A Priddy-type Koszulness criterion for non-locally finite algebras

Abstract

Abstract. A celebrated result by S. Priddy states the Koszulness of any locally finite homogeneous PBW-algebra, i.e. a homogeneous graded algebra having a Poincaré-Birkhoff-Witt basis. We find sufficient conditions for a non-locally finite homogeneous PBW-algebra to be Koszul, which allows us to completely determine the cohomology of the universal Steenrod algebra at any prime.Introduction. The notion of Koszul algebra, introduced by S. Priddy in [15] in particular to construct resolutions for the Steenrod algebra, has led to remarkable achievements in the study of associative algebras defined by quadratic relations. The Koszulness condition provides decisive information to solve several basic problems in that context. [14] gives a beautiful and comprehensive account of the impact of Koszul algebras in several areas of mathematics. Such algebras arise in fact in algebraic geometry, representation theory, non-commutative geometry, number theory, and obviously algebraic topology.In this paper we deal with homogeneous algebras A isomorphic to a quotient of the form T (V )/J(R), where T (V ) = i T i is the tensor algebra over a K-vector space V with basis X = {x i | i ∈ I}, I is a (not necessarily bounded) totally ordered set, and J(R) is the two-sided ideal of relations generated by some R ⊂ T 2 = V ⊗ V .Note that all the Koszulness criteria listed for example in [9] concerning the Hilbert series of A become meaningless if I is not finite; even Priddy's criterion, i.e. the existence of a Poincaré-Birkhoff-Witt basis [15], only holds if the algebra has an internal degree induced by a map g : I n → Z and it is locally finite with respect to length and g (see [15]). It follows that there are examples of homogeneous quadratic algebras whose Koszulness cannot be checked using directly the criteria listed in [9] and [14]: Poisson enveloping algebras of Poisson algebras with generators indexed by Z and quadratic brackets (see [11] for the definition), infinite quantum grassmannians (see

Stability in the high-dimensional cohomology of congruence subgroups

“…We now follow Brunetti and Ciampella's argument [BC07,Theorem 2.5] Proof. Because B n * (St) is supported in homological degrees n, it is enough to prove that H s (B n * (St)) = 0 for s < n. We define a k-linear map Φ :…”

Length-preserving monomorphisms for some algebras of operations

“…In both cases such structure relies on the existence of a strict monomorphism λ : Q(2) −→ Q (2) that preserves the length of each monomial.…”

“…Recently the first and the second author have determined a fractal structure for Q (2) following two different approaches (see [3,11]). In both cases such structure relies on the existence of a strict monomorphism λ : Q(2) −→ Q (2) that preserves the length of each monomial.…”

“…In [20] Peter May introduced an F p -algebra Q( p) known by different names such as the algebra of all generalized Steenrod operations [19] and the universal Steenrod algebra (see [2][3][4][5][6][7][8][9][10][11][12][13][15][16][17][18]). It is the algebra of cohomology operations in the category C( p, ∞) of H ∞ -ring spectra.…”

“…On that occasion Sam Gitler encouraged the speaker to explore the existence in Q( p) of sequences of smaller subalgebras in order to circumvent problems arising from its 'bigness'. Such suggestion led along the years to many insights concerning the universal Steenrod algebra, among which its koszulity (see [2,6,9]), and more recently the existence of a fractal structure for p = 2 (see [3,11]). Some results in this paper are also inherent to that very hint.…”