奈良教育大学学術リポジトリNEAR
Some Topological Properties of the Derivation Algebra D(K/k) for a Field Extension K/k
著者 KIKUCHI Teppei
journal or
publication title
奈良教育大学紀要. 自然科学
volume 31
number 2
page range 1‑4
year 1982‑11‑25
URL http://hdl.handle.net/10105/2308
Some Topological Properties of the Derivation
Algebra &(K/k) for a Field Extension K/k
Teppei KIKUCHI
(Department of Mathematics, Nara University of Education, Nara, Japan) (Received April 17, 1982)
Let K be a field extension of a field k, and @(K/k) be its derivation algebra. In the previous paper [1], we gave some topological considerations on @(K/k). And we showed, for example, that @(K/k) is dense in Homt(Z,K) with respect to the finite topology if and only if the center Z(3)(K/k)) of 2>{K/k) coincides with k. On the other hand, we
know already that 3{K/k) ‑9(K/k.) and Z(9(K/k,))⊃k, (where k, denotes the separable algebraic closure of k in K) and that if [K:k ] is finite then 3>{K/k) is discrete. Con‑
sequently &(K/k) is discrete if [K:k,〕 is finite.
In this paper first we show that the converses of these are false. That is, we show the existence of an example K/k such that Z(3>(/k))キks and 3>(K/k) is discrete nevertheless ¥̲K:k, ] is not finite.
Next we shall prove that, under the assumption Z(S>{K/k)) ‑k, @{K/k) is discrete if and only if ¥̲K:k] is finite.
Finally we shall notice that, under a natural isomorphism <p:RomK(KョK, K)CSH.omk (K, K) such that <p(f)(*) ‑/(7㊤x), <p‑1(@(K/k)) is the totality of the iiT‑linear maps f:K
㊥K‑→K which are continuous with respect to the /K/*‑adic topology of見④K and the discrete topology of K.
Notation and terminology are the sarrle as [1], [2] and [3]. I､he finite topology of Homk(K, K) or <&(K/k) defined in [1] will be called the finite ^‑topology in this paper.
㊤ means always (x)k.
1. An example
Let K be a field extension of a field k, and h, be the separable algebraic closure of k in K. First we notice the following.
LEMMA 1. Let F be an intermediate field between k and K. Then thefiniteF‑topol‑
ogy of B‑omF(K, K) coincides with the relative topology induced by the finite k‑topology of Horn*(if, K), that is, HomF(if, K) is a subspace of Horn,(if, K).
This is obvious by the fact ｣/J?OI‑:jy,‑) ‑｣/サ(#,・;.)>,・)nHorrid(K, K) for any pair of finite
l
2
subsets {#,‑} and {>>,‑} (7∈/) of K.
Teppei Kikuchi
COROLLARY 2. (1) The set ^(K/k) is identical with @(K/ks) and its finite k‑topol‑
ogy coincides with its finite k,‑topology. (2) If [K:k, ¥ is finite, then 3>(K/k) is discrete.
PROOF. (1) The factョ(K/k)‑S(K/ks) has been already proved in [3] and the rest is obvious because 2(K/k.) is a subspace of 2>(K/k) in general by Lemma 1. (2) If [K:k, ¥ is finite, then <2(K/k,) is discrete. Hence 3i(K/k) is also discrete due to (1).
q.e.d.
Now a question arises: if 2){K/k) is discrete, is [K:k,] or [K:k ¥ finite?
On the other hand we know that Z(9(K/k))‑Z(9(K/k,))⊃k,. Then is the con‑
verse of this true? That is, does it hold that Z(@(K/k))‑k, in general?
Next example shows that the answers of the both questions are negative.
EXAMPLE 3. Let ko be aperfect field of characteristic p>0, and x and y be inde‑
pendent variablesoverk｡. Set k‑k｡(x,y) and K‑k(yp , yp H, y♪‑",‑). Then we have the following.
(1) K is purely inseparable over k, hence k,‑k.
(2) [K:k]‑‑‑.
(3) 9(K/k) ‑Z(9(K/k)) ‑‑=Kキk.
(4) 2{K/k) is discrete.
PROOF (1) and (2) are trivial. By definition we have K‑k｡(x) h(.y)* 叩, KP叫‑k.
(y)♪‑∽ and k(Kf伽)‑K Hence, for any positive integer n, we have IK/k⊂IK/f because
each element l㊤a‑a㊨l(a∈K) is written in the form of 冒 b,a⑪Ci‑Ci㊤l)p with bi∈kl
and cj∈K. This means that IK/kp ‑/*/* and hence the null map is the only one high order derivation of a over k. Thus we have 5d{K/lx) ‑K and consequently Z(@(K/k))‑K, which proves (3). To prove (4), we have only to show that the set {o} is open in K¥, with respect to the topology induced from the finite ^‑topology of Homi(it, K). However this is obvious from the fact Ui{a;o)r¥∬lx‑{o} for any non‑zero a∈K. q.e.d.
2. Discreteness of 2{K/k)
Above example shows that the discreteness of 2(K/k) does not imply the finiteness of [K:k] in general. Now we shall prove, if K/k satisfies the condition Z(S>(K/k))‑k, the discreteness of 2(K/k) is equivalent to the finiteness of [K:k ].
LEMMA 4. Let K be a field extension ofa field k. Then the following^ are equiva‑
lent.
(1) 3)(K/k) with the finite k‑topology is discrete.
(2) {o} is open in 2>{K/k) with the finite k‑topology.
(3) lhere exists afinitesubset {xu…,xn) ofKsuch that Uk(xh…,xn¥O,…,0)(3>{K/k)
‑{o}.
PROOF. The implications (1)^>(2)<>(3) are obvious. (3悼(1) is also immediate, because Uk(xh‑ xn; 0,‑, 0)〔3{K/k)‑ {o) implies Ut(xltH.. *.;/(*i)....,/(* ))(3{K/k) ‑ {f) for each /∈3>(K/k), which means each one point set is open inョ(K/k).
LEMMA 5. Let K be a field extension ofa field k. Then Horn*(if, K) with the finite k‑fopology is a Hausdorff space.
PROOF. Let /and g be elements of Horn*(if, K) and /キg. Choose an element x∈
Ksuch that f(x)キg(x). Then we have Uk(x;f(x))nUt(x;g(x))‑￠ q.e.d.
Now suppose that if is a field extension of a field k such that Z(S>(K/k))‑k and 9) (A/ft) is discrete with respect to the finite ^‑topology. Then there exists a finite subset
{*!,.‑,xn} of if such that Uk(xi xn;O,…,0)nS>(K/k)‑{o} by Lemma 4. However, by
Cor. 7 0f our previous paper [1], @(K/k) is dence in Homk(K,K). Hence Uk{xu‥.xn;
0,...,0)(9>(K/k) is dence in Uk(xi,...,xn;0,.‥,o), that is, {o} is dence in Uk(xlt.‥,xn;O,…,0).
Consequently, as {o} is closed in Ut(xi,…,xn;0,...,0) by Lemma 5, we have Uk(xl xn;O,…,
o)‑{o}, which means that Kis a vector space over k generated by a finite set {xu...,x九).
Hence [a:ォ] must be finite.
Conversely if [K:k] is finite, then Homk(K, K) with the finite ^‑topology is discrete.
Hence @(K/k) is also discrete.
Thus we obtain the following.
THEOREM 6. Let K be a field extension of a field k such that Z(3>(K/k))‑k.
Then <2>(K/k) with the finite k‑topology is discrete if and only if [a:ォ]<oo.
COROLLARY 7. Let K be a field extension of a field k such that Z{2(K/k))‑k.
Then the followings are equivalent.
(1) 3>(K/k) with the finite k‑topology is discrete.
(2) LK:K]<‑.
(3) Hamk(K, K) with the finite k‑topology is discrete.
(4)ョ(K/k) ‑Horn*(K, K).
(5) ix/t is a nilpotent ideal of K(*)K.
K is a purely inseparable finite extension over k.
PROOF. As seen above (1), (2) and (3) are equivalent. By our previous paper [2], (4),
(5) and (6) are equivalent without assumption Z(@(K/k)) ‑k. Because @(K/k) is dense
in Homk(K,K), (3)⇔(4) is obvious. Finally (6)l=>(2) is trivial, q.e.d.
4 Teppei Kikuchi
