Form of an (algebraic) structure

2020 Mathematics Subject Classification: Primary: 12-XX [MSN][ZBL]

Let $k'/k$ be an extension of fields, and let $X$ be some "object" defined over $k$. For example, $X$ could be a vector space together with a quadratic form, a Lie algebra over $k$, an Azumaya algebra over $k$, an algebraic variety over $k$, an algebraic group over $k$, a representation of a finite group in a $k$-vector space, etc. A form of $X$ over $k$, more precisely, a $k'/k$-form, is an object $Y$ of the same type over $k$ such that $X$ and $Y$ become isomorphic over $k'$, i.e. after extending scalars from $k$ to $k'$ the objects $X$ and $Y$ become isomorphic. Let $E(k'/k,X)$ denote the set of $k$-isomorphism classes of $k'/k$ forms of $X$. If now $k'/k$ is a Galois extension, then under suitable circumstances one has a bijection between $E(k'/k,X)$ and the Galois cohomology group $\def\H{ {\rm H}}\H^1(\def\Gal{ {\rm Gal}}\Gal(k'/k),\def\Aut{ {\rm Aut}}\Aut_{k'}(X))$ (cf. Galois cohomology), where $\Aut_{k'}(X)$ is the group of automorphisms over $k'$ of $X$. Consider, for instance, the case where the object $X$ is a finite-dimensional algebra $A$ over $k$. Then $B$ is a form of $A$ if $A\otimes_k k' \simeq B\otimes_k k'$ as $k'$-algebras. Let $\def\a{\alpha}\a$ be an automorphism of $A$ over $k'$, i.e. an isomorphism of $k'$-algebras $\a:A\otimes_k k'\to B\otimes_k k'$, and let $s\in\Gal(k'/k)$. Then $s(\a) = (1\otimes s)\circ \a\circ(1\otimes s^{-1})$ is another $k'$-automorphism of $A$. This defines the action of $\Gal(k'/k)$ on $\Aut_k'(A)$. Now let $B$ be a form of $A$. The set of $k'$-isomorphisms $B\otimes_k k'\to A\otimes_k k'$ is naturally a principal homogeneous space over $\Aut_k'(A)$ and thus defines an element of $\H^1(\Gal(k'/k),\Aut_k'(A))$. This mapping is a bijection in this case. More generally one has such a bijection for the case that the structure $X$ is a vector space $V$ together with a $(p,q)$-tensor (the previous case corresponds to the case of a $(2,1)$-tensor). (To prove surjectivity one needs the generalization of Hilbert's theorem 90: $\H^1(\Gal(k'/k),\def\GL{ {\rm GL}}\GL_n(k'))=0$, cf. Galois cohomology) For the case of algebraic groups over $k$ cf. Form of an algebraic group.

For the case of algebraic varieties over $k$ one has that $E(k'/k,X)\to \H^1(\Gal(k'/k),\Aut_k'(X))$ is injective and that it is bijective if $X$ is quasi-projective.

The concept of forms makes sense in a far more general setting, e.g. in any category with base change, i.e. with fibre products. Indeed, let $\def\fC{ {\mathfrak C}}\fC$ be such a category, and $S$ an object in $\fC$. An object over $\fC$ is a morphism in $\fC$, $X\to S$. Let $f: S'\to S$ be a morphism in $C$. Base change from $S$ to $S'$ gives the pullback (fibre product) $X_S'=X\times_S S'$ defined by the Cartesian square $$\def\mapright#1{\xrightarrow{#1}} \def\mapdown#1{\Big\downarrow\rlap{\raise2pt{\scriptstyle{#1}}}} \begin{array}{ccc} X_{S'}& \mapright{} & X \\ \mapdown{} & & \mapdown{} \\ S'& \mapright{} & S \end{array}$$ (In case $S' = \def\Spec{ {\rm Spec}}\Spec(k')$, $S = \def\Spec{ {\rm Spec}}\Spec(k)$ and $\fC$ is, for instance, the category of (affine) schemes this corresponds to extending scalars.)

An object $Y\in\fC_{/S}$ is now an $S'/S$-form of $X\in\fC_{/S}$ if the objects $X_{S'}$ and $Y_{S'}$ are isomorphic over $S'$. For an even more general setting cf. [Gr].

A related problem (to that of forms) is the subject of descent theory. In the setting of a category with base change as above this theory is concerned with the question: Given $Z\in\fC_{/S'}$, does there exists an $X$ over $S$ such that $X_{S'}$ is isomorphic over $S'$ to $Z$, and what properties must $Z$ satisfy for this to be the case.

Below this question is examined in the following setting: $R$ is a commutative algebra (with unit element) and $S$ is a commutative $R$-algebra. Given a module $M$ over $S$ the question is whether there exists a module $N$ over $R$ such that $M\simeq N_S = N\otimes_R S$ (as $S$-modules). Below all tensor products $\otimes$ are tensor products over $R$: $\otimes_R$. If $M$ is of the form $N_S$ there is a natural isomorphism of $S\otimes S$ modules $S\otimes N_S \to N_S\otimes S$ given by $\def\o{\otimes} s_1\o n\o s_2\mapsto n\o s_1\o s_2$. Let $M$ be an $S$-module. A descent datum on $M$ is an isomorphism of $S\o S$ modules $g:S\o M \to M\o S$ such that $g_2 = g_3 g_1$. Here $g_1, g_2, g_3$ are the three natural $S\o S\o S$-module homomorphisms defined by $g$, where $g_i$ is the identity on factor $i$ and given by $g$ on the other two components:

$$g_1 : S\o S\o M\to S\o M\o S,$$

$$g_2 : S\o S\o M\to M\o S\o S,$$

$$g_3 : S\o M\o S\to M\o S\o S.$$ The faithfully flat descent theorem now says that if $S$ is faithfully flat over $R$ and $g$ is a descent datum for $M$ over $S$, then there exists an $R$-module $N$ and an isomorphism $\eta : N_S\to M$ such that the following diagram commutes

$$\begin{array}{ccc} S\o N_S & \mapright{1\o \eta} & S\o M\\ \mapdown{} & & \mapdown{g}\\ N_S\o S & \mapright{\eta\o 1} & M\o S\\ \end{array}$$ where the left vertical arrow is the descent datum on $N_S$ described above. Moreover, the pair $(N,\eta)$ is uniquely defined by this property. One defines $N$ by an invariance property: $N=\{x\in M : x\o 1 = g(1\o x)\}$ (which is like invariance under the Galois group in the case of Galois descent).

There is a similar theorem for descent of algebras over $S$.

In algebraic geometry one has for instance the following descent theorem (a globalization of the previous one for algebras). For a morphism of schemes $f : Y\to X$, consider the fibre products $Y\times_X Y$ and $Y\times_X Y\times_X Y$ and let $p_{ij} : Y\times_X Y\times_X Y \to Y\times_X Y$ be the projections $(y_1,y_2,y_3)\mapsto (y_i,y_j)$, $3\ge i>j \ge 1$; and $p_i: Y\times_X Y \to Y$ the projections $(y_1,y_2)\mapsto y_i,$, $i=1,2$. Let $f:Y\to X$ be faithfully flat and compact. Then to give a scheme $Z$ affine over $X$ is the same as to give a scheme $Z'$ affine over $Y$ together with an isomorphism $\def\a{\alpha}\a: p_1^*Z'\to p_2^*Z'$ such that $p_{31}^*(\a) = p_{32}^*(\a)p_{21}^*(\a)$.

The theory of descent is quite general and includes such matters as specifying a section of a sheaf by local sections and the construction of locally trivial fibre bundles by glueing together trivial bundles $U_i\times F\to U_i$ over the elements of an open covering $\{U_i\}$ of $X$. Indeed, let $X'$ be the disjoint union of the $U_i$ and $p:X'\to X$ the natural projection. Giving glueing data $\a_{ij}(U_i\cap U_j)\times F \to (U_j\cap U_i)\times F$ is the same as giving an isomorphism $\a: p_1^*E'\to p_2^* E'$, where $E'$ is the trivial vector bundle $X'\times F$ with fibre $F$ and the compatibility of the glueing data amounts to the condition $p_{31}^*(\a) = p_{32}^*(\a) p_{21}^*(\a) $.

For a treatment of forms of Lie algebras (over fields) cf. [Ja], for Lie algebras over characteristic zero fields and the modular case (i.e. over fields of characteristic $p>0$) cf. [Se2]. For a quite comprehensive treatment of descent and forms cf. [KnOj].

A form of an object is also occasionally called a twisted form.

In the case of descent with respect to a Galois field extension $k\subset k'$ (or $\Spec(k') \to \Spec(k)$) one speaks of Galois descent.

References

[Gr]	A. Grothendieck, "Revêtements étales et groupe fondamental", SGA 1960–1961, Exp. VI: Categories fibrées et descente, IHES (1961) MR2017446 MR0354651 MR0217088 MR0217087 Zbl 1039.14001
[Ja]	N. Jacobson, "Lie algebras", Dover, reprint (1979) pp. Chapt. X ((also: Dover, reprint, 1979)) MR0148716 {{MR|0143793} Zbl 0121.27504 {{Zbl}0109.26201}}
[KnOj]	M.-A. Knus, M. Ojanguren, "Théorie de la descent et algèbres d'Azumaya", Springer (1974) MR417149
[Mu]	J.P. Murre, "Lectures on an introduction to Grothendieck's theory of the fundamental group.", Tata Inst. Fund. Res. (1967) pp. Chapt. VII MR302650
[Se]	J.-P. Serre, "Cohomologie Galoisienne", Springer (1973) MR0404227 Zbl 0259.12011
[Se2]	G.B. Seligman, "Modular Lie algebras", Springer (1967) pp. Chapt. IV MR0245627 Zbl 0189.03201
[Se3]	J.-P. Serre, "Groupes algébrique et corps des classes", Hermann (1959) pp. Chapt. V, Sect. 20 MR0103191