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2 votes
0 answers
47 views

Is a triangulated category admitting a tilting object triangle equivalent to the unbounded derived category of the endomorphism ring of this object?

Let $\mathcal{T}$ be a triangulated category. We call an object $G$ tilting if $G$ is compact, that is, $\mathrm{Hom}_{\mathcal{T}}(G, -)$ preserves all set-indexed coproducts; $G$ is a generator, ...
Chen Yifan's user avatar
10 votes
1 answer
1k views

What's the relationship between spherical twist functors and tilting?

I've been reading about connections between Coxeter groups and preprojective algebras, and I keep running into two operations on the derived categories of preprojective algebras which seem very ...
Will Dana's user avatar
  • 453
5 votes
1 answer
212 views

On tilting and cotilting modules

Let A be an Artin algebra and assume all modules are basic, then a classical result says that tilting modules $T$ are in bijection with complete cotorsion pairs $(T^{\perp}, \check{ add(T)})$ (with ...
Mare's user avatar
  • 26.5k
2 votes
1 answer
213 views

How to get $Hom_A(M,N) \cong Hom_{B^{op}}(Hom_A(N,T),Hom_A(M,T))$?

I am reading the paper"Dominant dimensions, derived quivalences and tilting modules", the link is here:http://link.springer.com/article/10.1007/s11856-016-1327-4. On page 22,Lemma 4.2 says that let M ...
Xiaosong Peng's user avatar
1 vote
1 answer
215 views

The projective modules of an algebra and the tilting module?

Let A be an algebra. We denote by by A-proj the full subcategory of A-mod consisting of projective modules. An A-module T is called a tilting module if $proj.dim(_{A}T)=n < \infty$, $Ext_{A} ^{j} (...
Xiaosong Peng's user avatar
6 votes
1 answer
775 views

Socle of tilting modules in the BGG category $\mathcal{O}$ over a semisimple Lie algebra

Suppose that $\mathfrak{g}$ is a finite dimensional, complex, semisimple Lie algebra. Let $\mathcal{O}$ be the BGG category over $\mathfrak{g}$. Tilting module theory play an important role in the ...
Steven's user avatar
  • 159
2 votes
0 answers
203 views

Could Partial Tiltings be studied as Almost Complete Tiltings?

The first part of what follows is a brief recap of the definitions, setting and motivations for my questions. Experts can find the questions at the end. Here $k$ denotes an algebraically closed field,...
Kaveh's user avatar
  • 493