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  "body": "\n## Summary\n\nTilting theory compares the module categories of two algebras $A$ and $B = \\End_A(T)$ by means of a tilting module $T$.\nIt generalizes [[Morita equivalence|RD5WKA1VK7]], which is the case where $T$ is a progenerator.\nThe two module categories are not equivalent in general, but they share large pieces, and by Happel's theorem their bounded [[derived categories|5ST7NVJFSM]] are equivalent.\nIt grew out of the [[reflection functors|MG4RNMA8AR]] of Bernstein, Gelfand and Ponomarev.\n\n## Explanations\n\n### L1 human\n\nTwo algebras can have different modules and still be closely related.\nTilting theory gives a way to pass from one to the other through a special module $T$, the tilting module.\nPart of the modules of the first algebra correspond to part of the modules of the second, and the rest is \"tilted\" across.\nAt the level of derived categories, which keep track of complexes of modules, the two algebras look exactly the same.\nThis lets one transfer hard questions from a complicated algebra to a simpler one.\n",
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