# Questions tagged [tilting]

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

31
questions

8
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1
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### What is the remaining difficulty in the proof of the Humphreys conjecture (on the support variety of tilting modules)?

Humphreys conjecture describes the support variety of tilting modules using the correspondence between two-sided cells and nilpotent orbits. But the support variety can also be given by Lusztig–Vogan ...

2
votes

1
answer

152
views

### References on tilting distributions

I would be interested in any book, paper, or other reading material that gives a comprehensive treatment og tilted distributions using the following notion of "tilting" (or equivalent):
...

3
votes

1
answer

248
views

### Perverse tilting sheaves

In the article titled Tilting Exercises (See http://arXiv.org/abs/math/0301098v3) the authors define a notion of tilting perverse sheaves on an algebraic vareity $X$ with respect to stratification $\...

2
votes

0
answers

100
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### A question about t-structures in derived category

Let R be a ring and $_{R}P$ be a projective module, my question is whether $P^{\perp_{>0}}:=\{X\in D(R)|Hom(P,X[i])=0, i>0\}$ is an aisle i.e. if $(P^{\perp_{>0}}, (P^{\perp_{>0}})^{\perp}...

3
votes

1
answer

139
views

### On the definition and an example of silting/tilting subcategories in a triangulated categories according to a paper by Aihara and Iyama

In the paper "Silting mutation in triangulated categories" by Aihara and Iyama, I stumbled upon this nice definition( Definition 2.1) of a tilting/silting subcategory of a triangulated ...

10
votes

1
answer

451
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### 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 ...

5
votes

1
answer

202
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

644
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### 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

310
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

185
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

202
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

193
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

372
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

750
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

197
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

510
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

319
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

205
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

323
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

366
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

492
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

215
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

679
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

266
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

201
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?

12
votes

1
answer

2k
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### 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

173
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

713
views

### tilting module

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

4
votes

0
answers

532
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### 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

323
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, ...