general, if \(S\) is a scheme and \(g \in \mathrm{GL}_n(S)\), we can form the schematic centralizer \(Z_G(g)\); this is a finitely presented closed subscheme of \(\mathrm{GL}_{n, S}\) which represents the functor

\[ T \mapsto \{h \in \mathrm{GL}_n(T): hg_T = g_Th\}. \]

Using the fact that \(n\) is the minimal dimension of a schematic centralizer over a field, it follows from Chevalley’s theorem on upper semicontinuity of fiber dimension that the regular locus in \(\mathrm{GL}_n\) is open. We will say that \(g\) is fiberwise regular if for every \(s \in S\) the element \(g_s \in \mathrm{GL}_n(k(s))\) is regular. We have an entirely similar definition for sections of \(\mathrm{SL}_n\).

Theorem. If \(g \in \mathrm{GL}_n(S)\) is fiberwise regular, then the schematic centralizer \(Z_{\mathrm{GL}_n}(g)\) is flat. The same statement holds for \(\mathrm{SL}_n\) in place of \(\mathrm{GL}_n\).

Proof. First, we need only prove the claim for \(\mathrm{GL}_n\). Indeed, note that every centralizer \(Z_{\mathrm{SL}_n}(g)\) contains the (flat) center \(\mu_{n, S}\). Thus flatness of \(Z_{\mathrm{SL}_n}(g)\) is equivalent to flatness of \(Z_{\mathrm{SL}_n}(g)/\mu_{n, S}\). This latter group scheme is isomorphic to \(Z_{\mathrm{GL}_n}(g)/\mathbf{G}_{m, S}\), where \(\mathbf{G}_{m, S}\) is the center of \(\mathrm{GL}_{n, S}\). By the same reasoning, flatness of this latter group scheme is equivalent to flatness of \(Z_{\mathrm{GL}_n}(g)\).

Now we prove the claim for \(\mathrm{GL}_n\). We may consider only the universal case that \(S\) is equal to the (open) regular locus of \(\mathrm{GL}_n\), and thus assume that \(S\) is noetherian and reduced. In this case, there is a remarkable result due to Grothendieck, called the valuative criterion of flatness. This says that if \(X\) is a finite type \(S\)-scheme for a reduced noetherian scheme \(S\), then \(X\) is \(S\)-flat if and only if, for every valuation ring \(A\) and every map \(\mathrm{Spec} A \to S\), the base change \(X_A\) is \(A\)-flat. (See EGA IV, Thm. 11.8.1 for a slightly more general statement.) Thus we may in fact assume that \(S\) is the spectrum of a DVR \(A\). In this case we have the following useful lemma.

Lemma. Let \(A\) be a DVR and let \(X\) be a finite type \(A\)-scheme. Suppose

(1) The map \(X(A) \to \pi_0(X_s)\) is surjective,

(2) Every connected component of \(X_s\) is irreducible,

(3) \(X_s\) and \(X_\eta\) are equidimensional of the same dimension \(d\),

(4) The special fiber \(X_s\) is reduced.

Then \(X\) is \(A\)-flat.

Proof. Let \(Z\) be the closure of the generic fiber \(X_\eta\) in \(X\), so that \(Z\) is an \(A\)-flat closed subscheme of \(X\) (since \(A\) is a DVR!). To prove that \(X\) is \(A\)-flat, it suffices to prove that the map \(Z \to X\) is an isomorphism, and the fibral isomorphism criterion (see EGA IV, Cor. 17.9.5) allows us to reduce to proving that the map \(Z_s \to X_s\) of special fibers is an isomorphism. By (1), \(Z_s\) meets every connected component of \(X_s\). Since \(Z\) is flat, \(Z_s\) is equidimensional of dimension \(d\), and thus by (2) and (3), the map \(Z_s \to X_s\) is surjective. By (4), it follows that \(Z_s \to X_s\) is an isomorphism. The Lemma is proved.

When \(X = Z_{\mathrm{GL}_n}(g)\), every condition in this lemma is satisfied: (1) holds because both fibers are connected and the identity section of \(G\) factors through \(X\); (2) holds because any connected finite type group scheme over a field is irreducible; (3) holds by assumption; and (4) holds because the special fiber is smooth. Thus indeed the Theorem holds.

Remarks. (a) Since smoothness over a base is equivalent to flatness + smoothness of the fibers (in the presence of local finite presentation), one sees that under the hypotheses of the Theorem \(Z_{\mathrm{GL}_n}(g)\) is always smooth, and \(Z_{\mathrm{SL}_n}(g)\) is smooth whenever \(n\) is invertible on the base.

(b) I do not know whether one can remove the use of the valuative criterion in the above proof. One may be tempted to use instead miracle flatness (see Matsumura, Commutative Ring Theory, Thm. 23.1), but to apply this one needs to first show that the schematic centralizer \(Z_{\mathrm{GL}_n}(g)\) is Cohen-Macaulay. This might be possible via an explicit computation (especially for small \(n\)), but some care is needed since there is no access to the Jordan decomposition over a general base, and without a priori knowledge of flatness it is not clear how to deduce Cohen-Macaulayness purely fibrally.

(c) This theorem, suitably formulated, remains true for arbitrary simply connected semisimple group schemes over a general base (and even in slightly greater generality), but considerably more technical input is required for its proof.

Now that we have our Theorem, we move on to the promised applications.

For the first example, let \(A\) be a DVR and let \(g \in \mathrm_{SL}_2(A)\) be a matrix with regular unipotent special fiber and regular semisimple generic fiber. Explicitly, we may let

\[ g = \begin{pmatrix} 1 + \pi & 1 \\ 0 & (1 + \pi)^{-1} \end{pmatrix} \]

where \(\pi\) is a uniformizer of \(A\). It is easy to check that \(Z_{\mathrm{SL}_2}(g_s) \cong \mu_2 \times \mathbf{G}_a\) and \(Z_{\mathrm{SL}_2}(g_\eta) \cong \mathbf{G}_m\), so that altogether \(G := Z_{\mathrm{SL}_2}(g)/\mu_2\) has special fiber \(\mathbf{G}_a\) and generic fiber \(\mathbf{G}_m\), providing our first example.

Remark. There are other natural ways of constructing such a group scheme: for example, one can use the torsion component of the Picard scheme of a nodal cubic degenerating to a cuspidal cubic over a DVR.

For the second example, we will construct a regular section \(g\) of \(\mathrm{SL}_3\) over a regular dimension \(2\) \(k\)-scheme \(S\) such that only finitely many fibers of \(g\) are unipotent modulo the center. (By the first Lemma, the regular unipotent locus in \(\mathrm{SL}_3\) is locally closed and irreducible of codimension \(2\), so it is perhaps evident that such examples will exist by Bertini’s theorem.) If \(k\) is not of characteristic \(3\), then \(G = Z_{\mathrm{SL}_3}(g)\) is smooth over \(S\) by Remark (a), and in particular normal. Moreover, over a field of characteristic not \(3\), if \(h\) is a regular element then \(Z_{\mathrm{SL}_3}(h)\) is disconnected if and only if \(h\) is unipotent modulo the center, so all but finitely many fibers of \(G\) are connected. So \(G - G^0\) is closed of codimension \(2\) in \(G\), and thus Hartogs’ lemma implies regular functions on \(G^0\) are the same as regular functions on \(G\). Thus \(\Gamma(G, \mathcal{O}_G) = \Gamma(G^0, \mathcal{O}_G)\), so \(G^0\) cannot be affine since \(G^0 \neq G\)

Now we construct our example. Suppose that \(k\) does not have characteristic \(3\) and consider \(A = k[x, y, (1 + x)^{-1}, (1 + y)^{-1}]\). Let \(g \in \mathrm{SL}_3(A)\) be given by

\[ g = \begin{pmatrix} 1 + x & 1 & 0 \\ 0 & 1 + y & 1 \\ 0 & 0 & (1 + x)^{-1}(1 + y)^{-1} \end{pmatrix}. \]

Then \(g\) is unipotent modulo the center if and only if \(x = y \in \mu_3 - 1\) (in which case it is regular). In this case, the centralizer of \(g\) is isomorphic to \(\mu_3 \times \mathbf{G}_a^2\). Moreover, \(g\) is semisimple if and only if its eigenvalues are all distinct, i.e., \(1 + x \neq 1 + y\) and \(1 + x \neq (1 + x)^{-1}(1 + y)^{-1}\) and \(1 + y \neq (1 + x)^{-1}(1 + y)^{-1}\). We consider the centralizer in case \(1 + x = 1 + y\) and \(x \not\in \mu_3 - 1\), i.e., \(\mathfrak{p} \subset A\) is a prime ideal containing \(x - y\) and not \(x\). (The other non-unipotent non-semisimple cases are entirely similar.) In this case, \(g_{k(\mathfrak{p})}\) is conjugate to an element of the form

\[ h = \begin{pmatrix} \alpha & 1 & 0 \\ 0 & \alpha & 0 \\ 0 & 0 & \alpha^{-2} \end{pmatrix}, \]

where \(\alpha \neq 1\). The centralizer of \(h\) is isomorphic to \(\mathbf{G}_m \times \mathbf{G}_a\), as one may readily check. In particular, \(Z_{\mathrm{SL}_3}(g_{k(\mathfrak{p})})\) is connected. The centralizer of a regular semisimple element is a torus, so in fact every fiber of \(Z_{\mathrm{SL}_3}(g)\) is connected except the fiber over \((0, 0)\). (In general, using a Jordan decomposition argument as above, one can show that if the characteristic of \(k\) does not divide \(n\) then \(Z_{\mathrm{SL}_n}(g)\) is connected if and only if \(g\) is not unipotent modulo the center.) To summarize:

  • Over every geometric point of \(\{(x, x): 1 + x \in \mu_3(k)\}\), \(Z_{\mathrm{SL}_3}(g)\) is isomorphic to \(\mu_3 \times \mathbf{G}_a^2\);
  • If \(f = 1 + x\) and \(g = 1 + y\) then over \(V(f - g, f - f^{-1}g^{-1}, g - f^{-1}g^{-1}) - \{(x, x): 1 + x \in \mu_3(k)\}\), every geometric fiber of \(Z_{\mathrm{SL}_3}(g)\) is isomorphic to \(\mathbf{G}_m \times \mathbf{G}_a\);
  • Over every other geometric point of \(\mathrm{Spec} A\), \(Z_{\mathrm{SL}_3}(g)\) is isomorphic to \(\mathbf{G}_m^2\).

Thus the argument of the previous paragraph shows that \(G^0\) is not affine.

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