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We give several characterizations of Zimmermann's uniform chain condition on matrix subgroups. In particular, we prove that a family of modules satisfies it precisely when the direct sum of their pure injective envelopes splits in their direct product. We also prove that a weaker non-uniform version of this condition is equivalent to Σ-algebraic compactness of reduced products. [ABSTRACT FROM PUBLISHER]

This work examines the possible projectivity of 2-modular block parts of nonprojective Lefschetz characters over 2-local geometries of several sporadic groups. Previously known results on M12, J2, and HS are mentioned for completeness. The main new results are on the sporadic groups Suz, Co3, Ru, O'N, and He. For each group, the Lefschetz character is calculated, and its 2-modular block parts are examined for projectivity. In each case it is confirmed that a nonprincipal block part contains a nonprojective summand. The case of O'N is additionally found to have a nonprojective summand in its principal block part. Nineteen of the sporadic groups (including many previously known cases) are categorized into three classes based on projectivity properties of their Lefschetz characters. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic 0, and let A = k[x, y]/(f) be a quasi-homogeneous plane curve. We show that for any graded torsion free A-module M, there exists a natural graded integrable connection, i.e., a graded A-linear homomorphism ∇: Derk(A) → Endk(M), that satisfies the derivation property and preserves the Lie product. In particular, a torsion free module N over the complete local ring [image omitted] admits a natural integrable connection if A is a simple curve singularity, or if A is irreducible and N is a gradable module. [ABSTRACT FROM AUTHOR]