We introduce the generalized Serre functor S on a skeletally-small Hom-finite Krull-Schmidt triangulated category C. We prove that its domain Cr and range Cl are thick triangulated subcategories. Moreover, the subcategory Cr (resp. Cl) is the smallest additive subcategory containing all the objects in C which appears as the third term (resp. the first term) of some Aulsander-Reiten triangle in ...

Assuming only a very rudimentary knowledge of enriched category theory — V-categories, V-functors, and V-natural transformations for a closed, symmetric monoidal, complete and cocomplete category V — we introduce weighted limits and colimits, the appropriate sort of limits for the enriched setting. This “modern” approach was introduced to the author through talks by Mike Shulman at the Category...

In this paper, we principally explore flat modules over a commutative ring with identity. We do this in relation to projective and injective modules with the help of derived functors like Tor and Ext. We also consider an extension of the property of flatness and induce analogies with the “special cases” occurring in flat modules. We obtain some results on flatness in the context of a noetherian...

This note establishes internal criteria on a category C and a separator 2 in C which characterize the condition that the 2-induced covariant hom-functor /i2: C -> Set is (epi, mono-source)-topological. Introduction. Hoffmann [4] showed how topological functors (of Herrlich [2]) may be recovered from factorizations of sources and sinks in the domain category. This process was extended to (F, M)-...

Suppose X is a smooth projective scheme of finite type over a field K, E is a locally free OX -bimodule of rank 2, A is the non-commutative symmetric algebra generated by E and ProjA is the corresponding non-commutative P -bundle. We use the properties of the internal Hom functor HomGrA(−,−) to prove versions of Serre finiteness and Serre vanishing for ProjA. As a corollary to Serre finiteness,...

The category is introduced as an ordered 5-tuple of the form 〈O,M,dom,cod, ·, id〉 where O (objects) and M (morphisms) are arbitrary nonempty sets, dom and cod map M onto O and assign to a morphism domain and codomain, · is a partial binary map from M×M to M (composition of morphisms), id applied to an object yields the identity morphism. We define the basic notions of the category theory such a...

The Yoneda Lemma tells us that the Hom bifunctor is “non-degenerate” in a similar way. (a) For each object X ∈ C verify that hX := HomC(X,−) defines a functor C → Set. (b) Given two objects X,Y ∈ C state what it means to have hX ≈ hY as functors. (c) Given two objects X,Y ∈ C and an isomorphism of functors hX ≈ hY , prove that we have an isomorphism of objects X ≈ Y . [Hint: Let Φ : hX ∼ −→ hY ...

Abstract. Let X be a smooth scheme of finite type over a field K, let E be a locally free OX -bimodule of rank n, and let A be the non-commutative symmetric algebra generated by E. We construct an internal Hom functor, HomGrA(−,−), on the category of graded right A-modules. When E has rank 2, we prove that A is Gorenstein by computing the right derived functors of HomGrA(OX ,−). When X is a smo...

Crossed complexes have longstanding uses, explicit and implicit, in homotopy theory and the cohomology of groups. It is here shown that the category of crossed complexes over groupoids has a symmetric monoidal closed structure in which the internal Hom functor is built from morphisms of crossed complexes, nonabelian chain homotopies between them and similar higher homotopies. The tensor product...