Let C be an abelian category and consider its ind-category Ind(C): (1) Is Ind(C) always abelian? (If not, what conditions are needed?) (2) Is C⊆Ind(C) a Serre subcategory? In other words, if we have a short exact sequence: 0⟶M′⟶M⟶M″ in Ind(\mathcal C), then M\in \mathcal C iff M’ and M” are in \mathcal C? Answer … Read more
Recall the following localization theorem, as stated in Hsiang’s Cohomology Theory of Compact Transformation Groups: Theorem. Let G=(S1)k be a torus, X a paracompact G-space with finite cohomological dimension, and F the fixed point set of X. Then the following localized restriction homomorphism S−1H∗G(X;Q)→S−1H∗G(F;Q), where S=H∗(BG;Q)−{0}=Q[t1,…,tk]−{0}, is an isomorphism. Something similar happens when G is … Read more
I have a monad T:C→C on a (Grothendieck) abelian category which preserves filtered colimits and direct sums (but is not exact). There is a finite collection G of compact, projective objects which generate C in the sense that every object X is presented as a cokernel R1→R0→X→0 of objects R1 and R0 which are (possibly … Read more
Recall that, if R is a commutative ring, then a suitably finite R-module M is projective if and only if the localization Mm is a direct sum of finitely many copies of Rm for every maximal ideal m⊆R. Consider the following alternate condition for an R-module M: each Mm is a direct sum of finitely … Read more
Let C be an abelian category and S be a full subcategory of C. We call S a Serre subcategory of C if S is closed under forming subobjects, quotients, and extensions. For a Serre subcategory S we could form the quotien category C/S and have the quotient functor p:C→C/S, which is obviously exact. We … Read more
Let k be a field and C be a dg-category over k. It is standard to define the homotopy category H0(C) as the category consisting the same objects as C but morphisms between two objects x and y are defined as H0(C)(x,y):=Z0(C)(x,y)/B0(C)(x,y). We could also consider the category Z0(C) and the class W of homotopic … Read more
I am reading Bousfield’s paper entitled “The localization of spectra with respect to homology” (MSN). In that paper, Corollary 6.13 states that, if a ring spectrum E has countable homotopy and satisfies some vanishing conditions in the associated Adams spectral sequence, then the localization is equivalent to nilpotent completion. So, my question is the following: … Read more
Let $\mathbb{S}$ be the sphere spectrum. We can localize the category of based spaces, $Top_*$ at a homology theory, and hence at $\mathbb{S}$. Equipping $Top_*$ with the Quillen model structure (weak homotopy equivalences, and Serre fibrations), and Bousfield localizing at $\mathbb{S}$ gives a model structure with the weak equivalences the isomorphisms on stable homotopy groups, … Read more
Lets take a ring R and an ideal p⊂R, and call them an L-pair (just for brevity) if p-adic completion of any projective module is again projective (as R-module); and L-ring is a ring which is an L-pair with any of its finitely generated ideals. (I’m not sure, but have a feeling that looking at … Read more
Let C be a category and suppose B⊆C is a full subcategory. Let i:B⟶C denote the inclusion functor. Suppose that S⊆MorC is a class of morphisms in C. Then we get a functor ˜i:B[(S∩MorB)−1]⟶C[S−1] If S satifies the Ore conditions, and if for every morphism s:M⟶N in S with M∈B there exists a morphism u:N⟶P … Read more
