Reducedness, formal smoothness and approximation in characteristic p
Reducedness, formal smoothness and approximation in characteristic p
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特征 p 的简化、形式平滑和近似
DOI：
10.1080/00927879508825309
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发表时间：
1995
影响因子：
0.7
通讯作者：
T. Dumitrescu
中科院分区：
文献类型：
--
作者：
T. Dumitrescu
对于正素数特征p的noether环的同态u: a + B，我们将a在a中，B在B中，由w (a@ B)= u (a) bP给出的同态w: ACP) B-+ B@)联系起来，其中a@ = a是a的Frobenius自同态给出的a代数。&（[15],[5]）建立了u的正则性与w的平坦性之间的等价性，证明了u是约简的（p < 0.05）。u是局部光滑的)当且仅当w是内射且B/Im (w)是平a模（见图2）。w的完成是平的)。当u是域扩展时，我们初步证明了环a @) B的noether性与u的不完备模的有限B维性之间的已知等价性。作为应用，在正素数特征的情况下，我们给出了两个定理的新证明[17，them]。11和[13]，Prop. 2.41。本文所考虑的环都是交换的，酉的，并且具有特征p（一个固定素数正数）。我们的符号和术语将遵循[I 11；同样，我们将把“如果且仅当”缩写为“iff”。对于环同态u: a—+ B，我们将w= WB/ a: a@) B-+ B (P)由w (a@ B)= u (a) bP给出，其中a@)= a是由a的Frobenius自同态给出的a代数（通常用f表示）。同态序列s: B-9 C= A@)@ A B, w: C-+ B@)= B是[4]术语中f -对象的一个典型例子，即ws= f和sw= f，特别是由于谱上的诱导映射
To a homomorphism u: A+ B of noetherian rings of positive prime cha-racteristic p, we associate the homomorphism w: ACP) B-+ B@) given by w (a@ b)= u (a) bP for a in A and b in B, where A@)= A is the A-algebra given by the Frobenius endomorphism of A. In the spirit of the theorem of N. Radu and M. And. & ([15],[5]) establishing the equivalence between the regularity of u and the flatness of w, we prove that u is reduced (resp. u is local and formally smooth) if and only if w is injective and B/Im (w) is a flat A-module (resp. the completion of w is flat). When u is a field extension we prove elementarily the known equivalence between the noetherianity of the ring A@) B and the finite B-dimensionality of the imperfection module of u. As applications, in the case of positive prime characteristic, we present new proofs for two theorems [17, thm. 11 and [13, Prop. 2.41.All rings considered in this paper are commutative, unitary and having characteristic p (a fixed prime positive number). Our notation and terminology will follow [I 11; also, we shall abbreviate" if and only if'by" iff". TO a ring homomorphism u: A--+ B we associate the homomorphism w= WB/A: A@) B-+ B (P) given by w (a@ b)= u (a) bP for a in A and b in B, where A@)= A is the A-algebra given by the Frobenius endomorphism of A (always denoted by f or fA). The sequence of homomorphisms s: B-9 C= A@)@ A B, w: C-+ B@)= B is a canonical example of F-object in the terminology of [4], that is ws= f and sw= f. In particular, since the induced map on spectra