Questions tagged [tilting]

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

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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}...
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3 votes
1 answer
102 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 ...
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10 votes
1 answer
337 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 ...
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5 votes
1 answer
181 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 ...
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4 votes
1 answer
510 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 ...
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7 votes
1 answer
292 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 $...
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1 vote
1 answer
162 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 ...
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  • 420
2 votes
1 answer
190 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 ...
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1 vote
1 answer
180 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} (...
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4 votes
1 answer
334 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 ...
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6 votes
1 answer
691 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 ...
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  • 159
2 votes
0 answers
193 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,...
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  • 463
6 votes
1 answer
479 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 ...
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4 votes
0 answers
311 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 $B$ ...
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3 votes
1 answer
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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 ...
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  • 237
3 votes
0 answers
306 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 ...
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  • 1,495
4 votes
2 answers
354 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 ...
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  • 237
3 votes
1 answer
468 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 ...
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1 vote
1 answer
208 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 ...
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5 votes
2 answers
644 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 ...
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7 votes
0 answers
263 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 ...
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2 votes
2 answers
195 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?
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12 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 ...
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0 votes
0 answers
172 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 ...
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  • 159
1 vote
2 answers
697 views

tilting module

is any indecomposable projective-injective A-module a direct summand of tilting module
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  • 19
3 votes
0 answers
509 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 ...
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3 votes
1 answer
316 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 $...
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  • 43
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, ...
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