hcju/stacksqa huggingface dataset:
The existence of the functor $Lg_!$ and adjointness to $g^*$ is Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-existence-derived-lower-shriek}. (For the case of abelian sheaves use the constant sheaf $\mathbf{Z}$ as the structure sheaves.) Moreover, it is computed on a complex $\mathcal{H}^\bullet$ by taking a suitable left resolution $\mathcal{K}^\bullet \to \mathcal{H}^\bullet$ and applying the functor $g_!$ to $\mathcal{K}^\bullet$. Since $g^{-1}g_!\mathcal{K}^\bullet = \mathcal{K}^\bullet$ by Cohomology of Stacks, Lemmas \ref{stacks-cohomology-lemma-lisse-etale-modules} and \ref{stacks-cohomology-lemma-lisse-etale} we see that the final assertion holds in each case.
Pick a scheme $U$ and a surjective smooth morphism $x : U \to \mathcal{X}$. By Modules on Sites, Definition \ref{sites-modules-definition-site-local} there exists an \'etale (resp.\ fppf) covering $\{U_i \to U\}_{i \in I}$ such that each pullback $f_i^{-1}\mathcal{H}$ has a global presentation (see Modules on Sites, Definition \ref{sites-modules-definition-global}). Here $f_i : U_i \to \mathcal{X}$ is the composition $U_i \to U \to \mathcal{X}$ which is a morphism of algebraic stacks. (Recall that the pullback ``is'' the restriction to $\mathcal{X}/f_i$, see Sheaves on Stacks, Definition \ref{stacks-sheaves-definition-pullback} and the discussion following.) After refining the covering we may assume each $U_i$ is an affine scheme. Since each $f_i$ is smooth (resp.\ flat) by Lemma \ref{lemma-lisse-etale-functorial-derived} we see that $f_i^{-1}Lg_!\mathcal{H} = Lg_{i, !}(f'_i)^{-1}\mathcal{H}$. Using Cohomology of Stacks, Lemma \ref{stacks-cohomology-lemma-check-lqc-fbc-on-covering} we reduce the statement of the lemma to the case where $\mathcal{H}$ has a global presentation and where $\mathcal{X} = (\Sch/X)_{fppf}$ for some affine scheme $X = \Spec(A)$. \medskip\noindent Say our presentation looks like $$ \bigoplus\nolimits_{j \in J} \mathcal{O} \longrightarrow \bigoplus\nolimits_{i \in I} \mathcal{O} \longrightarrow \mathcal{H} \longrightarrow 0 $$ where $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{lisse,\etale}}$ (resp.\ $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{flat,fppf}}$). Note that the site $\mathcal{X}_{lisse,\etale}$ (resp.\ $\mathcal{X}_{flat,fppf}$) has a final object, namely $X/X$ which is quasi-compact (see Cohomology on Sites, Section \ref{sites-cohomology-section-limits}). Hence we have $$ \Gamma(\bigoplus\nolimits_{i \in I} \mathcal{O}) = \bigoplus\nolimits_{i \in I} A $$ by Sites, Lemma \ref{sites-lemma-directed-colimits-sections}. Hence the map in the presentation corresponds to a similar presentation $$ \bigoplus\nolimits_{j \in J} A \longrightarrow \bigoplus\nolimits_{i \in I} A \longrightarrow M \longrightarrow 0 $$ of an $A$-module $M$. Moreover, $\mathcal{H}$ is equal to the restriction to the lisse-\'etale (resp.\ flat-fppf) site of the quasi-coherent sheaf $M^a$ associated to $M$. Choose a resolution $$ \ldots \to F_2 \to F_1 \to F_0 \to M \to 0 $$ by free $A$-modules. The complex $$ \ldots \mathcal{O} \otimes_A F_2 \to \mathcal{O} \otimes_A F_1 \to \mathcal{O} \otimes_A F_0 \to \mathcal{H} \to 0 $$ is a resolution of $\mathcal{H}$ by free $\mathcal{O}$-modules because for each object $U/X$ of $\mathcal{X}_{lisse,\etale}$ (resp.\ $\mathcal{X}_{flat,fppf}$) the structure morphism $U \to X$ is flat. Hence by construction the value of $Lg_!\mathcal{H}$ is $$ \ldots \to \mathcal{O}_\mathcal{X} \otimes_A F_2 \to \mathcal{O}_\mathcal{X} \otimes_A F_1 \to \mathcal{O}_\mathcal{X} \otimes_A F_0 \to 0 \to \ldots $$ Since this is a complex of quasi-coherent modules on $\mathcal{X}_\etale$ (resp.\ $\mathcal{X}_{fppf}$) it follows from Cohomology of Stacks, Proposition \ref{stacks-cohomology-proposition-lcq-flat-base-change} that $H^p(Lg_!\mathcal{H})$ is quasi-coherent.
Since $\epsilon^*$ is exact it is clear that we obtain a diagram as in the statement of the lemma. We will show the middle vertical arrow is an equivalence by applying Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-compare-topologies-derived-adequate-modules} to the following situation: $\mathcal{C} = \mathcal{X}$, $\tau = fppf$, $\tau' = \etale$, $\mathcal{O} = \mathcal{O}_\mathcal{X}$, $\mathcal{A} = \mathcal{M}_\mathcal{X}$, and $\mathcal{B}$ is the set of objects of $\mathcal{X}$ lying over affine schemes. To see the lemma applies we have to check conditions (1), (2), (3), (4). Conditions (1) and (2) are clear from the discussion above (explicitly this follows from Cohomology of Stacks, Proposition \ref{stacks-cohomology-proposition-lcq-flat-base-change}). Condition (3) holds because every scheme has a Zariski open covering by affines. Condition (4) follows from Descent, Lemma \ref{descent-lemma-quasi-coherent-and-flat-base-change}. \medskip\noindent We omit the verification that the equivalence of categories $\epsilon^* :  D_{\mathcal{M}_\mathcal{X}}( \mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) \to D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ induces an equivalence of the subcategories of complexes with parasitic cohomology sheaves.
Consider the morphism of ringed topoi $g : \Sh(\mathcal{X}_{flat, fppf}) \longrightarrow \Sh(\mathcal{X}_{fppf})$. Set $E' = Lg_!g^{-1}E$ and let $P$ be the cone on the adjunction map $E' \to E$. Since $g^{-1}E' \to g^{-1}E$ is an isomorphism we see that $P$ is an object of $D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ by Lemma \ref{lemma-derived-quasi-coherent} (2)(b). Finally, $\Hom(E', P') = \Hom(Lg_!g^{-1}E, P') = \Hom(g^{-1}E, g^{-1}P') = 0$ as $g^{-1}P' = 0$. \medskip\noindent Uniqueness. Suppose that $E'' \to E \to P'$ is a second distinguished triangle as in the statement of the lemma. Since $\Hom(E', P') = 0$ the morphism $E' \to E$ factors as $E' \to E'' \to E$, see Derived Categories, Lemma \ref{derived-lemma-representable-homological}. Similarly, the morphism $E'' \to E$ factors as $E'' \to E' \to E$. Consider the composition $\varphi : E' \to E'$ of the maps $E' \to E''$ and $E'' \to E'$. Note that $\varphi - 1 : E' \to E'$ fits into the commutative diagram $$ \xymatrix{ E' \ar[d]^{\varphi - 1} \ar[r] & E \ar[d]^0 \\ E' \ar[r] & E } $$ hence factors through $P[-1] \to E$. Since $\Hom(E', P[-1]) = 0$ we see that $\varphi = 1$. Whence the maps $E' \to E''$ and $E'' \to E'$ are inverse to each other.
We have to show that $Rf_*E$ is an object of $D^{+}_{\mathcal{M}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y})$ for $E$ in $D^{+}_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$. This follows from Cohomology of Stacks, Proposition \ref{stacks-cohomology-proposition-lcq-flat-base-change} and the spectral sequence $R^if_*H^j(E) \Rightarrow R^{i + j}f_*E$. The case of parasitic modules works the same way using Cohomology of Stacks, Lemma \ref{stacks-cohomology-lemma-pushforward-parasitic}. The final statement is clear from the definition of $H^i$ in (\ref{equation-Hi-quasi-coherent}).
To prove the first statement, we have to show that $f^*E$ is an object of $D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ for $E$ in $D_{\mathcal{M}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y})$. Since $f^* = f^{-1}$ is exact this follows immediately from the fact that $f^*$ maps $\mathcal{M}_\mathcal{Y}$ into $\mathcal{M}_\mathcal{X}$. \medskip\noindent Set $\mathcal{D} = D_{\mathcal{M}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y})$. Let $S$ be the collection of morphisms in $\mathcal{D}$ whose cone is an object of $D_{\mathcal{P}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y})$. Set $\mathcal{D}' = D_\QCoh(\mathcal{O}_\mathcal{X})$. Set $F = q_\mathcal{X} \circ f^* : \mathcal{D} \to \mathcal{D}'$. Then $\mathcal{D}, S, \mathcal{D}', F$ are as in Derived Categories, Situation \ref{derived-situation-derived-functor} and Definition \ref{derived-definition-right-derived-functor-defined}. Let us prove that $LF(E)$ is defined for any object $E$ of $\mathcal{D}$. Namely, consider the triangle $$ E' \to E \to P \to E'[1] $$ constructed in Lemma \ref{lemma-bousfield-colocalization}. Note that $s : E' \to E$ is an element of $S$. We claim that $E'$ computes $LF$. Namely, suppose that $s' : E'' \to E$ is another element of $S$, i.e., fits into a triangle $E'' \to E \to P' \to E''[1]$ with $P'$ in $D_{\mathcal{P}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y})$. By Lemma \ref{lemma-bousfield-colocalization} (and its proof) we see that $E' \to E$ factors through $E'' \to E$. Thus we see that $E' \to E$ is cofinal in the system $S/E$. Hence it is clear that $E'$ computes $LF$. \medskip\noindent To see the final statement, write $B = q_\mathcal{X}(H)$ and $A = q_\mathcal{Y}(E)$. Choose $E' \to E$ as above. We will use on the one hand that $Rf_{\QCoh, *}(B) = q_\mathcal{Y}(Rf_*H)$ and on the other that $Lf^*_\QCoh(A) = q_\mathcal{X}(f^*E')$. \begin{align*} \Hom_{D_\QCoh(\mathcal{O}_\mathcal{X})}(Lf^*_\QCoh A, B) & =  \Hom_{D_\QCoh(\mathcal{O}_\mathcal{X})}(q_\mathcal{X}(f^*E'), q_\mathcal{X}(H)) \\ & =  \colim_{H \to H'} \Hom_{D(\mathcal{O}_\mathcal{X})}(f^*E', H') \\ & = \colim_{H \to H'} \Hom_{D(\mathcal{O}_\mathcal{Y})}(E', Rf_*H') \\ & = \Hom_{D(\mathcal{O}_\mathcal{Y})}(E', Rf_*H) \\ & = \Hom_{D_\QCoh(\mathcal{O}_\mathcal{Y})}(A, Rf_{\QCoh, *}B) \end{align*} Here the colimit is over morphisms $s : H \to H'$ in $D^+_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ whose cone $P(s)$ is an object of $D^+_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$. The first equality we've seen above. The second equality holds by construction of the Verdier quotient. The third equality holds by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-adjoint}. Since $Rf_*P(s)$ is an object of $D^+_{\mathcal{P}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y})$ by Proposition \ref{proposition-derived-direct-image-quasi-coherent} we see that $\Hom_{D(\mathcal{O}_\mathcal{Y})}(E', Rf_*P(s)) = 0$. Thus the fourth equality holds. The final equality holds by construction of $E'$.
Choose a commutative diagram $$ \xymatrix{ U \ar[d]_p \ar[r]_\alpha & V \ar[d]^q \\ X \ar[r]^a & Y } $$ with $U$, $V$ schemes and $p$, $q$ surjective and \'etale as in Spaces, Lemma \ref{spaces-lemma-lift-morphism-presentations}. By More on Morphisms, Theorem \ref{more-morphisms-theorem-openness-flatness} the set $U' = \{u \in |U| : p^*\mathcal{F}\text{ is flat over }V\text{ at }u\}$ is open in $U$. By Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-flat-module} the image of $U'$ in $|X|$ is the set of the theorem. Hence we are done because the map $|U| \to |X|$ is open, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-topology-points}.
Choose a diagram as in Lemma \ref{lemma-flat-on-fibres-at-point} part (3). It follows from the definitions that this reduces to the corresponding theorem for the morphisms of schemes $U \to V \to W$, the quasi-coherent sheaf $a^*\mathcal{F}$, and the point $u \in U$. Thus the theorem follows from the corresponding result for schemes which is More on Morphisms, Theorem \ref{more-morphisms-theorem-criterion-flatness-fibre}.
Choose a diagram as in Lemma \ref{lemma-flat-on-fibres-at-point} part (3). It follows from the definitions that this reduces to the corresponding theorem for the morphisms of schemes $U \to V \to W$, the quasi-coherent sheaf $a^*\mathcal{F}$, and the point $u \in U$. Thus the theorem follows from the corresponding result for schemes which is More on Morphisms, Theorem \ref{more-morphisms-theorem-criterion-flatness-fibre-Noetherian}.
Let $f = \pi \circ f'$ be the factorization of Lemma \ref{lemma-stein-universally-closed}. Note that besides the conclusions of Lemma \ref{lemma-stein-universally-closed} we also have that $f'$ is separated (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-compose-after-separated}) and finite type (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-permanence-finite-type}). Hence $f'$ is proper. By Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-proper-pushforward-coherent} we see that $f_*\mathcal{O}_X$ is a coherent $\mathcal{O}_Y$-module. Hence we see that $\pi$ is finite, i.e., (2) holds. \medskip\noindent This proves all but the most interesting assertion, namely that the geometric fibres of $f'$ are connected. It is clear from the discussion above that we may replace $Y$ by $Y'$.  Then $Y$ is locally Noetherian, $f : X \to Y$ is proper, and $f_*\mathcal{O}_X = \mathcal{O}_Y$. Let $\overline{y}$ be a geometric point of $Y$. At this point we apply the theorem on formal functions, more precisely Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-formal-functions-stalk}. It tells us that $$ \mathcal{O}^\wedge_{Y, \overline{y}} = \lim_n H^0(X_n, \mathcal{O}_{X_n}) $$ where $X_n = \Spec(\mathcal{O}_{Y, \overline{y}}/\mathfrak m_{\overline{y}}^n) \times_Y X$. Note that $X_1 = X_{\overline{y}} \to X_n$ is a (finite order) thickening and hence the underlying topological space of $X_n$ is equal to that of $X_{\overline{y}}$. Thus, if $X_{\overline{y}} = T_1 \amalg T_2$ is a disjoint union of nonempty open and closed subspaces, then similarly $X_n = T_{1, n} \amalg T_{2, n}$ for all $n$. And this in turn means $H^0(X_n, \mathcal{O}_{X_n})$ contains a nontrivial idempotent $e_{1, n}$, namely the function which is identically $1$ on $T_{1, n}$ and identically $0$ on $T_{2, n}$. It is clear that $e_{1, n + 1}$ restricts to $e_{1, n}$ on $X_n$. Hence $e_1 = \lim e_{1, n}$ is a nontrivial idempotent of the limit. This contradicts the fact that $\mathcal{O}^\wedge_{Y, \overline{y}}$ is a local ring. Thus the assumption was wrong, i.e., $X_{\overline{y}}$ is connected as desired.
We may apply Lemma \ref{lemma-stein-universally-closed} to get the morphism $f' : X \to Y'$. Note that besides the conclusions of Lemma \ref{lemma-stein-universally-closed} we also have that $f'$ is separated (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-compose-after-separated}) and finite type (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-permanence-finite-type}). Hence $f'$ is proper. At this point we have proved all of the statements except for the statement that $f'$ has connected geometric fibres. \medskip\noindent It is clear from the discussion that we may replace $Y$ by $Y'$.  Then $f : X \to Y$ is proper and $f_*\mathcal{O}_X = \mathcal{O}_Y$. Note that these conditions are preserved under flat base change (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-proper} and Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-flat-base-change-cohomology}). Let $\overline{y}$ be a geometric point of $Y$. By Lemma \ref{lemma-characterize-geometrically-connected-fibres} and the remark just made we reduce to the case where $Y$ is a scheme, $y \in Y$ is a point, $f : X \to Y$ is a proper algebraic space over $Y$ with $f_*\mathcal{O}_X = \mathcal{O}_Y$, and we have to show the fibre $X_y$ is connected. Replacing $Y$ by an affine neighbourhood of $y$ we may assume that $Y = \Spec(R)$ is affine. Then $f_*\mathcal{O}_X = \mathcal{O}_Y$ signifies that the ring map $R \to \Gamma(X, \mathcal{O}_X)$ is bijective. \medskip\noindent By Limits of Spaces, Lemma \ref{spaces-limits-lemma-proper-limit-of-proper-finite-presentation-noetherian} we can write $(X \to Y) = \lim (X_i \to Y_i)$ with $X_i \to Y_i$ proper and of finite presentation and $Y_i$ Noetherian. For $i$ large enough $Y_i$ is affine (Limits of Spaces, Lemma \ref{spaces-limits-lemma-limit-is-affine}). Say $Y_i = \Spec(R_i)$. Let $R'_i = \Gamma(X_i, \mathcal{O}_{X_i})$. Observe that we have ring maps $R_i \to R_i' \to R$. Namely, we have the first because $X_i$ is an algebraic space over $R_i$ and the second because we have $X \to X_i$ and $R = \Gamma(X, \mathcal{O}_X)$. Note that $R = \colim R'_i$ by Limits of Spaces, Lemma \ref{spaces-limits-lemma-descend-section}. Then  $$ \xymatrix{ X \ar[d]  \ar[r] & X_i \ar[d] \\ Y \ar[r] & Y'_i \ar[r] & Y_i } $$ is commutative with $Y'_i = \Spec(R'_i)$. Let $y'_i \in Y'_i$ be the image of $y$. We have $X_y = \lim X_{i, y'_i}$ because $X = \lim X_i$, $Y = \lim Y'_i$, and $\kappa(y) = \colim \kappa(y'_i)$. Now let $X_y = U \amalg V$ with $U$ and $V$ open and closed. Then $U, V$ are the inverse images of opens $U_i, V_i$ in $X_{i, y'_i}$ (Limits of Spaces, Lemma \ref{spaces-limits-lemma-descend-opens}). By Theorem \ref{theorem-stein-factorization-Noetherian} the fibres of $X_i \to Y'_i$ are connected, hence either $U$ or $V$ is empty. This finishes the proof.
Choose an affine scheme $V$ and a surjective \'etale morphism $V \to X$. Because strict transform commutes with \'etale localization (Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-strict-transform-local}) it suffices to prove the result with $X$ replaced by $V$. Hence we may assume that $X \to B$ is representable (in addition to the hypotheses of the lemma). \medskip\noindent Assume that $X \to B$ is representable. Choose an affine scheme $W$ and a surjective \'etale morphism $\varphi : W \to B$. Note that $X \times_B W$ is a scheme. By the case of schemes (More on Flatness, Theorem \ref{flat-theorem-flatten-module}) we can find a finite type quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_W$ such that (a) $|V(\mathcal{I})| \cap |\varphi^{-1}(U)| = \emptyset$ and (b) the strict transform of $\mathcal{F}|_{X \times_B W}$ with respect to the blowing up $W' \to W$ in $\mathcal{I}$ becomes flat over $W'$ and is a module of finite presentation. Choose a finite type sheaf of ideals $\mathcal{J} \subset \mathcal{O}_B$ as in Lemma \ref{lemma-push-ideal}. Let $B' \to B$ be the blowing up of $\mathcal{J}$. We claim that this blowup works. Namely, it is clear that $B' \to B$ is $U$-admissible by our choice of ideal $\mathcal{J}$. Moreover, the base change $B' \times_B W \to W$ is the blowup of $W$ in $\varphi^{-1}\mathcal{J} = \mathcal{I}\mathcal{I}'$ (compatibility of blowup with flat base change, see Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-flat-base-change-blowing-up}). Hence there is a factorization $$ W \times_B B' \to W' \to W $$ where the first morphism is a blowup as well, see Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-blowing-up-two-ideals}). The restriction of $\mathcal{F}'$ (which lives on $B' \times_B X$) to $W \times_B B' \times_B X$ is the strict transform of $\mathcal{F}|_{X \times_B W}$ (Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-strict-transform-local}) and hence is the twice repeated strict transform of $\mathcal{F}|_{X \times_B W}$ by the two blowups displayed above (Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-strict-transform-composition-blowups}). After the first blowup our sheaf is already flat over the base and of finite presentation (by construction). Whence this holds  after the second strict transform as well (since this is a pullback by Divisors on Spaces, Lemma \ref{spaces-divisors-lemma-strict-transform-flat}). Thus we see that the restriction of $\mathcal{F}'$ to an \'etale cover of $B' \times_B X$ has the desired properties and the theorem is proved.
Let $S$ be a scheme. Let $f : X \to Y$ be a radicial morphism of algebraic spaces over $S$. It is clear from the definition that given a morphism $\Spec(K) \to Y$ there is at most one lift of this morphism to a morphism into $X$. Hence we conclude that $f$ is universally injective by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-universally-injective}.
Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is stable under base change and composition and holds for closed immersions. Assume $f : X \to Y$ has $\mathcal{P}$ and is universally injective. Then, in the situation of Definition \ref{definition-radicial} the morphism $(\Spec(K) \times_Y X)_{red} \to \Spec(K)$ is universally injective and has $\mathcal{P}$. This reduces the problem of proving $$ \mathcal{P} + \text{universally injective} \Rightarrow \text{radicial} $$ to the problem of proving that any nonempty reduced algebraic space $X$ over field whose structure morphism $X \to \Spec(K)$ is universally injective and $\mathcal{P}$ is representable by the spectrum of a field. Namely, then $X \to \Spec(K)$ will be a morphism of schemes and we conclude by the equivalence of radicial and universally injective for morphisms of schemes, see Morphisms, Lemma \ref{morphisms-lemma-universally-injective}. \medskip\noindent Let us prove (1). Assume $f$ is decent and universally injective. By Decent Spaces, Lemmas \ref{decent-spaces-lemma-base-change-relative-conditions}, \ref{decent-spaces-lemma-composition-relative-conditions}, and \ref{decent-spaces-lemma-properties-trivial-implications} (to see that an immersion is decent) we see that the discussion in the first paragraph applies. Let $X$ be a nonempty decent reduced algebraic space universally injective over a field $K$. In particular we see that $|X|$ is a singleton. By Decent Spaces, Lemma \ref{decent-spaces-lemma-when-field} we conclude that $X \cong \Spec(L)$ for some extension $K \subset L$ as desired. \medskip\noindent A quasi-separated morphism is decent, see Decent Spaces, Lemma \ref{decent-spaces-lemma-properties-trivial-implications}. Hence (1) implies (2). \medskip\noindent Let us prove (3). Recall that the separation axioms are stable under base change and composition and that closed immersions are separated, see Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-base-change-separated}, \ref{spaces-morphisms-lemma-composition-separated}, and \ref{spaces-morphisms-lemma-immersions-monomorphisms}. Thus the discussion in the first paragraph of the proof applies. Let $X$ be a reduced algebraic space universally injective and locally separated over a field $K$. In particular $|X|$ is a singleton hence $X$ is quasi-compact, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-space}. We can find a surjective \'etale morphism $U \to X$ with $U$ affine, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}. Consider the morphism of schemes $$ j : U \times_X U \longrightarrow U \times_{\Spec(K)} U $$ As $X \to \Spec(K)$ is universally injective $j$ is surjective, and as $X \to \Spec(K)$ is locally separated $j$ is an immersion. A surjective immersion is a closed immersion, see Schemes, Lemma \ref{schemes-lemma-immersion-when-closed}. Hence $R = U \times_X U$ is affine as a closed subscheme of an affine scheme. In particular $R$ is quasi-compact. It follows that $X = U/R$ is quasi-separated, and the result follows from (2).
The question is \'etale local on $Y$ by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-universally-injective-local}. Hence we may assume that $Y$ is a scheme. Then $Y$ is in particular decent and by Decent Spaces, Lemma \ref{decent-spaces-lemma-conditions-on-point-in-fibre-and-qf} we see that $f$ is locally quasi-finite. Let $y \in Y$ be a point and let $X_y$ be the scheme theoretic fibre. Assume $X_y$ is not empty. By Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-locally-quasi-finite-over-field} we see that $X_y$ is a scheme which is locally quasi-finite over $\kappa(y)$. Since $|X_y| \subset |X|$ is the fibre of $|X| \to |Y|$ over $y$ we see that $X_y$ has a unique point $x$. The same is true for $X_y \times_{\Spec(\kappa(y))} \Spec(k)$ for any finite separable extension $\kappa(y) \subset k$ because we can realize $k$ as the residue field at a point lying over $y$ in an \'etale scheme over $Y$, see More on Morphisms, Lemma \ref{more-morphisms-lemma-realize-prescribed-residue-field-extension-etale}. Thus $X_y$ is geometrically connected, see Varieties, Lemma \ref{varieties-lemma-characterize-geometrically-disconnected}. This implies that the finite extension $\kappa(y) \subset \kappa(x)$ is purely inseparable. \medskip\noindent We conclude (in the case that $Y$ is a scheme) that for every $y \in Y$ either the fibre $X_y$ is empty, or $(X_y)_{red} = \Spec(\kappa(x))$ with $\kappa(y) \subset \kappa(x)$ purely inseparable. Hence $f$ is radicial (some details omitted), whence universally injective by Lemma \ref{lemma-radicial-implies-universally-injective}.