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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $A$ be a finite-dimensional [[associative algebra|3CTW1P4HBP]] over $k$.\nA finitely generated left $A$-module $T$ is a *tilting module* if:\n\n1. the [[projective dimension|MR9QCH1PNA]] of $T$ is at most $1$;\n2. $\\operatorname{Ext}^1_A(T, T) = 0$, where $\\operatorname{Ext}$ is the [[Ext functor|4P7GBWJ4RR]];\n3. the number of isomorphism classes of [[indecomposable|E2P29GCD9T]] direct summands of $T$ equals the number of isomorphism classes of [[simple|G00C22AEDH]] $A$-modules.\n\n*Tilting theory* studies the relation between $A$ and the [[endomorphism ring|AWSBZWWMG9]] $B = \\End_A(T)$ of a tilting module $T$, through the functors $\\Hom_A(T, \\cdot)$ and $\\operatorname{Ext}^1_A(T, \\cdot)$.\n\n## Remarks\n\nExample: $T = A$ is a tilting module, since it is projective and its indecomposable summands are the indecomposable projectives, one for each simple module.\nExample: let $A$ be the path algebra of $1 \\to 2$, with $S_2 = P_2$, $P_1$ and $S_1$ as indecomposables, and let $T = P_1 \\oplus S_1$.\nThe projective resolution $0 \\to P_2 \\to P_1 \\to S_1 \\to 0$ gives clause 1.\nClause 2 holds since $P_1$ is projective and injective and $\\operatorname{Ext}^1_A(S_1, S_1) = 0$, and clause 3 holds with $2$ summands.\nNon-example: $T = S_1 \\oplus S_2$ fails clause 2, since the non-split sequence $0 \\to S_2 \\to P_1 \\to S_1 \\to 0$ gives $\\operatorname{Ext}^1_A(S_1, S_2) \\ne 0$.\nHappel proved that for a tilting module $T$ the bounded derived categories of $\\operatorname{mod} A$ and $\\operatorname{mod} B$ are equivalent as triangulated categories.\n",
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