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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, $A$ a finite dimensional algebra over $k$, and $n$ is the number of non-isomorphic simple right modules over $A$.

A right $A$-module $X$ is called $partial$ $tilting$ if

  • $pd_A(X)\leq 1$, and
  • $Ext{^1}(X,X)=0$.

Moreover, a partial tilting module $T$ is $tilting$ if it also satisfies one (therefore both) of the followings:

  • $T$ has exactly $n$ non-isomorphic direct summands;
  • There exists an exact sequence $0 \rightarrow A \rightarrow T_0 \rightarrow T_1 \rightarrow 0$, with $T_0$ and $T_1$ in $add(T)$.

In the paper "Tilted Algebras" by K. Bongartz (1981), it is shown that every partial tilting $X$ in $mod-A$ could be completed to a tilting module $T=X\oplus Z$, where $Z$ is given as the middle term of a short exact sequence in $Ext^1(X^n, A)$. This completion however, is not necessarily unique.

In the paper "Almost complete tilting modules", by D. Happel and L. Unger (1989), it is shown that for each module in this special class (i.e., partial tilting modules with exactly $n-1$ non-isomorphic summands), the completion to a tilting could be done in at most two different ways and also under which condition it would be unique.

With that in mind, my questions are the followings:

  • Question 1: Although the completion of a partial tilting to a tilting is not unique, is there any canonical choice in the set of all possible complements?"
  • Question 2: Given an arbitrary partial tilting module $X$ over $A$, is it always possible to find an algebra $B$ and a functor $\mathcal F: mod(A) \rightarrow mod(B)$, such that $\mathcal F(X)$ is (almost complete) tilting over $B$?
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  • $\begingroup$ For question 2 you can chose $B={\rm{End}}_A(T)$ and $\mathcal F ={\rm{Hom}}(T,-)$. Then $\mathcal F(T)=B$ is a projective $B$-module. $\endgroup$ Sep 4, 2016 at 10:09

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