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Q1: My first question is about defining the category $\text{IndCoh}(S)$ for a $DG$ scheme $S$. So in page $18$ of this paper, they are defined as being the ind-completion of the category $\text{Coh}(S)$. Here $\text{Coh}(S) \subset \text{QCoh}(S)$ is defined as the full subcategory of objects with bounded cohomological amplitude and coherent cohomologies. What is the ind-completion?

Q2: My second question is about the singular support of an ind-coherent sheaf. For a $DG$ scheme $Z$, the scheme of singularities $\text{Sing}(Z)$ is defined in page $19$ of this paper . In page $41$ of the same paper, given $\mathcal{F} \in \text{IndCoh}(Z)$, the singular support of $\mathcal{F}$ is defined as $\text{supp}_{A}(\mathcal{F})$

with $A=\Gamma(\text{Sing}(Z),\mathcal{O}_{\text{Sing}(Z)})$; . How should I correctly interpret and understand this definition of singular support?

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There is a lot of information and references about Ind-categories here: for example. Perhaps you could make your questions a bit more specific? – Sam Gunningham Jun 25 '12 at 18:27
Thanks! The information there to some extent answers my first question. I'm still unsure about my second question, though. – Vinoth Jun 26 '12 at 8:33
up vote 7 down vote accepted

(Hopefully t3uji, tony pantev or Greg Stevenson will chime in with a more authoritative answer, but in the meanwhile..)

The notion of singular support of a coherent sheaf is an analog of the notion of singular support of a constructible sheaf or D-module. Let's quickly recall the latter: given a sheaf we can measure its failure to be locally constant at a particular point in a particular codirection. Namely given a covector at a point, i.e., a hyperplane in the tangent space, you ask if the sheaf behaves locally constantly moving off this hypersurface (this is a generalization of the Cauchy-Kovalevski theorem in PDE). You can measure this by taking a local function with the given covector as its differential at our point and calculating relative cohomologies (Morse groups) of its level sets near this point and "seeing if anything happens". The singular support (or microlocal support) is the collection of all covectors where our sheaf is not locally constant - ie all points and directions where "something interesting happens".

A very nice recent idea of several people (Isik, Arinkin-Gaitsgory, and others following on Orlov's work on categories of singularities --- someone who knows the history better please correct) is that one can do a very similar operation for coherent sheaves.

Recall that on a smooth variety any coherent sheaf (or bounded coherent complex) is quasiisomorphic to a perfect complex (bounded complex of vector bundles). This fails precisely at singular points of varieites by a theorem of Serre. Orlov introduced the category of singularities of a variety as the quotient of the bounded derived category by perfect complexes --- i.e., a measure of "how and where" the variety is singular (this category of singularities is supported at the singular locus). This is intimately related to the theory of matrix factorizations.

The new notion of singular support is an "individual" version of this construction for lci schemes (schemes with cotangent complex in degrees -1,0) - i.e. one looks at a specific coherent sheaf (or bounded complex) and attaches to it its "microlocal support" --- naively speaking, the collection of points and (degree -1) codirections where the sheaf fails to be perfect (see section 0.3.7 for a way to make this intuition precise, using a description of our scheme as a local complete intersection). One can also define the singular support as an honest support for the sheaf, when considered as a module for the Hochschild cohomology sheaf of the variety (self-Ext of the identity). One can also think roughly about representing covectors as differentials of functions, and then passing to categories of matrix factorizations of this function and seeing whether our sheaf survives - again, whether the sheaf is "interesting" at a given point and codirection.

Edit: Let me add a little about the role of Hochschild cohomology. By (one) definition the Hochschild cohomology of a scheme is the self-Ext of the identity functor of the (dg) derived category. In other words, the Hochschild cohomology tautologically acts by endomorphisms of every sheaf, in a way compatible with all morphisms. One can say more -- the Hochschild cohomology can be identified with the enveloping algebra of a Lie algebra structure on the shifted tangent complex $T[-1]$, which acts on every object via the construction of Atiyah classes (see Kapranov's paper on Rozansky-Witten theory for example, Markarian's preprint on HH and many more recent papers, the latest word maybe Calaque-van den Bergh). This can be nicely interpreted in terms of derived loop spaces -- $T[-1]$ is the Lie algebra of the free loop space, and Hochschild cohomology is its "group algebra"..

In our case we are interested in an lci scheme, and in particular the action of the top (+1) piece of the tangent complex, which after shift to $T[-1]$ lives in degree 2, or after taking enveloping algebra lives in even degree Hochschild cohomology. The singular support is then the support of the action of this commutative algebra (piece of even $HH^*$) on the sheaf. In other words, any sheaf on $X$ tautologically has a bigger action of a not quite commutative algebra, the Hochschild cohomology, but there's a commutative piece we can single out in there in the lci case and then take usual support.

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Thanks! Why is the sheaf $F$ a module over Hochschild cohomology? I'm still not quite sure how $supp_A(F)$ is defined; I see that pg 45 describes a map $A→HH^{\bullet \text{even}}(Z)$. – Vinoth Jul 1 '12 at 10:24
Hi - tried to clarify a little, though maybe still cryptic - I think though the definition they ascribe to Drinfeld in section 0.3.7 is the most intuitive and certainly most "microlocal" one.. – David Ben-Zvi Jul 1 '12 at 18:39
Thanks. I'll try to digest this. – Vinoth Jul 3 '12 at 7:04

About the Ind completions of a category $\mathcal{C}$ :

Let $\mathcal{C}^>:=Fun(\mathcal{C}^{op}\to Set)$ the presheaves category.

we have the yoneda full embedding $Y: \mathcal{C}\to \mathcal{C}^>: X\mapsto h_X$.

For $P\in \mathcal{C}^>$ we indicate by $\int P$ the the comma category $\mathcal{C}\downarrow P$, with objects the couples $(X,x)$ with $X\in \mathcal{C},\ x\in P(X)$ and morphisms $f: (X, x)\to (Y, y)$ the $f\in \mathcal{C}(X, Y)$ such that $P(f)(y)=x$, we have the natural functor $\pi_P: \int P\to \mathcal{C}: (X, x)\mapsto X$ and is a foundamental theorem that $P\cong colim_{(X, x)\in \int P}\ h_X= colim\ Y\circ \pi $

we call $P$ flat if there is a final functor $\phi: I\to \int P$ (i.e. for any $\xi\in \int P$ the comma category $\xi\downarrow \phi$ is nonempty and connected) with $I$ a small filtered category (i.e. for any $\forall i, j\in I: \exists k\in I:\ I(i, k)\cup I(j, k)\neq\emptyset$ and for $\phi, \psi: I\to j$ exist $\theta: j\to k$ with $\theta\circ\phi=\theta\circ\psi$).

A flat functor $P$ preserve finite limit:

we can write $P\cong colim_{i\in I} h_{X_i}$ then

$P(Y)=colim_{i\in I}\mathcal{C}(Y, X_i)$ then if $Y= colim_{j\in J}$ then

$lim_{j\in J}P(Y_j)\cong lim_{j\in J}\ colim_{i\in I}\mathcal{C}(Y_j, X_i)\cong$ $\ colim_{i\in I}\ lim_{j\in J}\mathcal{C}(Y_j, X_i)\cong $ $colim_{i\in I}\mathcal{C}(colim_{j\in J} Y_j, X_i)\cong $ $colim_{i\in I}\mathcal{C}(Y, X_i)\cong P(Y)$.

Let $P_1(\mathcal{C})\subset\mathcal{C}^>$ the full subcategory of the flat presheaves. The category $P_1(\mathcal{C})$ has the small filtrant colimits (no too easy to prove).

Of course any representable is flat, then we have the Yoneda immersion $Y: \mathcal{C}\to P_1(\mathcal{C})$ A flat presheaf $P$ preserve finite colimits:

let $J$ a finite diagram, for any $P\in P_1(\mathcal{C})$ we have:

$(Y(colim_{j\in J}Y_j), P)\cong P(colim_{j\in J}Y_j)\cong$ $ lim_{j\in J}P(Y_j)\cong lim_{j\in J}(h_{Y_j}, P)\cong $ $(colim_{j\in J} h_{Y_j}, P)$

from Yoneda lemma follow that $colim_{j\in J}Y_j\cong colim_{j\in J} h_{Y_j}$ in $P_1(\mathcal{C})$.

We have the following universal property: let $F: \mathcal{C}\to \mathcal{A}$ where $\mathcal{A}$ has the filtered little colimits, then exist unique (to isomorphisms) the $F_1: P_1(\mathcal{C})\to \mathcal{A} $ preserving the small filtered colimits, with $F=F_1\circ Y$ (let $F_1:=Lan_{Y} F$ the left Kan extension of $F$ respect to $Y$).

Now, gived a $P\in P_1(\mathcal{C})$ we represent it as a filtered system $(X_i)_{i\in I}$

(with $P\cong colim_{i\in I}h_{X_i}$)

how we could represent a morfisms $\Phi: P\to Q$ (let fixed a representation $(Y_j)_{j\in J}$ of $Q$) ?

for $i\in I$ considering $h_{X_i}\to P\to Q\cong colim_{j\in J}h_{Y_j}$

by yoneda lemma, and the definition of colimit in $Set$, this has a factorization on some $f(i)\in J$ i.e. like some $\Phi_i: h_{X_i}\to h_{Y_{f(i)}}\to Q$

Then choosing for any $i\in I$ as above we have a map $f: I\to J$ and a famili o morphisms $(\Phi_i: X_i\to Y_{f(i)})_{i\in I}$, with the following property:

gived a $\phi: i\to i'$ we have a $k\in J$ and two morphisms $\psi: \Phi(i)\to k,\ \psi': \Phi(i')\to k$ such that $\psi'\circ\Phi_{i'}\circ X_\phi= \psi \circ \Phi_i$ (where $X_\phi: X_i\to X_{i'}$ is the canonical morphism).

Vice versa such data detecting a unique morphisms $\Phi: P\to Q$ that has such data as representation as above.

If $\Phi: P\to Q$ and $\Psi: Q\to R$ have the representation $(f, \phi)$ and $(g, \psi)$ (and $R\cong colim_{k\in K}h_{Z_k}$ then the composition $\Psi\circ \Phi$ is $(g\circ f, \psi\ast_{f, g} \phi)$ where $(\psi\ast_{f, g} \phi)_i: X_i\to Y_{f(i)}\to Z_{gf(i)}$.

The idetity $1_P: P\to P$ as the representation $(1_I, (1_: i\to i)_{i\in I}$.

Then we get a new category $Ind(\mathcal{C})$ equivalent to $P_1(\mathcal{C})$ with object the filtered diagrams in $\mathcal{C}$ like $(X_i)_{i\in I}$ and for morphism the representation $(f, \Phi)$ as above, with the above compositions and identities.

For the universal property above, follow a natural equivalence $P_1(P_1(\mathcal{C}))\cong P_1(\mathcal{C})$ then we have a natural equivalence $Ind(Ind(\mathcal{C}))\cong Ind(\mathcal{C})$.

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