Let $A$ be a finite dimensional k-algebra standardly stratified for a partial order $\leqslant$ and $\Delta$ be the direct sum of all standard modules. In this paper we study the extension algebra $E = Ext^{\ast}_A (Δ,Δ)$ of standard modules, characterize the stratification property of $E$ for $\geqslant$ and $\geqslant ^{op}$, and obtain a sufficient condition for $E$ to be a generalized Koszul algebra (in a sense which we define).

Let $A = \bigoplus _{i \geqslant 0} A_i$ be a graded locally finite $k$-algebra where $A_0$ is a finite-dimensional algebra whose finitistic dimension is 0. In this paper we develop a generalized Koszul theory preserving many classical results, and show an explicit correspondence between this generalized theory and the classical theory. Applications in representations of certain categories and extension algebras of standard modules of standardly stratified algebras are described.

We introduce the notions of a D-standard abelian category and a K-standard additive category. We prove that for a finite dimensional algebra A, its module category is D-standard if and only if any derived autoequivalence on Ais standard, that is, isomorphic to the derived tensor functor by a two-sided tilting complex. We prove that if the subcategory of projective A-modules is K-standard, then the module category is D-standard. We provide new examples of D-standard module categories.

We study endomorphism algebras of 2-term silting complexes in derived categories of hereditary finite dimensional algebras, or more generally of Ext-finite hereditary abelian categories. Module categories of such endomorphism algebras are known to occur as hearts of certain bounded t-structures in such derived categories. We show that the algebras occurring are exactly the algebras of small homological dimension, which are algebras characterized by the property that each indecomposable module either has injective dimension at most one, or it has projective dimension at most one.

For smooth compact oriented Riemannian manifolds$M$of dimension 4$k$+2,$k$≥0, with or without boundary, and a vector bundle$F$on$M$with an inner product and a flat connection, we construct a modification of the Hodge star operator on the middle-dimensional (parabolic) cohomology of$M$twisted by$F$. This operator induces a canonical complex structure on the middle-dimensional cohomology space that is compatible with the natural symplectic form given by integrating the wedge product. In particular, when$k$=0 we get a canonical almost complex structure on the non-singular part of the moduli space of flat connections on a Riemann surface, with monodromies along boundary components belonging to fixed conjugacy classes when the surface has boundary, that is compatible with the standard symplectic form on the moduli space.