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Katzarkov-Kontsevich-Pantev define a smooth dg ($\mathbb{C}$-)algebra $A$ to be a dg algebra which is a perfect $A \otimes A^{op}$-module. They say that an $A$-module $M$ is perfect if the functor $Hom(M,-)$ from ($A$-modules) to ($\mathbb{C}$-modules) preserves small homotopy colimits. (Definition 2.23 of KKP) *

Kontsevich-Soibelman define smooth dg (or $A_\infty$) algebras in the same way; but they define perfect $A$-modules to be ones which are quasi-isomorphic to a direct summand of an extension of a sequence of modules each of which is quasi-isomorphic to a shift of $A$. (Definition 8.1.1 of KS) **

First question: I have trouble wrapping my head around what it means for a functor to preserve small homotopy colimits. I don't understand the definition from Kontsevich-Soibelman, either. What are the "moral" meanings of these conditions?

Second question: Presumably these two conditions are equivalent. How do you prove this?

Third question: If $A$ is a commutative $(\mathbb{C}$-)algebra, then presumably smoothness of $A$ as a dg algebra is equivalent to smoothness of $\operatorname{Spec} A$ as a ($\mathbb{C}$-)scheme. (Maybe you need $A$ to be finite type?) How do you prove this? Furthermore, Example 8.1.4 of KS gives some more examples of dg algebras that are allegedly smooth (for instance free algebras $k\langle x_1, \dots, x_n \rangle$). Again, I don't know how to prove that these are in fact smooth, and Kontsevich-Soibelman don't seem to provide proofs; or maybe I overlooked something.

*The answers below indicate that this is the standard definition of compact (also known as small) module.

**The answers below indicate that this is the standard definition of perfect module.

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A good reference is D. Kaledin, Seoul lectures and to smaller extent his Tokyo lectures "Homological methods in non-commutative geometry"_, pdf, TeX; the Seoul lectures are based much on the sharp new insights from Toen's paper arXiv:math/0609762 which are in turn partly motivated by some conjectures of Kaledin. –  Zoran Skoda Jul 23 '10 at 11:02
Thanks, Zoran! I was aware of Kaledin's Tokyo lecture notes, but not his Seoul lecture notes. –  Kevin H. Lin Jul 23 '10 at 21:44
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2 Answers 2

up vote 23 down vote accepted

The condition of Hom(M,-) being a continuous functor, i.e. preserving (small homotopy) colimits is equivalent (in the present stable setting) to the maybe more concrete condition of preserving arbitrary direct sums. The issue is not finite direct sums, that's automatic (since the derived Hom is an exact functor, it automatically commutes with FINITE colimits). This is better known as a compact object of the dg category (see n-lab for lots of discussion). This is a strong finiteness condition on a module. Its importance in algebraic geometry was realized by Thomason (in a dream form of his friend Trobaugh), who proved some amazing fundamental results about the behavior (and abundance) of compact/perfect objects on schemes.

One can think of Hom(M,-) as the functional on the category defined by M (Hom being a kind of inner product), and this is saying the functional is "continuous".. There's a discussion of the different common notions of finiteness for modules and their properties (including the foundational results of Thomason and Neeman) in Section 3.1 of this paper (sorry for the self-referencing -- none of this is in the least original, but it's a convenient discussion with plentiful references). This includes an explanation of why compactness is the same as being in the thick subcategory category generated by the free module, which is your definition of perfect from K-S (ie built out of the free module by taking sums, cones, summands), and also the same as being a dualizable object with respect to tensor product in case your category is the derived category of quasicoherent sheaves (ie this is a "commutative" notion not applicable in the NC setting you're discussing).

By the way the definition from K-S is the standard definition of perfection, that from K-K-P is the standard definition of compactness.. there are settings where the two notions don't agree (for example for sheaves on the classifying space of a finite group in a modular characteristic, or on BS^1) [hence our terminology of "perfect stack" in the paper with Francis and Nadler, which is a stack where these notions for sheaves agree and these nice finite objects generate].

The fact that perfection of the diagonal (ie A as an A-bimodule) is equivalent to smoothness in the case of a scheme is a reformulation of the homological criterion of Serre for smoothness of a point in terms of finite Tor amplitude of the skyscraper at the point - ie we're saying all points are smooth all at once.

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Thank you, Prof. Ben-Zvi! I think I should be spending more time over at the n-Lab... –  Kevin H. Lin Apr 8 '10 at 16:50
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As David explained above we have two notions: perfect objects and compact(small) objects. They coincide for derived categories of DG modules over a DG algebra and, more generaly, over a DG category. Proof can be found, for example, in Bernhard Keller paper link text Section 5.3.

For answer on the third question you can look paper of Rouquier link text and Lemma 7.2 there which says that if $A$ is a finite dimensional $k$-algebra or a commutative $k$-algebra essentially of finite type such that for given $V$ a simple A-module, $Z(End_A(V ))$ is a separable extension of $k$. Then, $pdim_{A^{en}} A = gldim A$.

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Thanks a lot for the references! –  Kevin H. Lin Apr 8 '10 at 16:49
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