'homotopy' between morphisms of a 'topological' or 'algebraic' category (Stanley-Reisner ring)

Question

In what follows, a homotopy is a congruence $\simeq$ on a given category. Given such a homotopy, objects $X$ and $Y$ of the given category are homotopy equivalent when there exist morphisms $f\!:X\rightarrow Y$ and $g\!:Y\rightarrow X$ with $gf\simeq id_X$ and $fg\simeq id_Y$.

Q1: Given abstract simplicial complexes $\Delta,\Delta'$ and simplicial maps $f,g\!:\Delta\rightarrow\Delta'$, is there some (nontrivial yet meaningful) notion of a homotopy $h$ or homotopicness $\simeq$ between $f$ and $g$?

Of course this should be a mathematical object with 'finite information' (the usual continuous homotopies between continuous maps $f,g\!: |\Delta|\rightarrow|\Delta'|$ in general cannot be specified by finite amount of data). I'm asking for a discrete analogue of a continuous homotopy between continuous maps. A desired property would also be that if $f\simeq g$, then $H_n(f)=H_n(g)$.

Q2: Let $K$ be a field and denote $K[x_1,\ldots,x_n|p_1,\ldots,p_k]:=K[x_1,\ldots,x_n]/(p_1,\ldots,p_k)$ for given polynomials $p_1,\ldots,p_k\in K[x_1,\ldots,x_n]$. Given morphisms of $K$-algebras $f,g\!: K[x_1,\ldots,x_m|p_1,\ldots,p_k]\rightarrow K[y_1,\ldots,y_n|q_1,\ldots,q_l]$, is there some (nontrivial yet meaningful) notion of a homotopy $h$ or homotopicness $\simeq$ between $f$ and $g$? What about the special case when $K=\mathbb{Z}_2,\mathbb{Q},\mathbb{R},\mathbb{C}$?

Q3: If the answer to Q2 is no, then is there any meaningful notion of a homotopy equivalence between $K[x_1,\ldots,x_m|p_1,\ldots,p_k]$ and $K[y_1,\ldots,y_n|q_1,\ldots,q_l]$? Such a homotopy equivalence is required to satisfy that if $\Delta$ and $\Delta'$ are homotopy equivalent simplicial complexes, then their Stanley-Reisner rings $K[\Delta]$ and $K[\Delta']$ are also homotopy equivalent. What about the special case when $K=\mathbb{Z}_2,\mathbb{Q},\mathbb{R},\mathbb{C}$? What about the special case when $p_i,q_j$ are square-free monomials?

Basically I'd like to know how the homotopy equivalence of simplicial complexes $\Delta,\Delta'$ affects the rings $K[\Delta],K[\Delta']$. For example, $(n\text{-simplex})\mathbb{B}^n\!\simeq\!\mathbb{B}^0\simeq \mathbb{I}_n(n\text{-path})$, so what algebraic property do $K[\mathbb{B}^n]=K[x_0,\ldots,x_n]$ and $K[\mathbb{B}^0]=K[x_0]$ and $K[\mathbb{I}_n]=K[x_0,\ldots,x_n|x_ix_j; i\!-\!j\!\geq\!2]$ share, whilst $\mathbb{B}^n\!\not\simeq\!\mathbb{S}^n$, so what algebraic property don't $K[\mathbb{B}^n]=K[x_0,\ldots,x_n]$ and $K[\mathbb{S}^n]=K[x_0,\ldots,x_n|x_0\cdots x_n]$ share?

Answer 1

Q1: yes, but I don't know a reference for this. Two simplicial maps $h, k : K \to L$ are contiguous if for each simplex $\sigma$ of $K$, there is a simplex $\sigma'$ of $L$ such that both $h(\sigma)$ and $k(\sigma)$ are faces of $\sigma'$. The following hold (I was given these as exercises in an algebraic topology course):

• Any two simplicial approximations of a continuous map $f : |K| \to |L|$ (where $|K|$ denotes the geometric realization) are contiguous.
• Any two contiguous maps induce homotopic maps $|K| \to |L|$.
• If $f, g : |K| \to |L|$ are homotopic, then there is a barycentric subdivision $K^{(N)}$ of $K$ and a sequence of simplicial maps $h_1, ... h_n : K^{(N)} \to L$ such that $h_1$ is a simplicial approximation to $f$, $h_n$ is a simplicial approximation to $g$, and each pair $(h_i, h_{i+1})$ is contiguous.

So the third condition can be taken as a combinatorial definition of homotopy. There is a better definition if we work with simplicial sets instead of simplicial complexes; see simplicial homotopy.

Q2: see $\mathbb{A}^1$-homotopy theory.

Q3: my impression is that simplices enter homotopy theory for a reason unrelated to the reason that simplices enter combinatorial commutative algebra. I could be wrong, though.

Answer 2

For Q2 I've heard the following definition (some years ago, so I might be wrong). The idea is to replace the unit interval $[0,1]$ in the usual definition of homotopy by the coordinate ring of the affine line, i.e. $\mathcal{O}(\mathbb{A}^1) = \mathbb{Z}[x]$. Thus, two ring homomorphisms $f,g : R \to S$ are called homotopic if there is a ring homomorphism $h : R \to S[x]$ such that $h(r)(0)=f(r)$ and $h(r)(1)=g(r)$.

Note that this relation is compatible with composition of ring homomorphisms (in the sense $f \simeq g \Rightarrow fs \simeq gs$ and $tf \simeq tg$), reflexive and symmetric (use $S[x] \to S[x], x \mapsto 1-x$). But it is not transitive, see below. In order to get a good notion of homotopy, one has to take the transitive closure. This is then a congruence relation on the category of rings.

The same definition applies to the category of $k$-algebras for an arbitrary (commutative) ring $k$. More generally, one may call two morphisms of $k$-schemes $f,g : X \to Y$ homotopic if there is a morphism of $k$-schemes $h : X \times \mathbb{A}^1 \to Y$ such that $h(-,0)=f$ and $h(-,1)=g$. Also note that using $C([0,1])$ instead of $\mathcal{O}(\mathbb{A}^1)$, one gets the well-known notion of homotopy between homomorphisms of Banach algebras (see e.g. here).

Transitivity. If $S$ is a commutative ring, the following are equivalent:

1. Homotopy is a transitive relation on $\hom(R,S)$ for every $R$.
2. There is a ring homomorphism $H : S[x] \times_{e_1,S,e_0} S[x] \to S[x]$ such that $e_0 H = e_0 p_1$ and $e_1 H = e_1 p_2$ (here, $e_i : S[x] \to S$ are given by $s \mapsto s$ and $x \mapsto i$, therefore $S[x] \times_{e_1,S,e_0} S[x] = \{(p,q) \in S[x] \times S[x] : p(1)=q(0)\}$).
3. There are two polynomials $p,q \in S[x]$ such that $pq=0$ and $p(0)=-1$, $p(1)=0$ and $q(0)=0$, $q(1)=1$.

In particular, this fails when $S$ is an integral domain.

Proof:

$1. \Rightarrow 2.$ Let $R=S[x] \times_{e_1,S,e_0} S[x]$. Then $e_0 p_1 \simeq e_1 p_1 = e_0 p_2 \simeq e_1 p_2$, hence also $e_0 p_1 \simeq e_1 p_2$, which is exactly 2.

$2. \Rightarrow 1.$ Let $f,g,h \in \hom(R,S)$ with $f \simeq g \simeq h$, say via $H_1,H_2 \in \hom(R,S[x])$ with $e_0 H_1 = f$, $e_1 H_1 = g$ and $e_0 H_2 = g$, $e_1 H_2 = h$. Because of $e_1 H_1 = e_0 H_2$ there is a unique $T \in \hom(R,S[x] \times_{e_1,S,e_0} S[x])$ with $p_1 T = H_1$ and $p_2 T = H_2$. Then $H_3 := H T \in \hom(R,S[x])$ satisfies $e_0 H_3 = f$ and $e_1 H_3 = h$ and therefore witnesses $f \simeq h$.

$2. \Leftrightarrow 3.$ There is an isomorphism of $S$-algebras $S[u,v]/(uv) \to S[x] \times_{e_1,S,e_0} S[x]$ given by $u \mapsto (x-1,0)$ and $v \mapsto (0,x)$. You can check this directly, or derive it from the algebro-geometric observation that the union of the two coordinate axes is a pushout or gluing of two affine lines at two points. Hence. $2.$ is equivalent to the existence of a ring homomorphism $H : S[u,v]/(uv) \to S[x]$ such that $e_0(H(u))=e_0(x-1)=-1$, $e_1(H(u))=e_1(0)=0$ and $e_0(H(v))=e_0(0)=0$, $e_1(H(v))=e_1(x)=1$. Using the universal properties of quotients and polynomial rings, we see that this is equivalent to the existence of a ring homomorphism $S \to S[x]$ (but this exists anyway) and two polynomials $p=H(u), q=H(v)$ with the properties as in $3.$ $\square$