Structure of Zariski-closed algebras

What is it about?

The article under review concerns representable algebras. Given an associative algebra $A$ over a field $F$, one says that $A$ is representable if $A$ can be embedded as a subalgebra of $M_n(K)$ for suitable $n$ and a suitable extension field $K\supseteq F$ (sometimes, a commutative $F$-algebra $C$ may be considered in place of $K$). Representable algebras are closely connected with PI-algebras, which also permeate the paper. Recall that $A$ is an algebra with polynomial identities (PI-algebra) if the ideal ${\rm Id}(A)$ of the free associative (noncommutative) algebra $F\langle X\rangle$, for $X=\{x_1,x_2,\dots\}$, composed of all polynomials $f(x_1,\dots,n_n)\in F\langle X\rangle$ such that $f(a_1,\dots,a_n)=0$ for all $a_1,\dots,a_n\in A$, is not trivial. When the base field $F$ is infinite, a common strategy to study ${\rm Id}(A)$ is to consider the algebra obtained by extending the scalars, that is, to consider the algebra $K\otimes_FA$ for some field $K\supseteq F$. In fact, the polynomial identities of $A$ are stable over an infinite field, that is, they still hold for the larger algebra $K\otimes_FA$, so ${\rm Id}(A)={\rm Id}(K\otimes_FA)$. Therefore, if $A$ is representable, it is possible to consider a finite-dimensional algebra $B$ over an algebraically closed field $K$ such that ${\rm Id}(A)={\rm Id}(B)$. The advantages should be clear: $K$ is algebraically closed, and the whole structure theory of finite-dimensional algebras is available. Namely, $B=KA$, the central extension of $A$ inside $M_n(K)$. This approach cannot be satisfying if $F$ is a finite field: identities are no longer stable, so $A$ may have identities that do not hold after a scalar extension. For instance, if $\vert F\vert=q=p^a$ then $x^q-x$ is an identity for $F$, but it is not an identity for $\overline{F}$. In this article, the authors change perspective, keeping a wider view in choosing a replacement for the original algebra $A$: instead of taking $KA$, they take the topological closure $A^{\text{cl}}$ of $A$ inside $M_n(K)$ endowed with the Zariski closure, a classical topology of "algebraic'' taste. This way they get a unifying approach with the finite-field case: indeed $A^{\text{cl}}\subseteq KA$, and $A^{\text{cl}}=KA$ if $\vert F\vert=\infty$, thus getting back the infinite case (Lemma 3.6), but they also prove that $A$ and $A^{\text{cl}}$ always generate the same variety over $F$ (Lemma 3.18), that is, ${\rm Id}(A)={\rm Id}(A^{\text{cl}})$. The algebra $A^{\text{cl}}$ is an excellent candidate to replace $A$. Indeed, any Zariski closed algebra has several important structural features of finite-dimensional algebras, listed in section 3.3 of the article, and the closure operator acts functorially, preserving all the significant algebraic structures and morphisms (section 3.1). For instance, the closure operator sends vector spaces to vector spaces, algebras to algebras, and a Zariski closed algebra has a Wedderburn decomposition and possesses Krull-Schmidt invariance. In the second part of the paper, namely from section 4 on, the authors go more deeply into the study of Zariski closed algebras: In section 4, they investigate the polynomial relations of a Zariski-closed algebra, that is, those polynomials in a fixed number of commutative variables vanishing on $A$. This is different from considering the polynomial identities of $A$, but all the same significant, because the polynomial identities of $A$ are in some sense determined by the polynomial relations of $A$. Moreover, the ideal they constitute in $K[\lambda_1,\dots,\lambda_n]$ is after all an ideal in a commutative algebra, hence more "manageable'' in principle. In section 5, the object is to find a "canonical" way of representing a Zariski-closed algebra inside $M_n(K)$. The inspiring idea is a celebrated theorem of Lewin and the use Giambruno and Zaicev made of it in their classification of minimal algebras. In section 6, the work done in the previous sections gives rise to a fairly detailed description of the generators of the polynomial relations of a Zariski-closed algebra $A$. Finally, in section 7, the authors carefully build a finitely generated generic algebra acting as a relatively free algebra of a variety generated by a Zariski-closed algebra. This object has to be available also when the base field $F$ is finite, and this causes many more difficulties than the usual construction. This leads also to the definition of PI-generic rank, generalizing what in ordinary PI-theory is known as "basic rank''. It turns out that these relatively free algebras have after all a nice description, and they play a key role in the proof of Specht's conjecture for affine PI-algebras in arbitrary characteristic. Reviewed by Vincenzo Carmine Nardozza

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