# Questions tagged [tilting]

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

25
questions

**5**

votes

**1**answer

149 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 ...

**4**

votes

**1**answer

370 views

### Graded quivers vs “ordinary” quivers and derived categories

I have heard the "slogan" that graded quivers are (derived) equivalent to ordinary quivers (with this "result" being attributed to Keller) and am looking for a precise statement and a reference.
By a ...

**7**

votes

**1**answer

267 views

### Number of tilting modules

Let $A=A_n$ be the algebra of upper triangular matrices over a field $K$ with $n$ simple modules.
It is a nice result that there are $C_{n+1}=1,2,5,14,...$ (Catalan numbers for $n \geq 1$) tilting $...

**1**

vote

**1**answer

146 views

### Generating $K^b(\mathrm{proj})$ as a triangulated category from a full subcategory

Let $K^b(\mathrm{proj}\, A)$ be the bounded homotopy category of chain complexes over $\mathrm{proj}\, A$. In Rickard's paper 'Derived categories and stable equivalence', he defines a tilting complex ...

**2**

votes

**1**answer

175 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 ...

**1**

vote

**1**answer

159 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} (...

**4**

votes

**1**answer

302 views

### Tilting modules in positive characteristic

Consider the category of finite-dimensional representations for the algebraic group $\mathrm{SL}(n)$ in characteristic $p$. I know very little about this but am told there is a highest weight category ...

**6**

votes

**1**answer

645 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 ...

**2**

votes

**0**answers

190 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,...

**6**

votes

**1**answer

421 views

### An equivalence of derived categories by Happel-Reiten-Smalø

I have a problem in understanding the proof of a theorem by Happel-Reiten-Smalø. The original reference is this article
http://arxiv.org/abs/0911.4473
.
I write down the text of the theorem and a ...

**4**

votes

**0**answers

270 views

### 2-periodic derived equivalence

Let $A$ and $B$ be finite-dimensional algebras with finite global dimension over some field (in fact I am thinking of rational incidence algebras of finite posets).
Suppose we know that $A$ and $B$ ...

**3**

votes

**1**answer

175 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

**0**answers

280 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 ...

**4**

votes

**2**answers

331 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

**1**answer

409 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 ...

**1**

vote

**1**answer

203 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 ...

**5**

votes

**2**answers

578 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 ...

**7**

votes

**0**answers

259 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 ...

**2**

votes

**2**answers

189 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?

**9**

votes

**1**answer

1k 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 ...

**0**

votes

**0**answers

171 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 ...

**1**

vote

**2**answers

685 views

### tilting module

is any indecomposable projective-injective A-module a direct summand of tilting module

**3**

votes

**0**answers

501 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 ...

**3**

votes

**1**answer

313 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

**1**answer

1k 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, ...