\(\mathbf{GL}\)-algebras in positive characteristic. I: The exterior algebra (Q6565773)

Let \(k\) be a field and, for a non-negative integer \(d\), \(\mathbf{Pol}_d\) denote the category of strict polynomial functors over \(k\) of degree (or weight) \(d\) in the sense of [\textit{E. Friedlander} and \textit{A. Suslin}, Invent. Math. 127, No. 2, 209--270 (1997; Zbl 0945.14028)]. This nice and important abelian category is equivalent to the category of polynomial representations of degree \(d\) of the algebraic group \(\operatorname{GL}_n(k)\) for \(n\ge d\), or to the category or modules over a suitable Schur algebra over \(k\). Let \(\mathbf{Pol}:=\prod_{d\in\mathbb{N}}\mathbf{Pol}_d\) be the category of (non homogeneous) strict polynomial (or, more accurately, analytic) functors over \(k\); one has a canonical exact functor (given by the direct sum for \(d\in\mathbb{N}\) of the forgetful functors), which is fully faithful if \(k\) is infinite, from \(\mathbf{Pol}\) to the category of all functors from finite dimensional \(k\)-vector spaces to \(k\)-vector spaces. By taking a colimit of evaluations, a functor in \(\mathbf{Pol}\) can be also seen as a representation, polynomial in a suitable meaning, of the linear group \(\operatorname{GL}_\infty(k):=\underset{n\in\mathbb{N}}{\mathrm{colim}}\operatorname{GL}_n(k)\).\N\NThe category \(\mathbf{Pol}\) has a symmetric monoidal structure given by the usual tensor product (of representations of \(\operatorname{GL}_\infty(k)\), or of functors to \(k\)-vector spaces, computed at the target). The author names (skew) \(\mathbf{GL}\)-algebra a (skew) commutative monoid object in this category (`skew' refers to the symmetric structure in which the braiding is twisted by the usual signs coming from the grading by weights). A typical example of a \(\mathbf{GL}\)-algebra (resp. skew \(\mathbf{GL}\)-algebra) is the symmetric (resp. exterior) algebra functor \(S^*=\bigoplus_{d\ge 0}S^d\) (resp. \(\Lambda^*=\bigoplus_{d\ge 0}\Lambda^d\)). The modules over such a (skew) \(\mathbf{GL}\)-algebra \(A\) form a nice abelian category \(\mathbf{Mod}_A\), endowed with a symmetric monoidal structure (the tensor product over \(A\)), with a lot of features similar to usual commmutative algebra.\N\NIf the characteristic of the field \(k\) is \(0\), it is classical (and easy to prove, using Schur-Weyl duality) that both categories \(\mathbf{Mod}_{S^*}\) and \(\mathbf{Mod}_{\Lambda^*}\) are equivalent to the category of \(FI\)-modules, that is, the category of functors from the category \(FI\) of finite sets with injections to \(k\)-vector spaces. This category has been extensively studied these last ten years. For example, it is known that the category of \(FI\)-modules is locally noetherian of Krull dimension \(1\), and a lot of homological properties have been proved using methods from combinatorics, commutative algebra or from the theory of polynomial functors. Nevertheless, in positive characteristic, the relations between representations of symmetric and linear groups are more subtle, so the previous equivalences fail, and one can not use \(FI\)-modules.\N\NIn the article under review, the author investigates the category \(\mathbf{Mod}_{\Lambda^*}\) (denoted by \(\mathrm{Mod}_R\) in the paper) when \(k\) has prime characteristic \(p\) (the field is also assumed to be algebraically closed, but it seems to be unnecessary), announcing for a forthcoming work that the behaviour of \(\mathbf{Mod}_{S^*}\) is largely different in this setting. The proven results are similar to some ones for \(FI\)-modules, but also for \(VI\)-modules (where \(VI\) denotes the category of finite-dimensional vector spaces over a finite field \(\mathbb{F}\)) in nondescribing characteristic (i.e. \(\mathrm{char}(\mathbb{F})\ne p\)) [\textit{R. Nagpal}, Algebra Number Theory 13, No. 9, 2151--2189 (2019; Zbl 1461.20004)].\N\NA first important theorem, proved by combinatorial methods à la Gröbner, is that \(\mathbf{Mod}_{\Lambda^*}\) is locally noetherian (\textit{Theorem 3.2}).\N\NTwo fundamental classes of objects appear in \(\mathbf{Mod}_{\Lambda^*}\): the \textit{semi-induced} modules, that is, the ones admitting a finite filtration whose subquotients are of the shape \(\Lambda^*\otimes F\) for some \(F\in\mathbf{Pol}\), and the \textit{torsion modules}, meaning that each element in their evaluations are annihilated by some power of the augmentation ideal of \(\Lambda^*\). It is proved (\textit{Proposition 2.4}) that a finitely generated object in \(\mathbf{Mod}_{\Lambda^*}\) is semi-induced if and only if it flat as a module over \(\Lambda^*\). It is also proved, in section 3.2, that the torsion modules are exactly the locally finite ones, and that the inclusion of the localising subcategory of torsion modules into \(\mathbf{Mod}_{\Lambda^*}\) preserves injective objects. The \textit{generic} category is by definition the quotient category; the author proves (\textit{Proposition 3.8}) that this generic category is locally finite, implying that \(\mathbf{Mod}_{\Lambda^*}\) has Krull dimension \(1\).\N\NAnother important tool in this work is given by the \textit{Schur derivative}, an exact endofunctor \(\mathbf{\Sigma}\) of \(\mathbf{Pol}\) with very nice properties (see section 2.3); \(\mathbf{\Sigma}(F)(V)\) is by definition the part of \(F(k\oplus V)\) on which the multiplicative group \(k^*\) acts (by multiplication on the factor \(k\), and trivially on \(V\)) with weight \(1\). This functor lifts to \(\mathbf{Mod}_{\Lambda^*}\); if \(M\) is an object of this category, one has a natural morphism \(M\to\mathbf{\Sigma}(M)\) whose cokernel is denoted by \(\mathbf{\Delta}(M)\) and whose kernel is a torsion module, so that \(\mathbf{\Delta}\) (which is called the \textit{difference functor}) induces a nice exact endofunctor of the \textit{generic} category (see section 4.1). The objects of \(\mathbf{Mod}_{\Lambda^*}\) which cancel \(\mathbf{\Delta}\) can be controled (section 4.2), and \(\mathbf{\Delta}^r(M)=0\) for \(r\) big enough for every finitely generated \(M\) in \(\mathbf{Mod}_{\Lambda^*}\), what allows inductive arguments using \(\mathbf{\Delta}\).\N\NWith this background, the article is ready to prove its \textit{shift theorem} (1.2), inspired by Nagpal (who proved such a result for \(FI\)-modules, and another one for \(VI\)-modules in nondescribing characteristic -- see [loc. cit.]): if \(M\) is a finitely generated object of \(\mathbf{Mod}_{\Lambda^*}\), then for \(l\in\mathbb{N}\) big enough, \(\mathbf{\Sigma}^l(M)\) is semi-induced. As a consequence, there is a finite complex \(0\to M\to P^0\to P^1\to\dots\to P^r\to 0\) in \(\mathbf{Mod}_{\Lambda^*}\) whose homology is torsion and in which each \(P^i\) is semi-induced (\(P^0\) is \(\mathbf{\Sigma}^l(M)\) for \(l\) big enough, and one can continue the resolution by suitable induction). It permits to exhibit a very small class of generators (in the triangulated sense) of the bounded derived category of finitely generated objects of \(\mathbf{Mod}_{\Lambda^*}\) (\textit{Theorem 5.4}). The article ends with several interesting homological consequences of this result, as the finiteness of the Castelnuovo-Mumford regularity for finitely generated objects of \(\mathbf{Mod}_{\Lambda^*}\).