Questions about tilting theory, including questions on tilting modules, tilting sheaves, tilting complexes, and tilting objects.

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7
votes
1answer
524 views

got any tricks to build up t-structures on derived categories?

Are there any good tricks to construct a heart of a t-structure? (I'm thinking on the derived category of coherent sheaves of some variety) I'll start with the only one I know. If $(T,F)$ is a ...
5
votes
0answers
194 views

Not isomorphic varieties with isomorphic tilting algebras

Let $X$ be a smooth projective variety over a field, than tilting object $T$ on $X$ is a perfect complex that is a compact generator of the derived category $\operatorname{D}(QCoh(X))$ and satisfies ...
4
votes
2answers
383 views

Examples of tilting objects that don't come from exceptional sequences

This is a question on geometric tilting theory. On smooth projective variety it is possible to define in general tilting object as perfect complex that satisfy some properties, but are there examples ...
4
votes
2answers
207 views

Torsion pairs and projective dimension

Let $A$ denote an algebra finite dimensional, basic, and connected algebra over a algebraically closed field $K$. We denote by $mod A$ the abelian category whose objects are finitely generated right ...
3
votes
1answer
260 views

Inverse of a tilting module

Let $k$ be a field, $A$ an associative unital $k$-algebra, $\operatorname{\mathsf{Mod}} A$ the category of left $A$-modules and $D^b(\operatorname{\mathsf{Mod}} A)$ the bounded derived category. Let ...
3
votes
1answer
753 views

Tensor product of sheaves and modules

Hello to all, I have been looking quite recently at the following theorem: Let $X$ be a projective variety and $T$ a tilting object for $X$. If $A:=End(T)$ is the associated endomorphism algebra, ...
3
votes
1answer
97 views

A canonical algebra of type $(2,2,r)$ is derived equivalent to a path algebra of type $\tilde{D}_{r+2}$ (references)

According to several articles I could find, a canonical algebra of type $(2,2,r)$ is derived equivalent to a path algebra of type $\tilde{D}_{r+2}$, where $r \geq 2$. I don't know how to obtain this ...
3
votes
0answers
336 views

Geometric picture behind tilting sheaves

I am trying to read "Tilting exercises" and have trouble to see any geometric pictures behind the formulas. So my questions are, how to think about tilting perverse sheaves? Are they just formal ...
2
votes
0answers
200 views

Global dimension of endomorphism algebra of a coherent sheaf

Let $X$ be a smooth projective variety over algebraically closed field of characteristic zero. In one of the versions of definition of tilting object $\mathcal{F}$ on $X$ there is a requirement that ...
1
vote
2answers
451 views

tilting module

is any indecomposable projective-injective A-module a direct summand of tilting module
1
vote
1answer
168 views

Compact generator of $D(\mathbb{P}^1)$

I suppose that Beilinson's compact generator (and, in fact, tilting object) $\mathcal{O} \oplus \mathcal{O}(1)$ in $D(\mathbb{P}^1)$ is the most well known example. I have the following simple ...
1
vote
1answer
105 views

Which class of finite dimension algebra has only trivial tilting modules?

I have already knowed that selfinjective algebras have only trivial tilting modules,but besides this,is there any more?
0
votes
1answer
178 views

Equivalence of definitions of Tilting

There seem to be two types of definitions for what is a tilting module (as a reference, Handbook of Tilting Theory). I believe that the original definition of Ringel is Def: T, a module over a ...
0
votes
0answers
152 views

For a pair of non-commutative ring $(R, S)$, is there a faithfully semidualizing $(R, S)$-bimodule?

Given a pair of non-commutative rings $(R, S)$, how to construct a faithfully semidualizing $(R, S)$-bimodule? Henrik Holm and Diana White introduced the concept of faithfully semidualizing ...