Operations on categories of modules are given by Schur functors (Q1743613)

The author studies functors between categories of finitely generated modules \(F_R:\mathrm{Mod}_{fg}\to\mathrm{Mod}_{fg}\) which are defined simultaneously for all commutative \(k\)-algebras \(R\) (here \(k\) is a commutative ring) and behave well with respect to base changes in the sense that, for \(k\)-homomorphisms \(R\to S\), and finitely generated \(R\)-modules \(M\), there are isomorphisms of \(S\)-modules \(F_R(M)\otimes_RS\mathop{\to}\limits^{\sim} F_S(M\otimes_RS)\) which are coherent in a suitable sense. Such functors are called \textit{operations over} \(k\). Examples include the \(n\)-th tensor power, the \(n\)-th symmetric power, and the \(n\)-th exterior power. More generally, for the symmetric group \(\Sigma_n\) and a finitely generated right \(k[\Sigma_n]\)-module \(V_n\), the functor \(M\mapsto V_n\otimes_{k[\Sigma_n]}M^{\otimes n}\) is an operation. With finiteness conditions, every direct sum of such operations is an operation as well. They are called \textit{Schur operations} by the author (or \textit{analytic functors} by Joyal) since, for a field \(k\), if the \(k\)-algebra \(R\) is fixed, these functors are called \textit{Schur functors} in representation theory. The finitely generating restriction in the paper is needed because Nakayama's lemma is used intrinsically in one of the central proofs. The main result of the paper states that, under certain assumptions on \(k\), every operation is a Schur operation: {Theorem}. Let \(k\) be a commutative \(\mathbb{Q}\)-algebra. Then every operation \(\mathrm{Mod}_{fg}\to\mathrm{Mod}_{fg}\) over \(k\) is isomorphic to a Schur operation \(M\mapsto \bigoplus_{n\geq 0}V_n\otimes_{k[\Sigma_n]}M^{\otimes n}\), for a sequence of finitely generated right \(k[\Sigma_n]\)-modules \(V_n\). A similar characterization holds for the categories of finitely presented modules \(\mathrm{Mod}_{fp}\to\mathrm{Mod}_{fp}\). This result fails in characteristics \(>0\). A similar result was obtained by \textit{I. G. Macdonald} [Symmetric functions and Hall polynomials. Oxford Clarendon Press (1979; Zbl 0487.20007); J. Pure Appl. Algebra 18, 173--204 (1980; Zbl 0455.18002)] regarding the so-called polynomial functors. The author adapts several techniques of Macdonald to the context of this paper to reduce the general case to the cases of homogeneous and multilinear operations. The classification of the latter is as follows: {Theorem.} Let \(k\) be a commutative ring and \(n\in\mathbb{N}\). Then every multilinear operation \(\mathrm{Mod}^n_{fg}\to\mathrm{Mod}_{fg}\) (functors that are \(\mathbb{R}\)-linear in every variable) over \(k\) is isomorphic to the operation \((M_1,\dots,M_n)\mapsto V\otimes_k(M_1\otimes_R\dots\otimes_RM_n)\), for a finitely generated \(k\)-module \(V\). Another characterization is as follows: {Theorem.} Let \(k\) be a commutative \(\mathbb{Q}\)-algebra. Then the category of bounded operations \(\mathrm{Mod}_{fg}\to\mathrm{Mod}_{fg}\) over \(k\) is equivalent to the category of finite sequences \((V_n)_{n\in\mathbb{N}}\) of finitely generated right \(k[\Sigma_n]\)-modules. A similar classification is true for bounded operations \(\mathrm{Mod}_{fp}\to\mathrm{Mod}_{fp}\). The author gives also a constructive proof of his Theorem 3.3 and consequently Theorem 3.8.