Timeline for intuition for hochschild homology
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 12, 2018 at 15:22 | history | edited | John Pardon | CC BY-SA 4.0 |
added caveat relevant if k is not a field
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| Jan 2, 2018 at 12:07 | vote | accept | Zbigniew | ||
| Dec 30, 2017 at 2:45 | comment | added | Pedro | This boils down to the fact that if $A$ is $k$-projective, then the bar resolution is a honest projective resolution of $A$ as a bimodule, and so doing $M\otimes B(A,A)\otimes N$ works to compute Tor. There is a silly cancellation (two $A$s for the two tensors over $A$) that gives the complex in this post. | |
| Dec 30, 2017 at 2:45 | comment | added | Pedro | @QiaochuYuan One usually defines Hochschild (co)homology using Hochschild's complex, which ends up computing a relative Tor (or Ext). When the algebra is flat (or projective) over the base, you get usual Tor (or Ext) --no hypothesis should be necessary on the modules. I guess it merely depends on what you want to call HH, but I would follow Hochschild's original definition. (...) | |
| Dec 30, 2017 at 1:11 | comment | added | Qiaochu Yuan | John is omitting the differential. Also I think this complex only computes the right thing if $M, N, A$ are all flat over the base commutative ring. | |
| Dec 30, 2017 at 1:08 | comment | added | darij grinberg | Wait. Are you saying the Tors are literally the homology of a chain complex of tensor products? Of actual tensor products over $\mathbb {Z} $ or whatever base ring we are working over? Or is the tensor product inside the direct sum some sort of shorthand or metaphor? | |
| Dec 30, 2017 at 0:24 | comment | added | Qiaochu Yuan | It means the derived tensor product: stacks.math.columbia.edu/tag/09LP | |
| Dec 27, 2017 at 10:46 | comment | added | Zbigniew | Thanks for this nicely explanation but I have some confusion: What did you mean by $B\otimes_A^{\mathbb L}{}$ I do not meet this notation before. | |
| Dec 27, 2017 at 10:40 | vote | accept | Zbigniew | ||
| Dec 31, 2017 at 15:08 | |||||
| Dec 26, 2017 at 16:04 | history | answered | John Pardon | CC BY-SA 3.0 |