**Q1:** My first question is about defining the category $\text{IndCoh}(S)$ for a $DG$ scheme $S$. So in page $18$ of [this paper](http://www.math.harvard.edu/~gaitsgde/GL/IndCohtext.pdf), 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](http://www.math.harvard.edu/~gaitsgde/GL/singsupp.pdf) . 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?



Defining ind-coherent sheaves and their singular support
(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.


